1. Introduction

In the present decade there will be an unprecedented growth of generated data, computations, instructions and operations. This growth may not compromise clean air, clean water and a sustainable energy and material usage, but rather facilitate these prerequisites for flora and fauna. Overall, global primary energy consumption rises due to the Internet [1] and Internet's own electrical energy consumption is also rising [2,3,4]. Plausibly the global primary energy and electricity consumption would rise even faster without the handprint of certain ICT solutions. The Internet Sector is one of few which might off-set its own electrical energy consumption and GHG supply, i.e., its handprint [5] is larger than its footprint. There are many indications (expected trends and estimates) showing that the Internet infrastructure will be able to provide solutions - within main Sectors such as Buildings, Transportation and Industry - which will help reduce (halt the increase of) the total global consumption of energy and materials [6]. Information and communications technologies (ICTs) can potentially contribute to reduce resource consumption through increased productivity in many Sectors by enabling total optimization and dematerialization, occasionally using artificial intelligence (AI) [7]. Internet's deflationary characteristics suggest that it has a handprint. AI and machine learning (ML) are cornerstones of intelligent ICT Solutions which make them unique compared to incremental improvements. For instance, products are replaced by virtual services, e.g., by using e-readers instead of paperbacks [8], transportation is avoided by online shopping or online chatting. This is more resource and energy efficient than before and entire sectors, like transport, industry, and agriculture can be optimized. Internet may foster new sustainable lifestyles which can lower the affluence despite certain rebound effects. E.g. e-reader adopters are yet to fully abandon paper books for e-books suggesting a total net increase [8]. The underlying idea is that e.g. total anthropogenic global GHG supply (TAGGHGS) can be significantly halted if existing and developing ICT Solutions are used in other sectors (and Internet itself [9]) to their "full potential" in a smart manner. Such solutions include Products-sold-as-Services, Smart Grid and Smart Metering. The avoidance potential is highly dependent e.g. on how large share of the building GHG supply can be reduced by Smart Metering [10,11].

Internet will use smart ICT solutions to keep its own material and electrical energy use under control. The smart ICT solutions could also be used in the rest of the society to reach environmental goals. Still, the Internet Sector itself has a huge responsibility to try to reach high annual electrical energy efficiency gains of \(\approx20\)% in data centers and networks. This seems to have been the case in the last decade. New technologies such as neuromorphic, reversible and superconducting computing as well as nanophotonics may help in this decade [12,13]. Several attributional life cycle assessments have shown that Internet's share of TAGGHGS may have been stable 2015-2020 [13]. However, although recent literature is divided[14], the trends of rampant instructions/second and slowing improvements of switching energy are very clear [12,14]. Related cryptocurrency mining electrical energy demand is on the rise but not necessarily the related GHG supply [15]. Therefore power saving is a highly efficient strategy for GHG supply reduction in the Internet itself and beyond. In this work, the potential TAGGHGS avoidance of using ICT Solutions for energy saving, compared to low adoption of ICT Solutions, is explored. An algorithm for the estimation is established.

1.1. Objectives

The objective of this prediction study is to estimate the changes (GHG is proxy) to occur between 2019 and 2030 if traditional technologies are replaced with Information and Communication Technology (ICT) technologies. Internet's scope according to [16,17] consists of the use stage of end-user consumer devices, network infrastructure and data centers as well as the production of hardware for all. The attributional LCA approach [16,17] may not be able to capture the actual GHG avoidance derived from the use of ICT solutions, as many of them have the ability to decrease the energy and material losses. Consequently, less energy and materials need to be produced and purchased by a final customer in order to consume the same quantity of product. A consequential LCA (CLCA) with a planetary system boundary is attempted for ICT solutions handprint.

1.2. Hypotheses

The hypothesis is that Internet's GHG supply will increase according to the expected scenario as outlined by Andrae [16]. Moreover, Internet's GHG supply is off-set already in 2020 by ICT Solutions and the handprint will be 6 times the footprint by 2030.

2. Materials and methods

The approach for estimating Internet's direct GHG supply is established while the handprint potential of ICT solutions for TAGGHGS is less clear. Here, for the sake of modelling, the World is divided into seven sectors - Industry, Buildings, Transport, Travel, Agriculture, Waste and Land use. Then several ICT Solutions ability to reduce TAGGHGS in each sector is estimated. The approach is very much simplistic as there are highly granular Input-Output models [9,18] which describe the economic flows of different sectors in the society. Therefore, the coupling of IO and LCA can be applied to model indirect impacts of changes in product inputs and outputs in several economic sectors [19]. The coupling of IO and LCA can cover all economic sectors in a large geographical boundary. All assumptions made are available in the Supplementary Information.

2.1. Description of method for estimating Internet GHG supply

The approach for Internet direct GHG supply follows the one outlined in [16] expected scenario. Table 1 shows some global trends assumptions derived from [13,16]. Trends for TAGGHGS are followed closely [20].

Table 1. Global Electricity demand and average GHG intensity 2019 to 2030.

Year	Total global	Total global internet	TWh renewable electricity	GHG intensity
	electricity demand (TWh)	electricity demand (TWh)	including Hydro (TWh)	in Gt CO2e/TWh
2019	27050	1950	7042	0.000545
2020	27188	1988	7265	0.000543
2021	27826	1986	7494	0.000542
2022	28467	1987	7731	0.000540
2023	29117	1997	7975	0.000538
2024	29775	2015	8227	0.000536
2025	30446	2046	8487	0.000535
2026	31179	2139	8755	0.000533
2027	31968	2288	9032	0.000533
2028	32813	2493	9318	0.000532
2029	33751	2791	9612	0.000532
2030	34718	3218	9916	0.000533

2.2. Handprint - description of method for estimating GHG supply reductions and power savings by ICT Solutions

The overall methodological approach for estimating electricity demand and GHG supply handprint by the Internet in other Sub-Sectors (Industry, Transportation, Buildings, Agriculture, Land use and Waste) is described below. Land use leads to increased GHG supply if new plants (e.g. trees) are not planted which can absorb \(CO_2\). Waste (management) is a relevant Sub-Sector of its own, e.g. landfill, recycling, incineration etc.

2.2.1. Consequential handprint LCA

The functional unit of the CLCA is: global demand of electricity and TAGGHGS. Figure 1 shows the principle of provided functions replacing more inefficient ways.

ICT Solutions are replacing traditional solutions in the CLCA. Increased digitalization leads to increased production of ICT Solutions which substitute products, energy, materials and land use. When less travel and transport are used, also less fuel is produced. To compensate for "missing" fuel, more electricity will be produced. These effects contribute to the view of Internet driving deflation.

2.3. Estimation of total anthropogenic global GHG supply

The GHG supply from the Sub-Sector Agriculture in 2030 is \(\approx8.4\) Gt as shown in Table 2 below. Table 2 shows that Internet's share of TAGGHGS without Internet handprint will be low \((\approx2\%)\). Moreover, without active Internet handprint, Industry's GHG supply will increase almost 20% between 2020 and 2030. Land use and waste environmental impacts are more challenging to reduce with ICT Solutions, but they still contribute to TAGGHGS which could rise between 2020 and 2030 [13].

Despite economic growth and rebound effects, TAGGHGS could likely be significantly halted if existing and future ICT Solutions are used in other sectors (and Internet itself). Table 2 shows the present estimations by Sector for approximate TAGGHGS.

Table 2. Estimation of total anthropogenic global GHG supply (Gigatonnes) by Sector 2019-2030.

GHG Supply  from Global Sectors and ICT	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
GHG\(_{i=\text{Industry}}\)	18.5	17.6	17.9	18.2	18.5	18.8	19.1	19.4	19.7	20.0	20.4	20.7
GHG\(_{i=\text{Building}}\)	13.7	13.4	13.6	13.9	14.2	14.5	14.7	15.0	15.3	15.6	15.9	16.2
GHG\(_{i=\text{Travel}}\)	4.2	3.0	3.2	3.3	3.5	3.7	3.8	4.0	4.1	4.3	4.4	4.6
GHG\(_{i=\text{Transport}}\)	6.2	4.4	4.6	4.9	5.1	5.3	5.6	5.8	6.0	6.3	6.5	6.7
GHG\(_{i=\text{Agriculture}}\)	6.7	6.8	7.0	7.1	7.3	7.4	7.6	7.7	7.9	8.1	8.3	8.4
GHG\(_{i=\text{Land use}}\)	5.4	5.4	5.4	5.5	5.5	5.6	5.6	5.6	5.7	5.7	5.8	5.5
GHG\(_{i=\text{Waste}}\)	1.7	1.7	1.8	1.8	1.8	1.9	1.9	1.9	2.0	2.0	2.1	2.1
GHG\(_{i=\text{Internet}}\)	1.06	1.08	1.08	1.07	1.07	1.08	1.09	1.14	1.22	1.33	1.48	1.71
GHG\(_{i=\text{Electricity}}\)	Part of (embedded) all sectors
Total GHG  Supply \((\text{TAGGHGS}_{t})\) without  GHG\(_{i-\text{internet}}\) and internet handprint (Gt)	56.4	52.4	53.5	54.7	55.9	57.1	58.3	59.5	60.8	62.0	63.3	64.6

Equation (1) below shows TAGGHGS in year t:

Equation (2) below shows the total global handprint of ICT Solution j in year t: where, \( \text{TAGGHGS}_{t}=\) Total anthropogenic GHG supply in year \(t\); GHG\(_{i,t}=\) Anthropogenic GHG supply from Sector type \(i\) in year \(t\); ICT\(_{hp,t} =\) ICT Solutions total global GHG handprint in year \(t\); \(j=\) ICT Solution type; \(i =\) Sector type; \(t=\) year; MVCV\(_{j,i,t}=\) Marginal Variable Change Vector of ICT Solution \(j\) in Sector \(i\) in year \(t\); F\(_{j,i,t}=\) Fraction of Sector \(i\) which is applicable to ICT Solution \(j\) in year \(t.\)

Here follows two examples which explain somewhat (2);

• In 2030, F is \(0.1\) for Travel Sector for "Video/telemeeting, air" as it is assumed that \(10\)% of the Travel sector GHG supply are air travel GHG supply which can be reduced (50%, \(\text{MVCV}=0.5\)) via "Video/telemeeting, air" ICT solutions. The minimum values for this case are \(\text{F}=0.01\) and \(\text{MVCV}=0.05\).
• In 2030, F is \(0.5\) for Building Sector for "Smart metering in Buildings" as it is assumed that \(50\)% of the Building sector GHG supply are electricity related GHG supply which can be reduced (10%, \(MVCV=0.1\)) via "Smart metering in Buildings" ICT solutions. The minimum values for this case are \(F=0.05\) and \(\text{MVCV}=0.01\). \(\text{MVCV}_{\text{Smart metering, Buildings,} 2020 =0.03}\) is reported in literature [21].

2.4. Division of GHG supply between electricity and other sources for Industry, Buildings, Travel and Transports

Table 2 shows that the GHG supply from Industry is \(\approx18\) Gt in 2020 with around 6 Gt related to electricity demand.

Table 3. Division of GHG supply between electricity and other sources between 2020 and 2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
Industry  Electricity  GHG, Gt  CO2e	6.2	6.2	6.2	6.3	6.3	6.3	6.4	6.4	6.4	6.5	6.5	6.5
Industry  Electricity  Use, TWh	12000	11400	11500	11600	11700	11800	1190	12000	12100	12200	12300	12400
Industry  Others,  Gt CO2e	12.0	11.4	11.7	11.9	12.2	12.5	12.8	13.0	13.3	13.6	13.8	14.1
Buildings  Electricity  GHG, Gt  CO2e	6.7	7.1	7.2	7.4	7.6	7.8	7.9	8.1	8.3	8.5	8.7	8.3
Buildings  Electricity  Use, TWh	12300	13000	13370	13740	14110	14480	14850	15220	15590	15960	16330	16700
Buildings  Others,  Gt CO2e	7.0	6.3	6.4	6.5	6.6	6.7	6.8	6.9	7.0	7.1	7.2	7.3
Travel  Electricity  GHG, Gt  CO2e	0.2	0.2	0.3	0.3	0.4	0.5	0.5	0.6	0.6	0.7	0.7	0.8
Travel  Electricity Use, TWh	400	400	510	620	730	840	950	1060	1170	1280	1390	1500
Travel  Others,  Gt  CO2e	4.0	2.8	2.9	3.0	3.1	3.2	3.3	3.4	3.5	3.6	3.7	3.8
Transports  Electricity,  GHG, Gt,  CO2e	0.2	0.2	0.2	0.3	0.3	0.3	0.4	0.4	0.4	0.5	0.5	0.5
Transports  Power,  TWh	400	400	460	520	580	640	700	760	820	880	940	1000
Transports  Others, Gt CO2e	6.0	4.2	4.4	4.6	4.8	5	5.2	5.4	5.6	5.8	6	6.2
Internet  Electricity  GHG, Gt  CO2e	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.1	1.2	1.3	1.5	1.7
Internet  Electricity  Use, TWh	1950	1988	1986	1987	1997	2015	2046	2139	2288	2493	2791	3218
TOTAL  GLOBAL  CO2e from  human  energy  conversion  activities, Gt CO2e	43.7	39.5	40.4	41.4	42.4	43.3	44.3	45.4	46.4	47.5	48.7	49.9

Transport Electricity GHG and Travel Electricity GHG are both \(\approx6\)% of respective Sectors total GHG, while Land use Electricity GHG is excluded as it is assumed close to zero. However, with the anticipated electrification of vehicles, both Transport and Travel Electricity GHG will increase [17].

Moreover, hydrogen production for fuel cell vehicles - and indirectly ammonia production for internal combustion engines - will add to the electricity demand of the Transport and Travel Sectors [13]. Hydrogen production will also add electricity demand in the Industry sector (e.g. Steel supply chain), but at the same time the net GHG supply may be reduced in e.g. Steel production [13].

Ammonia has potential for Travel and Transport as a more or less non-ICT based GHG Supply reducer as ammonia can be used in converted internal combustion engines [22]. It is especially useful to know Industry Electricity Use and Building Electricity Use to understand the effectiveness of different electrical energy efficiency strategies such as Smart Metering. Internet also has some GHG supply from other sources than electricity, such as from diesel generators [23] but they have been excluded. Table 3 shows the split between electricity GHG and other sources for each Sector.

Table 3 may help understand where electrification, hydrogen and ammonia solutions make the most sense in energy related activities.

2.5. Estimation of ICT Solutions handprints

They key question addressed here is: How much GHG supply can smart ICT Solutions help avoid, \(ICT_{hp,t}\), in other sectors of society each year between \(t=2019\) and \(t=2030?\) Despite large uncertainties, quite likely more GHG supply can be avoided than the Internet emits itself i.e., ICT\(_{hp,t} > \text{GHG}_{i=\text{Internet,}t}\), perhaps already for \(t=2022\). Table 4 shows the addressed Fraction (F) of each Sector (i) and how much the ICT technology (j) can reduce (MVCV).

Table 5 roughly outlines how ICT\(_{hp,t}\) could increase year by year from 2019 to 2030 as estimated in the present study to reach (at the most) \(\approx11\text{Gt}\) in 2030. However, likely some reduction of TAGGHGS has already occurred historically due to ICT solutions and sensitivity checks are performed in Section 4. Table 5 shows mainly future potential to 2030. In Sections 2.5.1 to 2.5.9 the numbers in Table 5 are explained.

Table 6 outlines roughly how electricity handprints (TWh) could increase linearly year by year from 2020 to 2030 as estimated in the present study to reach \(\approx 8497\) TWh in 2030. As soon as 2022 more TWh can be cut by the Internet than its own usage.

2.5.1. Smart Grid

Table 7 shows three examples of where Smart Grid can achieve transformation. Smart Grid savings are firstly that 50% of all Buildings GHG supply are applicable for 10% reduction each using Smart Metering. This means that \(0.1\times 0.5\times 16.2\) Gt\( = 0.81\) Gt can be avoided in 2030. Smart Metering makes the users aware of the power consumption [21]. Smart Grid savings can also be obtained from "Power grid optimization" which may reduce 50% (power load balancing) of power use in each case it is introduced, but perhaps only 10% of global electricity GHG supply is applicable for optimization in 2030, i.e., \(0.5\times0.1\times34718\) TWh\(\times0.000533 \text{Gt}/\text{TWh} = 0.92\) Gt.

Further Smart Grid related savings by fewer losses than traditional grids [26] are possible from facilitation of renewable energy sources. This is assumed to reduce 10% of the power used in each applicable case which may be 20% of all globally used electricity, \(0.1\times 0.2\times 34718 \text{TWh} \times 0.000533 \text{Gt}/\text{TWh} = 0.37 \text{Gt}\). AI-driven battery management (including self-repair) is part of the renewable energy story for ICT handprint. All in all in 2030, \(0.81+0.92+0.37 = 2.1 \)Gt savings in 2030 from Smart Grid.

2.5.2. Smart Agriculture

Table 8 shows an example in which Smart Agriculture can achieve transformation. Smart Agriculture can be used to reduce TAGGHGS by e.g. more surveillance and less manual inspection. It is estimated that 10% of GHG supply of the entire Agriculture sector can be reduced, \(0.10\times8.43 \text{Gt} = 0.84 \text{Gt}\). AI is especially useful in Agriculture [28]. Autonomous variable herbicide spraying can save \(>50\%\) of liquid applied per hectare [29].

Table 4. Evolution matrix for F and MVCV for \(t=2019\) to 2030 for chosen \(i\) and \(j.\).

\# \(j\)	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030	MVC \(v,j\)
1	0.005	0.05	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	10%
2	0.0001	0.001	0.02	0.04	0.06	0.08	0.1	0.12	0.14	0.16	0.18	0.2	10%
3	0.00005	0.0005	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	20%
4	0.00025	0.0025	0.05	0.1	0.15	0.2	0.25	0.3	0.35	0.4	0.45	0.5	10%
5	0.005	0.05	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	10%
6	0.005	0.05	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	10%
7	0.0005	0.005	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	10%
8	0.0005	0.005	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0. 1	20%
9	0.0025	0.025	0.05	0.1	0.15	0.20	0.25	0.3	0.35	0.4	0.45	0.5	20%
10	0.01	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	50%
11	0.01	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	50%
12	0.005	0.05	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	20%
13	0.001	0.01	0.02	0.04	0.06	0.08	0.10	0.12	0.14	0.16	0.18	0.2	50%
14	0.00005	0.0005	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008	0.009	0.01	10%
15	0.0015	0.015	0.03	0.06	0.09	0.12	0.15	0.18	0.21	0.24	0.27	0.3	50%
16	0.0005	0.005	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	20%
17	0.0025	0.025	0.05	0.1	0.15	0.2	0.25	0.3	0.35	0.4	0.45	0.5	20%
18	0.0005	0.005	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	50%
19	0.005	0.05	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	10%
20	0.0005	0.005	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1	10%
Autonomous intelligent tractors help avoid manual checks of the fields of cultivation													1
Facilitating renewable energy sources													2
Power grid optimization													3
Smart metering in Buildings													4
Car pools, leasing services, mobility-as-a-service													5
Selling products as services, servitization													6
Public travel suggestions													7
Fleet car management													8
Route optimization in Leisure													9
Video/telemeetings, car													10
Video/telemeetings, air													11
Office space													12
Teleworking, car													13
Hotels													14
Office energy use													15
Food store cooling energy													16
Route optimization in Logistics													17
Facilitate choosing train instead of car													18
AI enabled optical sorting													19
Forest monitoring with drones and sensors													20

Table 5. Estimation of GHG supply (Gigatonnes) reductions enabled by ICT Solutions 2019-2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
Intelligent Grid  Handprint, Gt	0.00	0.01	0.17	0.35	0.54	0.74	0.94	1.15	1.37	1.60	1.85	2.11
Intelligent  Agriculture Handprint, Gt	0.00	0.03	0.07	0.14	0.22	0.30	0.38	0.46	0.55	0.65	0.74	0.84
Intelligent Service Handprint, Gt	0.01	0.10	0.21	0.43	0.66	0.90	1.15	1.40	1.67	1.95	2.23	2.53
Intelligent Travel  Handprint, Gt	0.00	0.02	0.04	0.09	0.14	0.19	0.25	0.31	0.38	0.45	0.52	0.60
Intelligent Work  Handprint, Gt	0.05	0.33	0.38	0.46	0.54	0.63	0.73	0.82	0.93	1.03	1.14	1.26
Intelligent  Buildings Handprint, Gt	0.01	0.11	0.23	0.47	0.72	0.98	1.25	1.53	1.82	2.12	2.43	2.75
Intelligent Transport Handprint, Gt	0.00	0.03	0.07	0.15	0.23	0.32	0.42	0.52	0.63	0.75	0.88	1.01
Intelligent Circularity Handprint, Gt	0.00	0.01	0.02	0.04	0.05	0.07	0.10	0.12	0.14	0.16	0.19	0.21
Intelligent Land  Use Handprint, Gt	0.00	0.00	0.01	0.01	0.02	0.02	0.03	0.03	0.04	0.05	0.06	0.5
TOTAL SMART  savings per year\(ICT_{hp}\)	0.08	0.65	1.20	2.14	3.12	4.16	5.23	6.35	7.53	8.75	10.03	11.37

Table 6. Estimation of electricity handprint (TWh) enabled by ICT Solutions 2019-2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
Industry	1201	1150	1345	1559	1781	2014	2256	2510	2776	3058	3356	3376
Buildings	12	127	322	662	1020	1396	1789	2201	2630	3077	3542	4025
Travel	4.7	47	68	103	145	195	252	316	387	466	552	645
Transports	0.3	3.0	6.9	16	26	38	53	68	86	106	127	150
TOTAL savings TWh	1218	1327	1742	2339	2973	3643	4349	5095	5880	6706	7577	8497

Table 7. Smart Grid saving ICT Technologies.

t=2030		i	Buildings	Global electricity supply
Facilitating renewable energy sources	10% [24]			20% [24]
Power grid optimization	50% [24]			10% [24]
Smart metering in Buildings	10% [25]		50% [25]

Table 8. Smart Agriculture saving ICT technologies.

\(t=2030\)	\(i\)		Agriculture
\(j\)	MVCV		F
Autonomous intelligent tractors help avoid manual checks of the fields of cultivation	10% [27]		100%[27]

2.5.3. Smart Services

Smart Services is a rather wide concept for Smart ICT but mainly it is about virtualization and dematerialization. The link to servitization is very strong [29,30]. Table 9 shows two examples in which Smart Services can achieve transformation.

Table 9. Smart Services saving ICT Technologies.

\(t=2030\)		\(i\)	Travel	Industry
\(j\)	MVCV		F	F
Car pools, leasing services, mobility-as-a-service	10% [30]		100%[30]
Selling products as services, servitization	10% [31]			100% [31]

E-readers and audio books are examples of book-as-a-service (potentially) replacing physical books [8]. Global Change Mix Factors [32] may reveal to which degree this will materialize and then cause a rebound effect on F. Regarding financial products, digital solutions on blockchain may change the electricity demand of the financial systems [33]. Another example of smart service is e-commerce [34]. With the current rate of digitalization it seems likely that 100% of all Industry products could be sold as services in 2030. It is assumed that 10% of all travel transport and 10% of all Industry GHG supply can be avoided by selling products as services. All in all, \(0.1\times1\times4.6 \text{Gt} + 0.1\times1\times20.7 \text{Gt} = 2.5 \text{Gt}\).

2.5.4. Smart Travel

Smart Travel is about optimizing travel routes and vehicle sharing. It is assumed that three main mechanisms lead to savings; smart public travel, fleet car management and route optimization. Table 10 shows two examples in which Smart Travel solutions can achieve transformation.

Table 10. Smart Travel saving ICT Technologies.

\(t=2030\)		\(i\)	Travel
\(j\)	MVCV		F
Public travel suggestions	10% [35]		10% [35]
Fleet car management	20% [36]		10% [36]
Route optimization in Leisure	20% [37]		50% [37]

AI Taxi is an example of fleet car management [7]. The eighth and ninth handprints may be overestimated. Map route travel service handprints are already achieved. All in all, Smart Travel may reduce GHG supply by \((0.1\times0.1 + 0.2\times0.1 + 0.2\times0.5)\times4.6 \text{Gt} = 0.6 \text{Gt}.\)

2.5.5. Smart Work

Smart Work is about reducing business travel and commuting and the need for less hotel rooms and offices. In 2020 the airline passenger traffic shrunk 67% compared to 2019 and was reduced to 1999 levels [38]. No matter the reason, this suggests that digitalization tools could reach a very high implementation (high F) for video/telemeetings already in 2020 (Table 4). Table 11 shows five examples in which Smart Work solutions can achieve transformation.

Table 11. Smart Services saving ICT Technologies.

\( t=2030\)		\(i\)	Travel	Buildings
\(j\)	MVCV		F	F
Video/telemeetings, air	50% [39]		10% [39]
Video/telemeetings, car	50% [40]		10% [40]
Teleworking, car	50% [41]		20% [41]
Office space	20% [42]			100% [30]
Hotels	10% [43]			1% [43]

The tenth and eleventh services could be similar and double counted. A sensitivity analysis will include such issues. All in all, Smart Work may reduce GHG supply by \((0.5\times0.1 + 0.5\times0.1 + 0.5\times0.2)\times4.6 \text{Gt} + (0.2\times0.1 + 0.1\times0.01)\times16.2 \text{Gt} = 1.26 \text{Gt}\).

2.5.6. Smart Buildings

Smart ICT is facilitating automated heating, ventilation and air conditioning (HVAC) systems as well as light control. Via deep learning and cloud-based computing, ICT solutions autonomously optimizes existing HVAC control systems for lowest possible energy consumption. Table 12 shows two examples in which Smart Building solutions can achieve transformation.

Table 12. Smart Building saving ICT solutions.

\( t=2030\)		\(i\)	Buildings
\(j\)	MVCV		F
Office energy use	50% [39]		30% [44]
Food storage cooling energy	20% [40]		10% [45]

Office energy use here, and Smart metering in Section 2.5.1, may address somewhat similar flows. HVAC savings are however also related to thermal energy and not only electricity. All in all, Smart Building solutions may reduce GHG supply by \((0.5\times0.3 + 0.2\times0.1)\times16.2 \text{Gt} = 2.75 \text{Gt}.\)

2.5.7. Smart transports

Smart Transports is mainly about optimization of truck logistics and shifting transport from e.g. car to train [46]. Table 13 shows two examples in which Smart Transport solutions can achieve transformation.

Table 13. Smart Transport saving solutions.

\(t=2030\)		\(i\)	Transport
\(j\)	MVCV		F
Route optimization in Logistics	20% [47]		50% [47]
Facilitate choosing train instead of truck	50% [48]		10% [48]

All in all Smart Transport solutions may reduce GHG supply by \((0.2\times0.5 + 0.5\times0.1)\times4.5 \text{Gt} = 0.68 \text{Gt}\). Arguably F for ICT solution (Table 13, last row) may be lower than 10%. Another example of a Smart Transport solution is wireless vehicle-vehicle communication with Cooperative Adaptive Cruise Control which saves truck fuel [49].

2.5.8. Smart Circularity

Around 99% of everything that we buy becomes waste after 6 months. Some 2 billion tonnes of waste (garbage) is generated annually of which \(\approx2.5%\) is e-waste. Total Material Consumption per capita is also increasing. The effective material flow is much higher than the conventional weight flow. The so called Total Material Requirements per kg metal is increasing, i.e., the ore grades are diminishing. Ore grades (e.g. copper) is gradually decreasing 2.5% per annum, while production and energy consumption (and GHG supply) from mining is increasing [50]. Using AI software for optimization is likely a more fruitful route than new waste management technologies. In product design, AI may predict product design variables for GHG supply reduction and customer relevance. AI can help the Waste sector by Smart logistics (improvements in route planning), mobile collection of e-waste on demand [51], and via intelligent optical sorting machines.

Here the example of intelligent Optical sorting machines is used to exemplify how much the GHG supply from the Waste Management Sector can be reduced. AI helps capturing data from optical sorters from which machinery can learn and "make" decisions that optimize sorting [52]. AI-Powered Robot Picking is another example of Intelligent Circularity. Smart Circularity can be used in Waste Management to avoid GHG supply. Waste can be identified for its proper handling. It is assumed that in each case Smart Circularity is used 10% of the GHG supply can be avoided and that all waste related GHG supply can be addressed by intelligent sorting in 2030, i.e., \(0.1\times1\times2.1 \text{Gt} = 0.21 \text{Gt}\). One could argue that selling services instead of products is also smart circularity [31]. Nevertheless, Industry related material savings (servitization) is addressed by Smart Services in Section 2.5.3.

Route planning of garbage trucks is addressed by Smart Transport in Section 2.5.7. Table 14 shows an example in which a Smart waste management solution can achieve transformation.

Table 14. Smart Waste management solutions.

\( t=2030\)		\(i\)	Transport
\(j\)	MVCV		F
AI enabled optical sorting	10% [52]		100% [52]

All in all Smart Circularity solutions may reduce GHG supply by \(0.1\times1\times2.1\text{ Gt} = 0.21 \text{Gt}\).

2.5.9. Smart land use

AI can be used for managing sustainable land use [53]. Smart Forestry (IoT monitoring) can be used to reduce GHG supply caused by illegal tree cutting [54,55]. Table 15 shows an example in which a Smart land use solution can achieve transformation.

Table 15. Smart Land use solutions.

\(t=2030\)		\(i\)	Transport
\(j\)	MVCV		F
Forest monitoring with drones and sensors	10% [56]		10% [56]

All in all Smart Land use solutions may reduce GHG supply by \(0.1\times0.1\times5.8 \text{Gt} = 0.058 \text{Gt}\).

2.6. Total Internet GHG supply handprint

All in all, Internet's handprint will be 11.37 Gt in 2030 using assumed (very high) values for MVCV and F. However, when introducing smaller minimum values for MVCV and F, the handprint will be much smaller, but still safely higher than Internet's footprint.

2.7. Estimation of global GHG supply with and without ICT handprint and Share of electric power GHG of total global GHG

It is relevant to estimate to which degree the savings from ICT Solutions of Table 5 can reduce TAGGHGS (Table 2). In Table 16 is shown how TAGGHGS is slowed down. In 2020 the TAGGHGS is 0.2% (57.4 instead of 57.5) less thanks to ICT. In 2030 TAGGHGS could be around 17% (54.9 instead of 66.3) lower than business as usual. Table 16 shows the estimation of global GHG supply with and without ICT handprint and share of electric power GHG supply.

2.8. Handprint in each Sector

In this section the savings made possible by Smart Grid, Smart Travel, Smart Buildings etc. are allocated to each societal sector (Table 17). For example the savings by Smart Work are allocated to Travel if the Smart Work savings are travel related, and savings by Smart Grid are allocated to Buildings if the Smart Grid savings are Buildings related.

2.9. Share of each sectors GHG supply that can be cut by ICT Solutions

This section discusses the share of each sectors' GHG supply which could be cut year by year from 2019 to 2030. The shares are obtained by dividing the Handprint in the Sector (Table 17) with the Sector (Table 2). Table 18 the shares of each sectors GHG supply that can be cut over time are shown.

Table 16. Estimation of global GHG supply (Gigatonnes) with and without ICT handprint and Share of electric power GHG, 2019-2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
TAGGHGS (Gt)  without  ICT  enabling	57.5	53.4	54.6	55.8	57.0	58.2	59.4	60.7	62.0	63.3	64.8	66.3
TAGGHGS (Gt)  thanks  to   ICT  enabling	57.4	52.8	53.4	53.7	53.9	54.1	54.2	54.3	54.5	54.6	54.7	54.9
Global electric  power related  GHG supply	14.7	14.8	15.1	15.4	15.7	16.0	16.3	16.6	17.0	17.5	18.0	18.5

Table 17. Estimation of global GHG supply (Gigatonnes) with and without ICT handprint and Share of electric power GHG, 2019-2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
Handprint  in Industry	0.01	0.09	0.28	0.58	0.88	1.20	1.53	1.86	2.22	2.58	2.96	3.37
Handprint  in Buildings	0.01	0.13	0.33	0.67	1.03	1.39	1.78	2.17	2.58	3.01	3.45	3.90
Handprint  in Travel	0.05	0.35	0.42	0.55	0.70	0.85	1.01	1.18	1.36	1.56	1.76	1.98
Handprint  in Transports	0.00	0.03	0.07	0.15	0.23	0.32	0.42	0.52	0.63	0.75	0.88	1.01
Handprint  in Agriculture	0.00	0.03	0.07	0.14	0.22	0.30	0.38	0.46	0.55	0.65	0.74	0.84
Handprint  in Waste	0.00	0.01	0.02	0.04	0.05	0.07	0.10	0.12	0.14	0.16	0.19	0.21
Handprint  in Land use	0.00	0.00	0.01	0.01	0.02	0.02	0.03	0.03	0.04	0.05	0.05	0.06
TOTAL Internet  handprint per year	0.08	0.65	1.20	2.14	3.12	4.16	5.23	6.35	7.53	8.75	10.03	11.37

Table 18. Shares of each sectors GHG supply that can be cut by ICT Solutions between 2019 and 2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
Industry	0%	1%	2%	3%	5%	7%	9%	10%	12%	14%	16%	17%
Buildings	0%	1%	2%	5%	7%	10%	12%	14%	17%	19%	22%	24%
Travel	1%	12%	13%	17%	20%	23%	27%	30%	33%	36%	40%	43%
Transports	0%	1%	2%	3%	5%	6%	8%	9%	11%	12%	14%	15%
Agriculture	0%	1%	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%
Land use	0%	0%	0%	0%	0%	0%	1%	1%	1%	1%	1%	1%
Waste	0%	1%	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%

2.10. Effect of Internet handprint on global GHG supply

In Table 19 the handprint of ICT solutions on TAGGHGS is summarized. In 2030 at the most \(\approx17\)% of TAGGHGS (11.37 Gt of 66.3 Gt) can be reduced. In Section 4 it is elaborated with sensitivity analysis what speaks for 11.37 Gt and what speaks against.

\begin{table}[H] \begin{center}

Table 19. Effect of Internet handprint on global GHG supply from 2019 to 2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
Internet  infrastructure GHG Supply	1.06	1.08	1.08	1.07	1.07	1.08	1.09	1.14	1.22	1.33	1.48	1.71
Internet's handprint	0.08	0.65	1.20	2.14	3.12	4.16	5.23	6.35	7.53	8.75	10.03	11.37
Total global GHG supply without  Internet's handprint	57.5	53.4	54.6	55.8	57.0	58.2	59.4	60.7	62.0	63.3	61.8	66.3
Total global GHG supply with  Internet's handprint	57.4	52.8	53.7	53.9	54.1	54.2	54.3	54.5	54.6	54.6	54.7	57.9

Table 20. Specific ICT Solutions shares of total handprint in 2030.

	Effect pathway	Gt CO2e	Share of handprint  in 2030
SMART GRID
Smart metering in Buildings	less energy use	0.81	7.1%
Facilitating renewable energy sources	less energy use	0.37	3.3%
Power grid optimization	less energy use	0.93	8.2%
SMART AGRICULTURE
Autonomous intelligent tractors help avoid manual checks of the fields of cultivation.	less material use and land use	0.84	7.4%
SMART SERVICES
Car pools, leasing services, mobility-as-a-service	Fuel saving	0.46	4.0%
Products sold as Services	Material efficiency	2.07	18.2%
SMART TRAVEL
Public travel suggestions	Information availability	0.05	0.4%
Fleet car management	Fuel saving	0.05	0.4%
Route optimization in Leisure	Fuel saving		0.8%
SMART WORK
Video/telemeetings, air	Marginal effect on aviation	0.23	2.0%
Video/telemeetings, car	Fuel saving	0.23	2.0%
Teleworking, car	Fuel saving	0.46	4.0%
Office space	Energy saving	0.32	2.8%
Hotels	Energy saving	0.02	0.1%
SMART BUILDINGS
Office energy use	Thermal energy saving	2.43	21.4%
Food storage cooling energy	Electric power saving	0.32	2.8%
SMART TRANSPORT
Route optimization in Logistics	Fuel saving	0.67	5.9%
Facilitate choosing train instead of car	Information availability	0.34	3.0%
SMART CIRCULARITY
AI enabled optical sorting	Material recycling	0.21	1.8%
SMART LAND USE
Forest monitoring with drones and sensors	Forestation	0.06	0.05%
TOTAL		11.37	100%

\caption{Specific ICT Solutions shares of total handprint in 2030.}\label{t20} \end{center} \end{table}

2.11. Specific ICT Solution share of total handprint in 2030

Here specific ICT Solutions are listed in Table 20 and their share of total handprint. The largest reduction amount, 2.43 Gt and 21%, can be achieved from "Office energy use" and "Selling products as services, servitization" 2.07 Gt and 18%.

2.12. Individual products handprint

In the present decade there will be an unprecedented growth of generated data shown in Table 21 [16]. This can be used to estimate Internet's handprint per Byte. Individual product's services handprints can then be estimated if their bandwidths are known. Evidently, applicable handprint services need to have specific pathways for creating handprints.

Table 21. Effect of Internet handprint on global GHG supply from 2019 to 2030.

	2019	2020	2021	2022	2023	2024	2025	2026	2027	2028	2029	2030
Mobile  data  traffic  (EB/year)	354	549	829	1228	1825	2718	4057	6175	9290	13904	20767	31008
Fixed  data  traffic  (EB/year)	1964	2444	3054	3829	4817	6079	7693	9763	12420	15836	20234	25901
Within  and  between  Data  centers, (EB/year)	13004	16926	22011	28606	37120	48094	62208	80207	103278	132745	170229	217689
Global  Data  Center  IP Traffic, (EB/year)	15322	19919	25895	33663	43762	46890	73957	96145	124988	162484	211230	274599

3. Results

The order of magnitude of the GHG Supply handprint is reasonable. Figure 2 is a graphical representation of the present article. It seems plausible that Internet's handprint will be higher than its footprint. However, 11.37 Gt may be heavily overestimated.

4. Discussion

What would overthrow the results in the present research? Are there any new developments since 2015 which would falsify magnificent Internet handprints? What is only theory and what are the facts? Is here Internet's handprint massively exaggerated several times? One obvious issue is to which extent the digitalization has already been achieved, meaning that F's max value for t=2030 is already achieved. 2020 meant a huge leap forward in this respect for some ICT Solutions such as "Video/telemeetings, air" and "Video/telemeetings, car". Anyway the reduction of GHG supply observed in 2020 is not really caused by certain ICT solutions. However, they made e.g. working from home possible. Here the view is taken that ICT solutions number 10 and 11 in Table 4 are tested big-scale making the assumed 50% (F=0.5) cut of applicable Transport and Travel GHG Supply between 2019 and 2020 possible.

On the other hand it is not argued that Internet's handprint will lead to absolute smaller TAGGHGS, but rather a slow-down of the increase of TAGGHGS. 2020 will show this fact e.g. with fewer travels and more virtual meetings.

2020 showed that digital ICT technologies can be widely adopted very quickly. Smart Buildings is a very important area for ICT solution offsetting of TAGGHGS as Buildings use much energy. However, it may be argued that "Smart Metering in Buildings" and "Smart Buildings office energy use reduction" is equal. Also, old buildings may not be able to reduce energy use as much as newly constructed ones. These limitations are considered to be included in the sensitivity analysis in which F and MVCV are reduced 10 times. \(\text{MVCV}_{\text{Smart metering, Buildings,}2020}=0.\)03 [20] measurement is a good indication for the reasonableness of \(\text{MVCV}_{\text{Smart metering,Buildings,}2030}=0.1\).

Likewise, certain Smart Travel ICT Solutions may be similar to Smart Services for Travel. Regarding Smart Work ICT Solutions, "telemeeting" and "teleworking" seem identical with regards to Travel handprint.

If the biggest one - teleworking - is churned the total ICT potential is reduced by 0.48 Gt in 2030. Anyway, for Smart Work the anticipated 2030 savings may already have happened in 2020 due to the global macro changes in 2020. Temporary or permanent decline to a new baseline of airline and automotive travel and transport are the most obvious observations in 2020.

The effectiveness of AI to achieve savings depends greatly on sufficient data and the data scientists and engineers developing the AI software. AI has successfully been employed for forecasting the volume of waste which will be generated. This facilitate proper planning of landfill sites, recycling units, development as well as operation of garbage collection infrastructure. AI can cope especially well with historical data which are of nonlinear nature. Still several indications of savings exist.

ICT Product handprints

What is the link to ICT product related handprints? Dividing the total handprint in Table 5 (11.37 Gt) with the Global Data Center IP Traffic in Table 21 (268 ZettaByte) gives e.g. 0.039 kg CO2e/GigaByte for 2030. However, this intensity is quite rough but may be tested together with specific GigaByte/s bandwidth data.

The right performance for the right application

What performance is good enough for a certain application? Such questions are valid e.g. for Travel Electricity Use (Table 3) where Na-ion batteries (90-115 Wh/kg) could be enough for certain electric vehicles instead of Li-ion (100-265 Wh/kg) [57,58].

The global material efficiency/waste problem seems not to be solved effectively by improved local waste management (e.g. collection). Perhaps an AI optimization of total global supply chains - which targets waste minimization in production and Total Material Consumption/capita is more effective. Optimizing and predicting the whole nonlinear global societal system with Internet as a driver - markets, Input-Output, GHG supply, resources, costs, jobs, waste - is a daunting task which theoretically could better be managed with AI and humans instead of humans alone.

Sensitivity analyses

Without sensitivity checks by 2030 the handprint/footprint ratio will be around 7 (11.37/1.71). Using Monte Carlo simulation and maximum and minimum values of F, MVCV etc. gives an uncertainty spread such that GHGi=Internet,t=2020 = 1.17 Gt (Min 1.07 Gt Max 1.29 Gt) and ICThp,2020 = 0.35 Gt (Min 0.23 Gt Max 0.48 Gt). All assumptions are found in the Supplementary Information. Videomeetings represent almos t 50% of Internet's handprint in 2020 but just 4% in 2030 where instead Office energy use and Servitization of products dominate.

Using maximum and minimum values of F, MVCV etc. (see Supplementary Information) gives an uncertainty spread such that GHG\(_{i=\text{Internet,}t=2030} = 1.53\) Gt (Min 1.28 Gt Max 1.8 Gt) and ICT\(_{hp,2030} = 6.32\) Gt (Min 4.41 Gt Max 8.52 Gt). Compared to the original mean value (11.37 Gt), including the spread of input values reduces the mean value of ICT\(_hp,2030\) substantially due to ten times lower minimum values of F and MVCV. In 2020 the GHG supply by the Internet is some 70% higher than Internet's handprint. Hence, the GHG supply by the Internet is not off-set already in 2020 by ICT Solutions. Nevertheless, the handprint/footprint ratio will be around 4 (6.32/1.53) by 2030. Thus the handprint hypotheses in Section 1.2 are falsified.

5. Next steps

The present research confirms that the largest handprints from the Internet in 2030 will be in Buildings and Industry. Still, individual nations and service providers would like to estimate better specific handprints. Moreover, the measurable entities in the system should be identified more carefully for individual ICT Solutions such those related to 5G [59]. Additionally, the standardization of the handprint calculation for electronic products should be attempted.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

The class of Diophantine equations is classified in two categories, one is linear Diophantine equations and the other one is non-linear Diophantine equations. Both categories have numerous applications in solving the puzzle problems. Aggarwal et al., [1] discussed the Diophantine equation \(223^x+241^y=z^2\) for solution. Existence of solution of Diophantine equation \(181^x+199^y=z^2\) was given by Aggarwal et al., [2]. Bhatnagar and Aggarwal [3] proved that the exponential Diophantine equation \(421^p+439^q=r^2\) has no solution in whole number.

Gupta and Kumar [4] gave the solutions of exponential Diophantine equation \(n^x+(n+3m)^y=z^{2k}\). Kumar et al., [5] studied exponential Diophantine equation \(601^p+619^q=r^2\) and proved that this equation has no solution in whole number. The non-linear Diophantine equations \(61^x+67^y=z^2\) and \(67^x+73^y=z^2\) are studied by Kumar et al., [6]. They determined that the equations \(61^x+67^y=z^2\) and \(67^x+73^y=z^2\) are not solvable in non-negative integers. The Diophantine equations \(31^x+41^y=z^2\) and \(61^x+71^y=z^2\) were examined by Kumar et al., [7]. They proved that the equations \(31^x+41^y=z^2\) and \(61^x+71^y=z^2\) are not solvable in whole numbers.

Mishra et al., [8] gave the existence of solution of Diophantine equation \(211^{\alpha}+229^{\beta}=\gamma^2\) and proved that the Diophantine equation \(211^{\alpha}+229^{\beta}=\gamma^2\) has no solution in whole number. Diophantine equations help us for finding the integer solution of famous Pythagoras theorem and Pell's equation [9,10]. Sroysang [11,12,13,14] studied the Diophantine equations \(8^x+19^y=z^2\) and \(8^x+13^y=z^2\). He determined that \(\{x=1,y=0,z=3 \}\) is the unique solution of the equations \(8^x+19^y=z^2\) and \(8^x+13^y=z^2\). Sroysang [12] studied the Diophantine equation \(31^x+32^y=z^2\) and determined that it has no positive integer solution. Sroysang [13] discussed the Diophantine equation \(3^x+5^y=z^2\).

Goel et al., [15] discussed the exponential Diophantine equation \(M_{5}^p+M_{7}^q=r^2\) and proved that this equation has no solution in whole number. Kumar et al., [16] proved that the exponential Diophantine equation \((2^{2m+1}-1)+(6^{r+1}+1)^n=w^2\) has no solution in whole number. The exponential Diophantine equation \((7^{2m} )+(6r+1)^n=z^2\) has studied by Kumar et al., [17]. Aggarwal and Sharma [18] studied the non-linear Diophantine equation \(379^x+397^y=z^2\) and proved that this equation has no solution in whole number. To determine the solution of exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m\), \(n\) are whole numbers, in whole numbers is the main objective of this paper.

2. Preliminaries

Lemma 1. For any whole number \(m\), the exponential Diophantine equation \((2^{2m+1}-1)+1=z^2\) is not solvable in whole number.

Proof. For any whole number \(m\), \(2^{2m+1}\) is an even number so \((2^{2m+1}-1)+1=z^2\) is an even number implies \(z\) is an even number. Which means

Now, for the same \(m\), \(2^{2m+1}\equiv 2(mod3)\) implies; The result of (2) denies the result of (1), hence the exponential Diophantine equation \((2^{2m+1}-1)+1=z^2\) is not solvable in whole number.

Lemma 2. For any whole number \(n\), the exponential Diophantine equation \(1+(13)^n=z^2\) is not solvable in whole number.

Proof. For any whole number \(n\), \((13)^n\) is an odd integer, so \(1+(13)^n=z^2\) is an even integer, implies \(z\) is an even integer. Which means

Now, for the same \(n\) \(13^n\equiv1(mod3)\) implies; The result of (4) denies the result of (3), hence the exponential Diophantine equation \(1+(13)^n=z^2\) is not solvable in whole number.

Theorem 1. For any whole numbers \(m\) and \(n\), the exponential Diophantine equation \((2^{2m+1}-1)+(13)^{n}=z^{2}\) is not solvable in whole number.

Proof. We have following four cases;

• 1. If \(m=0\) then the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\) becomes \(1+(13)^n=z^2\), which is not solvable in whole numbers according to Lemma 2.
• 2. If \(n=0\) then the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\) becomes \((2^{2m+1}-1)+1=z^2\), which is not solvable in whole numbers according to Lemma 1.
• 3. If \(m,n\) are natural numbers then \((2^{2m+1}-1),(13)^n\) are odd numbers, so \((2^{2m+1}-1)+(13)^n=z^2\) is an even number, implies \(z\) is an even number. which means
  \begin{equation} \label{e5} z^2\equiv0(mod3)\ \ \ \text{or}\ \ \ z^2\equiv1(mod3). \end{equation}

  (5)
  Now, \(2^{2m+1} \equiv2(mod3)\) and \((13)^n\equiv1(mod3)\) implies
  \begin{equation} \label{e6} (2^{2m+1}-1)+(13)^n=z^{2}\equiv 2(mod3). \end{equation}

  (6)
  The result of (6) denies the result of (5) hence the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m\), \(n\) are positive integers, is not solvable in whole numbers.
• 4. If \(m,n=0\) then \((2^{2m+1}-1)+(13)^n=1+1=2=z^2\), which is impossible because \(z\) is a whole number. Hence the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m,n=0\) is not solvable in whole numbers.

3. Conclusion

In this paper, author fruitfully studied the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m,n\) are whole numbers, for its solution in whole number. Author determined that the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m,n\) are whole numbers, is not solvable in whole numbers.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

The concept of atom localization comes from the birth of quantum mechanics. In 1927, Heisenberg [1] uses the uncertainty principal and suggests that an atom can be localized. Interest in the precise and accurate position measurement of atom is because of its rich potential application [2,3,4]. Atomic localization has a wide range of application, mostly in Bose-Einstein condensation [5], laser cooling and trapping [6], atom nano-lithography [7] and center of mass wave function [8].

The main phenomenon in the field of Quantum Optics and Laser Physics is the quantum interference and atomic coherence, by which giant Kerr nonlinearity [9], spontaneous emission enhancement [10], Optical bistability (OB) [11], electromagnetically induced transparency (EIT) [12,13] and many more aspects are obtained. For the localization of an atom in one dimension (1D) based on the atomic coherence and quantum interference, much efforts has been done. For example, Zubairy and his colleagues estimated numerous localization systems for a two-level system by resonant fluorescence [14,15]. Agarwal and kapale [16] obtained the atom localization in 1D by using the coherent population trapping (CPT) for three-level atomic scheme in a lambda-type. According to Agarwal scheme, the population measurement could localize the atom. This population had the same fringe pattern in one of the ground state as presented by Febry Perot interferometer. Storey et al., [17] and Marte and Zoller [18] introduced atomic localization passing through the standing wave regime. By using quantum interference, Kien et al., [19] showed the polarized atom localization in 1D. Paspalakis and Knight [20] used quantum interference for atom localization. Qamar et al., [21] used resonance fluorescence for atom localization. For 1D atomic localization through the internal state of atom and the field dual measurement [22], the driving field of amplitude control and phase [23], an atom can be localized. Nha et al., [24] explained the 1D atom localization inside the Ramsey interferometer arrangement by the interaction of quantized atom in nature with a standing-wave field. Sahrai et al., [25] proposed that the position measurement of an atom depends on absorption of a weak probe filed, due to which the probability of an atom localization is 50%. For more precise localization of an atom, Liu et al., [26] used the interference of double dark resonance which showed that the probability of finding atom is 50%. While Wang and jiang [27] detected 100% probability of atom localization by using coherent methodology. Recently, Proite et al., [28], by using the method of electromagnetically induced transparency (EIT), conducted an experiment in which the atomic level population was localized. They gained the result that population of accurate hyperfine level might be strongly localized than that of spatial period.

In this paper, a theoretical approach has been presented for the physical realization of the atom localization by superposition of three standing wave fields in a proposed four levels atomic configuration. The most interesting result that we encounter with, is the variation of the bandwidth of the localized peak with time independent Rabi frequency. It seems clear from our theoretical results that without disturbing the probability amplitude the bandwidths of the localized peaks are reduced as the intensity of the space independent Rabi frequency is increased and thus the reduction in the uncertainty can be achieved very easily. Though our approach is a theoretical one, but still our findings are very interesting and can actively support the day to day progress in the field of research and modern technology.

We also investigated that in the localization peaks, the space independent Rabi frequency condenses the uncertainty principal without disturbing the amplitude probability. In addition with the space independent wave fields, the width of the half maximum decreases. Along with this, it is investigated that the Doppler broadened effect and collision decay rate, which disturb the localization amplitude probability and the localization peaks, tends to doublet. When amplitude probability of the localized peaks changes, the collision decay rate is to be zero.

2. Model and equations

The energy-level diagram of the atom field interaction that we are taking under consideration has been presented in Figure 1. The system is an experimental one, having three ground states and one excited state. The upper state \( \left\vert 4\right\rangle \) is coupled with lower state \( \left\vert 1\right\rangle \) by two coherent fields of Rabi frequencies \(\Omega_{1}\) and \(\Omega_{2}\), respectively. Whereas, the upper state \( \left\vert 4\right\rangle \) is coupled with lower state \( \left\vert 3\right\rangle \) by another coherent field of Rabi frequency \(\Omega_{3}\) and the probe field of Rabi frequency \(\Omega_{p}\) is applied between the upper state \( \left\vert 4\right\rangle\) and lower state \( \left\vert 2\right\rangle\). The \(\Delta_p\) is the detuning of weak probe field \(\Omega_p\), while the \(\Delta_{1,2,3}\) are the detuning of \(\Omega_{1,2,3}\) by taking \(\Delta_{1,2,3}\) zero i.e., \(\Delta_{1,2,3} = 0\). The equation of motion is carried with Hamiltonian which is the sum of self Hamiltonian and interaction Hamiltonian \(H=H_0+H_\imath\). The self Hamiltonian \(H_0\) can be written as:

While the interaction Hamiltonian \(H_i\) is written as;

For dynamic of a system the master equation is written as: To solve Equation (3) the following set of equations are obtained; Taken the above coupling equations in first order of perturbations. Initially the atoms are considered in the ground state \(\left|2\right\rangle\), i.e., the probability of the atoms in this state is 1. So \(\rho^{(0)}_{22}=1\) and \(\rho^{(0)}_{44}\)=\(\rho^{(0)}_{41}\)=\(\rho^{(0)}_{43}\)=0. For first order perturbation, the equation \(\rho=-M^{-1}. A\) is used for the solution of \(\rho_{24}\), where \(M\) is a \((3\times3)\) matrix and \(A\) is a constant. where \(T=\left(\gamma_2-2i\Delta_p\right)\left(\gamma_1+\gamma_2-2i\Delta_p\right)+\left(\Omega_1+\Omega_2\right)^2. \)

3. Space dependent susceptibility

A susceptibility is complex response function, which is used to characterize the medium. The real and imaginary part of complex susceptibility is associated with absorption and dispersion spectrum of the probe field and hence related to position information of atom. The susceptibility is given as; where \(N\) is the density of the medium.

4. Results and Discussion

The atom field interaction through experimental and theoretical research led us to the new kinds of observations in physical phenomena. For the development of present technologies, a single atom in a trap, quantum lithography, localization of atom in the sub half wave length domain and quantum computing play important role. In Equation (8), we indicated the main result for precise and accurate position of the atom. The rate of decay \(\gamma\) is taken to be \(1GHz\). In atomic system, the universal physical constants are \(\hbar, \epsilon_0=1\). The real and imaginary parts of susceptibility are anticipated for the atom position at the range of \(-\pi< kx< \pi\). The position distribution functions \(Re(\chi)\) and \(Im(\chi)\) are related to the localization information of an atom. When the nodes and antinodes coincide, they form the stationary wave which appear on the real and imaginary distributed function of the probe field. The superposition of standing wave field expresses remarkable results for the development of localization peak in the space period of \(-\pi< kx< \pi\) with one wavelength domain, where \(\Omega\) shows the magnitude of space independent Rabi frequency i.e., \(|\Omega_1|=|\Omega_2|=|\Omega_3|=\Omega\).

In this theoretical model, single localization peak is observed within position range of \(0< kx< \pm\pi\) or \(\lambda/2\) domain, where the phase are set to \(\pi/2, 3\pi/2\) with \(\eta_{1,3}=0.5\) and \(\eta_{2}=-0.5\). The probe detuning is taken \(\Delta_p=0\). Furthermore, when the strength of \(\Omega=|\Omega_{1,2,3}|\) increases the peak width becomes narrow and the uncertainty decreases. At a very high intensity, the peak bandwidth approaches to zero and minimum uncertainty is observed in the localized peaks as shown in Figures 2 and 3. When the phase is \(\pi/2\) then the single localized peak appears within the sub half wavelength domain \(-\pi< kx< 0\). If the phase is changed to \(3\pi/2\) then the single localized peak appears within sub half wavelength domain \(0< kx< \pi\) also with the change in phase from \(\pi/2\) to \(3\pi/2\) the position of single peak is shifted from left to right as shown in Figures 2 and 3. These results are applicable for the development of atomic nanolithography, trapping of neutral atoms, laser cooling, Bose-Einstein condensation and measurement of center of mass wave function.

In Figures 4 and 5 the plots are traced for localization peaks of atom. It is observed that two localization peaks appear when the direction of wave numbers \(\eta_{1,2}=1\). In this case, one localization appear within the position range \(0< kx< \pi\) on the probe distributed function and other localized peak is within the position range \(-\pi< kx< 0\). The phase varies from \(\pi/2\) to \(3\pi/2\) as discussed in Figures 2 and 3 using both real and imaginary distribution function. It seems that one localized peak is near to the origin position (\(kx=0\)) and the other peak is away from the origin. The position of the localized peaks are shifted with the variation of phase \(\varphi\). The width of the localization peaks decreases with the strength of the space independent Rabi frequency \(\Omega\). Figure 6 shows the localized patron at the phase \(\varphi=\pi\) within position \(-\pi< kx< \pi\), where the detuning is \(\Delta_p=0\gamma\). In this case, a single peak appears at the position \(kx=0\) with \(40\%\) probability in the imaginary part of susceptibility.

5. Conclusion

In this paper, we have discussed the localization of atom in a superposition of three standing wave fields. We used a tripod type atomic system driving by three standing wave fields and a probe field. A single localization peak is observed at a specific direction of the wave number and parameters within the sub half wavelength domain using both the probe absorption and Dispersion spectrums. Similarly, two localization peaks are observed within the one wavelength domain at other spectroscopic parameters and direction of the wave numbers. The bandwidth of localization peaks are decreased with the strength of space independent Rabi frequency. A sharp localized peak is obtained at very high intensity of space independent Rabi frequency with negligible uncertainty. These theoretical results are applicable for the development of atomic nanolithography, trapping of neutral atoms, laser cooling, Bose-Einstein condensation and measurement of center of mass wave function and appealing to the researchers to demonstrate experimentally and to improve the applied aspect of current technologies.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

The incidence of chronic kidney disease (CKD) has been increasing every year and has become a global health issue. It affects 8-16% of the world's population, especially in developing countries [1]. CKD is a kind of comprehensive kidney disease with progressive deterioration of renal function and systemic lesions. It was characterized by high prevalence, high mortality, low diagnosis and low awareness [2]. In 2002, the National Kidney Foundation (NKF) developed guidelines for CKD [3]. The criteria for this disease were as follows;

• (1)   The glomerular filtration rate (GFR) was less than \(60 mL/min\) per \(1.73 m^2\) for more than 3 consecutive months;
• (2)   Abnormal structure or function of the kidney was caused by various factors for more than 3 months;
• (3)   There were pathological abnormalities with or without a decrease in GFR, abnormal signs of kidney damage, or abnormalities in imaging.

Machine learning algorithm enables machines to learn from massive knowledge to the behavior rules and thinking patterns that are similar to human beings [4]. In recent years, the application frequency of machine learning algorithms in the medical field is increasing rapidly and the research depth is deepening continuously. At present, many scholars established correlation analysis and prediction models for some diseases, and the results were remarkable [5]. This study aims to compare the applicability of different algorithms to the prediction of CKD and to measure the effect of related indicators on CKD.

2. Materials and methods

2.1. Study population

This study was conducted in the Affiliated Hospital of Nanjing University of Chinese Medicine between January 2016 and December 2017. According to whether the patients had CKD, we divided them into two groups (CKD group and control group). We screened a total of 224 patients, 69 of whom were diagnosed with CKD. We selected basic clinical characteristics, laboratory findings and imaging characteristics, including a total of 23 indicators. All patients obtained written informed consent.

2.2. Statistics analysis

We used Python software (version 3.7; https://www.python.org) for statistics analysis. Continuous variables were represented by mean standard deviation, and discrete variables were represented by numbers. All features with a missing rate greater than 70% were selected to be excluded. Missing values were filled by means or modes. In statistical analysis, continuous variables and discrete variables used \(t\) test and chi-square test respectively. \(P\) value was considered statistically significant.

2.3. Model construction

Three commonly used machine learning classification algorithms were selected for modeling to predict CKD. They are extreme gradient boosting (XGBoost) [6], support vector machine (SVM) [7], and logistic regression [8]. XGBoost is a type of tree ensemble model. Its main idea is to continuously add trees and grow subtrees through feature splitting. Each time a subtree is added, a new function was learned, and the prediction results of these subtrees were used by the additive model. Adding together can continuously improve the accuracy of the model, realize the fitting of the residuals, and then accurately predict the results. Its objective function was calculated by the following formula: \begin{equation*}\label{2.1} Obj(\circleddash)=\sum_{i=1}^{n}l(y_{i}, \hat{y}_{i})+\sum_{k=1}^{K}\Omega(f_{k}), \end{equation*} where \(n\) is the number of samples, \(y_{i}\) is the true value of the \(ith\) sample, \(\hat{y}_{i}\) is the predictive value of the \(ith\) sample, \(l(y_{i}, \hat{y}_{i})\) is the error function of the model and \(\Omega(f_{k})\) is the regularization function of the model. The SVM algorithm uses the ideas of finding the maximum interval and projecting to higher dimensions to find a hyperplane with good data classification effect to realize the classification of data. Its model was solved based on the following formula: \begin{equation*}\label{2.2} min_{w,b} max_{a\geq 0} \frac{1}{2}||w||^{2}-\sum_{i=1}^n \alpha_{i}(y_{i}(w^{T}X_{i}+b)+1), \end{equation*} where \(w\) and \(b\) are the hyperplane parameter to be solved, \(\alpha\) is langrangian, \((X_{i},Y_{i})\) is the \(ith\) sample and \(n\) is the number of samples. Because SVM algorithm has a very good performance in dealing with nonlinear classification problems, so this model in medicine can be used to diagnose some diseases. Logical regression model is a very simple classification model. It introduces sigmoid function on the basis of regression model, realizes the mapping from predicted value to probability, and then can be used to deal with classification problems. The training model was calculated based on the following formula: \begin{equation*}\label{2.3} P_{y_{i}=1}=\frac{1}{1+e^{-W^{T}+X}}, \end{equation*} where \(\omega\) is the model coefficient to be solved, \((X_{i},Y_{i})\) is the \(ith\) sample. Although the logistic regression model has a simple structure, it is often used for its satisfactory effect on general problems and good explanation.

In the diagnosis of CKD, firstly, all 224 samples were divided into training set and test set according to the ratio of 80% and 20% respectively. The training set was used to fit and establish the above three models, while the test set was used to evaluate the effects of the established models. According to the evaluation results, the best performing model was selected among the three models as the machine learning diagnosis model for CKD.

3. Results

3.1. Clinical characteristics

According to statistical tests, 16 variables were considered statistically significant in the two groups. They were age, weight, BMI, haemoglobin, fasting blood glucose, serum creatinine, LDL-C, HDL-C, triglyceride, SBP, DBP, uric acid, leukocyte count, cIMT, PWV-ES, GFR. Summary of clinical characteristics is shown in Table 1. \begin{table}[] \begin{center}

Table 1. Summary of variables.

Variable	CKD group	Control group	P value
Gender
Male	37	62
Female	33	93	0.087
Age (year)	55.768\(\pm\) 14.009	43.403\(\pm\)12.895	<0.001
Height (cm)	166.841\(\pm\)7.455	165.391\(\pm\)7.558	0.185
Weight (kg)	68.191\(\pm\)12.282	59.830\(\pm\)8.443	<0.001
BMI (kg/m\textasciicircum{}2)	24.471\(\pm\)4.069	21.826\(\pm\)2.255	<0.001
Haemoglobin, g/l	121.899\(\pm\)23.233	139.572\(\pm\)14.606	<0.001
Fasting blood glucose, mmol/L	5.768\(\pm\)1.343	4.876\(\pm\)0.692	<0.001
Serum creatinine, \(\mu mol/L\)	156.437\(\pm\)148.807	67.087\(\pm\)15.270	<0.001
Total cholesterol, mmol/L	5.324\(\pm\)2.722	4.607\(\pm\)0.802	0.065
LDL-C, mmol/L	2.695\(\pm\)0.830	2.460\(\pm\)0.559	0.035
HDL-C,mmol/L	1.244\(\pm\)0.374	1.525\(\pm\)0.397	<0.001
Blood pressure, mmHG
SBP	135.217\(\pm\)18.189	117.628\(\pm\)14.399	<0.001
DBP	79.116\(\pm\)10.657	71.662\(\pm\)9.179	<0.001
Uric acid, \(\mu mol/L\)	396.259\(\pm\)146.634	278.394 \(\pm\)77.579	< 0.001
Leukosyte Count ,  \(10^9/L\)	6.362\(\pm\)2.196	5.700\(\pm\) 1.382	0.023
Erythrocyte Count, \(10^{12}/L\)	3.924\(\pm\)0.810	5.483\(\pm\)9.947	0.195
Alanine aminotransferase, \(\mu/L\)	25.391\(\pm\)16.038	23.086\(\pm\)19.991	0.400
Aspartate aminotransferase, \(\mu/L\)	23.493\(\pm\)12.830	22.314\(\pm\)11.286	0.491
cIMT, cm	0.062\(\pm\)0.030	0.045\(\pm\)0.002	< 0.001
PWV-BS, m/s	6.401\(\pm\)1.465	6.105\(\pm\)1.140	0.140
PWV-ES, m/s	9.079\(\pm\)2.033	7.184\(\pm\)2.034	< 0.001
GFR, \(mL/min/m^3\)	89.616\(\pm\)54.372	138.334\(\pm\)33.860	< 0.001

Data were numbers or mean value \(±\) standard deviation. BMI, Body mass index; LDL-C, Low-density lipoprotein-cholesterol; HDL-C, High-density lipoprotein-cholesterol; SBP, Systolic blood pressure; DBP, Diastolic blood pressure; cIMT, Carotid intima-media thickness; PWV-BS, Pulse wave velocity-beginning of systole; PWV-ES, Pulse wave velocity-end of systole; GFR, Glomerular filtration rate.

3.2. GFR in patients

GFR is an important indicator of CKD disease. According to the disease guidelines for CKD, patients are subdivided with GFR into five subgroups as follows' term:

• 1.   \(15 mL/min/m^3\),
• 2.   \(16-29 mL/min/m^3\),
• 3.   \(30-59 mL/min/m^3\),
• 4.   \(60-89 mL/min/m^3\),
• 5.   \(\geq 90 mL/min/m^3\).

Patients with a GFR value less than 60 typically had CKD. The overall levels of cIMT, PWV-ES and age in CKD group were higher than those in control group (Figure 1). However, there was no significant difference in PWV-BS values between the two groups (Figure 1).

3.3. Important features in the model

There are many factors that affect CKD, but there is still a wide variation in the importance of features. In this study, XGBoost, SVM and logistic regression were respectively used to analyze the importance of features. The importance of 16 features in different models is displayed in Figure 2. According to the results of the three models, cIMT was the most important among features. It was considered that the common feature among the top 9 important features of the three models respectively are the features that have a relatively large influence on CKD. Through the intersection processing, factors that have a greater impact on CKD are age, GFR, haemoglobin, LDL-C, uric acid, and cIMT. Figure 3 is the analysis result of the correlation among important variables. It can be found that the correlation between GFR and uric acid is -0.557, which indicated that there might be some correlation between them.

3.4. Model comparison

After modeling analysis, the performance of the three models on the test set is shown in Figure 4. In general, XGBoost is significantly better than other models (accuracy, 0.93; specificity, 0.89; sensitivity, 0.94; F1 score, 0.91; AUC, 0.99). The ROC curves of three models are shown in Figure 5. It can be seen that the effect of the logic effect is the worst, and the XGBoost effect is the best among three models.

4. Discussion

In this study, we established three CKD prediction models which are XGBoost, SVM and logistic regression. According to these three models, the ranking of feature importance under each model was analyzed. In order to measure the prediction effect of each model, we put forward four evaluation indicators (accuracy, specificity, sensitivity, F1 score and AUC) and ROC curve.

In previous studies, some scholars used artificial neural network, decision trees, and logistic regression to predict the survival of kidney dialysis [9]. A study used support vector machine and artificial neural network to construct predictive models of kidney disease. Its experimental results showed that the performance of artificial neural network is better than other algorithms, and it could obtain better accuracy and performance [10]. Baby and Vital [11] used AD trees, J48, Kstar, Naïve Bayes and Random forest in the prediction model of kidney disease and they found that the best methods were Kstar and Random forest. Although other studies separately developed methods with strong applicability for CKD, this study comprehensively considered the effect of the model, the interpretability and the applicability of the data. Therefore, one of each type of machine learning algorithm selected in this paper is more comprehensive and the selected model could not only predict disease but also evaluate feature importance. In current study, according to the effect of three models (XGBoost, SVM and logistic regression), XGBoost had the best performance for the prediction of CKD.

The study had several advantages; Firstly, applying machine learning model to the analysis of medical problems could promote the progress of automatic disease diagnosis of CKD. Secondly, using machine learning methods to extract important features, it is possible to find indicators that has a relatively large impact on CKD from a data perspective. Thirdly, tree model, SVM model and logistic regression model are respectively used to diagnose and predict CKD. These three algorithms covered most models with relatively good explanatory ability and could comprehensively analyze the performance of machine learning algorithms in CKD diagnosis.

There were some limitations; Firstly, the amount of data used in modeling was relatively small, so the model established might be insufficient in generalization ability, and more data should be used to further verify the effectiveness of the model. Secondly, the extracted important features had not been further combined with the medical feasibility analysis, and the analyzed indicators might be uncontrollable, so the feasibility of the treatment plan needed to be further considered. In addition, since the "black box" model could not measure the importance of each feature, some "black box" models (such as artificial neural network) were not analyzed and compared. These "black box" models might had better effects, but there is no further discussion here.

5. Conclusion

Through the establishment of three predictive models, XGBoost is the most suitable for the diagnosis of CKD. In the feature importance analysis of the three models, cIMT was found to be a strong predictor of CKD. Compared with PWV-BS, PWV-ES has more important effect on CKD and stronger correlation with GFR.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

In December 2018, the world's population was estimated to be 7.7 billion. Nigeria ranked number seven and was estimated to have an equivalent of 2.57% of the world population (Worldometers, 2018). Nigeria, being the most populous African country, has an urban population of 51.0% of its population and this has a direct link to the demands on food, clothing and shelters [1]. Urban cities like Lagos, Ibadan, Warri, Kano, Ibadan, Kaduna, Abuja, Calabar, etc, are housing a large number of people and due to their landmass, there are high demands on taller buildings in order to accommodate the population [2]. In most developing countries like Nigeria, Portland cement as a water-based binder in concrete is the most utilized construction material [3] and is not an environmentally friendly material because its production and usage emit over 5% of the total \(CO_2\) anthropogenic emissions [4]. In 2015, the Fredonia group projected that in 2019 the global demand for cement would reach 5.2 billion metric tonnes. The cost of construction continues to increase globally, especially in Nigeria where construction cost is one of the highest [5].

According to the National Ready Mixed Concrete Association, in mix designs, the binder content is largely a function of the amounts of carbon dioxide \(CO_2\) embodied in concrete. For every ton of cement manufactured, over half a ton of \(CO_2\) emissions is on the loose into the atmosphere which makes cement the third-largest \(CO_2\) producer in the world [6]. The mining process of aggregates (fine and coarse) has an adverse effect on the ecosystem around the operation areas [7]. Quarry activities have a major impacton the environment like engineering impact (chemical spills, noise, vibrations, erosion, dust, loss of habitat, etc), cascading impacts (removal of rock) and geomorphic impact [8].

Nigerian Environmental Society (NES), said that Nigeria as a nation generates over 60 million tonnes of waste annually with less than 10% waste management capacity [9]. One of the goals of sustainable construction is the reduction of cement usage in the production of concrete. Also, the reusability of industrial and agricultural waste materials in concrete production offers environmental gains and the preservation of natural resources [10].

Sawdust, an industrial waste generated from the timber industry, produced as wood chippings or loose particles from sawing of timber into desired or standard usage sizes [11]. Due to the nation's poor waste management techniques, sawdust poses a nuisance to the health of citizens and the environment. Steel-slag, a by-product of the steel, produced when scrap metals and irons are liquefied together with fluxes under oxidizing conditions by injecting an enormous amount of air or oxygen [12]. Cordeiro et al., [13] proposed the usage of pozzolanic and blended cement to reduce the use of Portland cement in concrete production. Abdullah et al., [14] supported the proposition by saying blended cement and concrete incorporated with pozzolanic materials have created an innovative solution for producing concrete with savings in energy, improvement in certain properties of hardened and fresh concrete (like extensibility, workability, heat of hydration, resistance from sulphate attack, and other environmental considerations, by decreasing \(CO_2\) emissions). Raheem et al., [15], investigated saw dust ash as a partial replacement for cement in concrete and they observed that the compressive strength decreased with increasing SDA replacement at early stages but improves significantly with curing age. It was concluded that 5% SDA substitution is adequate to enjoy the maximum benefit of strength gain. The inclusion of SDA into the concrete matrix causes little expansion due to low calcium content [3], saturating the cement mix with oxides such as \(K_2O\) and \(MgO\) in SDA which form composites that may inhibit the formation of strength-giving calcium silicate hydrates from cement hydration [16] and the optimum replacement level of 10% by volume SDA can be used to partially replace cement [17,18,19].

The usage of waste aggregates such as steel slag can help reduce the dependence on the natural rock (granite), hence preserving our natural resources, recycling and optimum utilization of by-products for economic, environmental and construction aims [20]. The investigation of the compressive strength of steel slag aggregate concrete by many researchers shows an increase in strength up to 75% replacement and a further increase in steel slag resulted in a decline in strength [20,21]. Thangaselvi [22] stated that the improvement in strength may be due to shape, size and surface texture of steel slag aggregates, which provide better bonding between the particles and cement paste. Awoyera et al., [23] studied the performance of steel slag aggregate concrete with varied water-cement ratio, observing an increase in strength gain due to steel slag aggregate inclusion and concluded that a rapid strength development can be obtained in concrete by reducing the water-cement ratio. Presently, the desire to drastically reduce \(CO_2\) emissions, high demand for cement and its energy consumption and the dependence on other natural resources in the production of high performing concrete have led to the search for innovative binders and aggregates with the view to produce high strength and more durable concrete. These materials range from industrial bye products like silica fume, steel slag, blast furnace slag and fly ash to agriculture wastes like sugarcane bagasse ash, rice husk ash, sawdust ash, and com cub ash. Therefore, it is valuable to study the performance of concrete having a certain percentage of sawdust ash as binder and steel slag aggregate as normal coarse aggregate regarding the strength and durability properties of concrete. Hence, exploring innovative materials incorporated into concrete production is of great significance to the civil engineers for sustainable development, sustainable concrete production, sustainable construction and hence, sustainable development.

2. Materials and methods

2.1. Materials

The binders used were ordinary Portland cement (OPC), which conforms to the relevant standards (ASTM C-150, BS 12 and BS EN 197-1) and sawdust ash. The sawdust ash was sieved using the sieve size \(90\mu m\) micron size. Coarse aggregates were crushed granite ranging from \(12.5mm\) to \(19mm\) sizes and steel slag aggregate collected from an iron producing company (Top Steel Nigeria Limited, Odogunyan, Ikorodu, Lagos State, Nigeria), mechanically crushed using mechanical crusher set from \(12.5mm\) to \(19mm\) aggregate size. The fine aggregate used was river sand gotten from River Ogun, which was free from organic matter and salt. The water used for this research was clean, portable and impurities-free obtained from University of Lagos Water Distribution System, which was in accordance with BS 3148.

2.2. Methodology

2.2.1. Mix design and sample preparation

A design mix proportions of \(343.17Kg/m^3\) cement, \(622.85Kg/m^3\) sand, \(1264.58Kg/m^3\) granite and \(188.74Kg/m^3\) water with W/C ratio of \(0.55\) for a target strength of \(30N/mm^2\). A total of 48Nos. \(150mm \times 150mm \times 150mm\) concrete cubes and \(32Nos\). \(150mm \times 300mm\) concrete cylinders specimens were produced. A concrete mixer was used in mixing the concrete constituents to produce freshly mixed concrete. The mixtures were poured into various moulds for different concrete elements and compacted using tapping rod and vibrating machine. The moulds used for the cubes and cylinders were smeared with oil and the specimens were produced. The specimens were demoulded after \(242\) hours and cured in potable and lagoon water until the age of the test of 7, 14, 21 and 28 days for the cubes and cylinders. The Control sample consists of cement, sand and granite while the composite sample consists of 90% cement, 10% sawdust ash, sand, 75% steel slag aggregate and 25% granite.

2.2.2. Testing procedure

Chemical analysis was carried out on the Portland cement and sawdust ash at the Department of Chemistry, University of Lagos. The physical properties of the research materials were investigated. The workability was determined in the fresh state of the composite concrete. A compressive strength test was conducted using Avery Universal Testing Machine having a loading rate of \(120 kN/min\) which was in accordance with BS EN \(12390-3\). Splitting tensile strength test was done in accordance with BS EN \(12390-6 \) and ASTM \(C496-96\) using a loading rate of \(120 kN/min\).

2.2.3. Mathematical model

Formulation of mathematical models

The results of the experimental data for the mechanical properties of the sawdust ash blended steel slag aggregate concrete were analyzed using a multivariate interpolation method to develop mathematical models for predicting parameters with respect to its variables.

Multivariate interpolation method

The algorithm for multivariate interpolation method (bilinear interpolation) for the value of the unknown function \(f\) at points \(x\) and \(y\). For a known value of \(f\) at four points; \[Q_{11}=(x_{1},y_{1}), \ \ \ Q_{12}=(x_{1},y_{2}), \ \ \ Q_{21}=(x_{2},y_{1}) \ \ \ \text{and} \ \ \ Q_{22}=(x_{2},y_{2}) . \] Linear interpolation in the x-direction gives \begin{align*} f(x,y_1)&=\frac{x_2-x}{x_2-x_1}f(Q_{11})+\frac{x-x_1}{x_2-x_1}f(Q_{21}),\\ f(x,y_2)&=\frac{x_2-x}{x_2-x_1}f(Q_{12})+\frac{x-x_1}{x_2-x_1}f(Q_{22}). \end{align*} Interpolating in the y-direction gives the estimate: \begin{equation*}\label{e3} f(x,y)=\frac{y_2-y}{y_2-y_1}f(x,y_1)+\frac{y-y_1}{y_2-y_1}f(x,y_2). \end{equation*}

Validation of mathematical models

The validation of the model was done by determining the percentage difference and comparing the predicted values on the basis of the model and those data obtained from the experiment using simple percentage difference formula; \begin{equation*}\label{e4} \text{Percentage Difference}=\frac{\text{Actual Result }-\text{ Model Result}}{\text{Actual Result}}\times 100\%. \end{equation*}

3. Results and discussion

3.1. Atomic absorption spectrometry (AAS) analysis

The chemical composition of the samples was determined by conduction AAS analysis and the results are presented in Table 1.

Table 1. Chemical composition of studied materials.

Sample	Sample Description	Chemical Constituents
		\(SiO_2\)	\(Al_2O_3\)	\(Fe_2O_3\)	\(CaO\)	\(MgO\)	\(Na_2O\)	\(K_2O\)	\(CaCO_3\)	\(SO_3\)	\(P_2O_5\)	\(L.O.I\)	Total \(SiO_2\) +\(Al_2O_3\)	Total \(SiO_3\)+ \(Al_2O_3\) +\(Fe_2O_3\)
Cement	Grey  Powdery Solid	2.85	5.26	1.35	64.12	2.72	0.76	0.48	11.45	1.2	0.3	0.1	8.11	9.46
SDA	Grey  Powdery Solid	31.29	27.38	2.29	10.72	9.2	6.63	10.45	19.14	0	0.5	2.9	59.3	61.59

The results from Table 1 show that the samples are similar physical observation (colour). The similarity in colour does not translate to the ability to perform alike but its performance is based on the degree of its constituents' chemical elements present in it. The combining percentage masses of silica \(SiO_2\), alumina \(Al_2O_3\) and ferric oxides \(Fe_2O_3\) give a total of 61.59% which is similar to those of class C type pozzolans according to ASTM C618-12a. \(CaO\) known for providing strength in cement was observed to be low in sawdust ash leading to a low strength performance and increase in setting time. The absence of \(SO_3\) in sawdust ash makes it unsound. The \(MgO\) was found to be within the limit range of less than 5% but the sawdust ash recorded 9.2% which can be attributed to the reduction of the strength of concrete (BS 12, 1996; [17]). The loss on ignition value for sawdust ash is within the limits of 3.0% set by BS 12, 1996. The percentages of Na2O and K2O known as the alkali oxides were observed to be large when compared to the standard range (BS 12, 1996), which resulted in some difficulties in regulating the setting time of cement paste.

3.2. Physio-chemical properties of water used in concrete curing

The Physio-chemical properties of the water used for curing were conducted at Central Research Laboratory, University of Lagos. The potable water (UNILAG Tap Water) and lagoon water used as a curing medium in this study were examined in terms of physio-chemical compositions, and all the results are reported in Table 2. According to BS 3148 (1980), any drinkable water, either treated or untreated for distribution through the public supply, is suitable for making concrete. The percentage of chloride content present in both water specimens was within tolerable range and the TDS value is not above \(2000 ppm\) which render the water specimens fit for making and curing concrete (BS 3148,1980). A pH range between \(6.0\) and \(8.0\) have no significant effect on the compressive strength of concrete but a high percentage of chloride and some other contents contribute significantly. The values of the sulphates and alkalinity of the two water specimens were within the acceptable limit (WHO; BS 3148, 1980). However, it was observed that lagoon water had high chloride, salinity, TDS, total hardness, alkaline and sulphate contents when compared to tap water.

Table 2. Physio-chemical properties of different curing medium.

Parameters	UNILAG Tap water	UNILAG Lagoon water	WHO Standard	BS Standard (BS 3148:1980)	Nigeria Standard
ph	6.4	6.3	6.5-8.5	6.0-8.0	6.5-8.5
Temperature(\(^oC\))	27.4	27.5	Ambient	Not mentioned	Not mentioned
Salinity \((ppm)\)	72	460	Not mentioned	Not mentioned	Not mentioned
Conductivity (\(us cm^{-1}\))	170	340	750	Not mentioned	1000
TDS \((mg/l)\)	125	245	1000	2000	500
Chloride \((mg/l)\)	64	140	250	500	250
Calcium Hardness \((mg/l)\)	48	64	Not mentioned	Not mentioned	150
Total hardness \((mg/l)\)	60	108	100	Not mentioned	Not mentioned
Acidity \((mg/l)\)	12	16	Not mentioned	Not mentioned	Not mentioned
Alkalinity \((mg/l)\)	20	96	100	1000	Not mentioned
Sulphates \((mg/l)\)	60	90	250	1000	100
Colour \((pcu)\)	179	792	15	Not mentioned	Not mentioned
Calcium \((mg/l)\)	7.5	8.86	200	Not mentioned	Not mentioned
Manganese \((mg/l)\)	0.32	0.22	0.5	Not mentioned	0.2
Copper \((mg/l)\)	0.54	0.54	2	Not mentioned	1
Zinc \((mg/l)\)	3.76	5.62	3-5	Not mentioned	3-5
Lead \((mg/l)\)	0.12	ND	0.01	Not mentioned	0.01
Iron \((mg/l)\)	14.98	108.3	0.3	Not mentioned	0.3
Cadmium \((mg/l)\)	0.04	0.21	0.003	Not mentioned	0.003

3.3. Physical properties of constituents

Sieve analysis/graduation of aggregates

The results of the sieve analysis for samples are presented in Figures 1 and 2. The results of the investigation on the physical properties of concrete constituents used for this research are presented in Table 3. The fineness of cement or any cementitious materials gives the total surface area of the cement or cementitious materials that is available for hydration. From Table 3, the percentage fineness recorded 53.93% and 37.18% for sawdust ash and cement respectively, indicating that the cement is finer than the sawdust ash. The finer the sample, the more reactive the sample and it offers a greater surface area of particles for hydration. Also, the specific gravities of cement and granite were higher than sawdust ash and steel slag respectively, which signify that cement and granite are denser. This implies that the more the percentage replacement of sawdust ash and steel slag in the concrete, the lesser the overall weight of the concrete and structure at large which would be an economic gain for massive structures such as storey buildings and bridges without compromising desired strength.

Table 3. Physical properties of studied materials.

Physical properties	Cement	Saw Dust  Ash (SDA)	Sand	Granite	Steel Slag
Coefficient of Uniformity \((C_u)\)	-	-	2.67	1.7	1.6
Coefficient of Curvature \((C_c)\)	-	-	1.04	0.99	1.06
microns sieve) (Moisture Content (Dry Density \((kg/m^3)\)	521.01	129.28	126.13	692.66	1098.13
Bulk Density \((kg/m^3)\)	1015.96	511.96	1186.90	1295.27	1229.91
Specific Gravity	2.90	2.11	2.50	2.85	2.64

The values of dry and bulk density for the concrete constituents are within the standard ranges (BS EN 197-1:2011, BS 12:1996), having steel slag classified as normal weight aggregate according to ASTM C330. The steel slag aggregate recorded a significant increase in moisture content which can be attributed to the presence of voids. The water absorption values for the coarse aggregates are within the standard range of 0.1-2.0%. The results of the aggregate impact value test, aggregate crushing value test and Los Angeles abrasion test revealed that granite has greater resistance, stronger and higher toughness when compared to steel slag.

3.4. Workability of pozzolan blended steel slag aggregate concrete

The results of the slump test and compacting factor test for steel slag aggregate-based concrete are presented in Table 4. From the Table 4, a stiff plastic was observed having no separation of coarse aggregate particles from the mortar matrix during placement. After placement, no notable bleeding was found for fresh concrete.

Table 4. Workability of specimens.

	Control	Composite
Slump (mm)	30	20
Compacting factor	0.834	0.825
Degree of Workability	Low	Low
	Stiff plastic	Stiff plastic

The concrete without sawdust ash and steel slag aggregate recorded higher workability compared to the composite mix having sawdust ash and steel slag aggregate. The reduced values of a slump and compacting factor for the composite can be attributed to the presence of water-absorbing properties of sawdust ash and steel slag which left little amounts of water in the mortar available for hydration in the concrete matrix [17,24].

3.5. Density of pozzolan blended steel slag aggregate concrete

The results of the density of sawdust ash blended steel slag aggregate concrete is presented in Figure 3.

Figure 3 shows the relationship between the control specimen and the composite for two different curing media. The density of concrete increases with an increase in the curing age. A slower rate of strength development was observed for control and composite samples cured in lagoon water due to the presence of some chemicals above the standard level of concentration. Lower densities were recorded for specimens cured in lagoon water. The density at 28 days for the control specimen was about \(2640kg/mm^3\) and \(2660kg/mm^3\) for the composite. The value of the density of sawdust ash blended steel slag aggregate concrete (composite) was 1.16% higher than the normal concrete with granite (control). The rate of deterioration on the density of the control specimens was 1.16% and 1.26% for composite at 28 days when relating the potable water curing to the lagoon water curing. In both curing media, the composite specimen performed better than the control due to the presence of steel slag aggregate which caused more increase in a unit weight of concrete [25]. The high-density values observed in composite concrete can be attributed to the high dry density of steel slag compared with limestone. Also, the nature of SDA as a good absorbent of moisture contributed to the increase in density [15].

3.6. Compressive Strength of pozzolan blended steel slag aggregate concrete

The results from the compressive strength test performed on the control and composite specimens are presented in Figure 4 for different curing mediums.

From Figure 4, the influence of curing age on the compressive strength of the control and composite specimens was illustrated and it was observed that as the curing age increases, the compressive strength increases for both samples. The compressive strength of the composite was observed to be higher than the control specimens in potable water curing and vice versa in lagoon curing. This was mainly due to cement hydration and the build-up of hydration products which filled up the available pore spaces inside the concrete matrix resulting in strength performance improvement [23]. The rate of internal strength development in the composite depends on the pozzolanic activities of minerals present in the sawdust ash and steel slag as well as their differences in the particle shape, surface texture and zonal composition of aggregates [26,27].

The strength gain in the control increased with increase in curing age of 10.03% of 7 days to 14 days curing up to 24.64% gain at 28 days curing in potable water which was observed higher than the specimens cured in lagoon water having 9.13% at 14 days to 23.91% at 28 days. A trend opposite to that of the potable water was observed in the lagoon water curing as the composite having a higher strength gain of 28.96% to that of 24.79% of the control at 28 days. The decrease in the strength of the composite can be attributed to the presence of chloride and sulphate ion which leads to expansion and weakens the bonds between the aggregate and the paste. Therefore, the development of crack formation within the concrete mass together with leaching action of the newly formed compounds would resulting the reduction of the strength. Nevertheless, the swift gain in strength for specimens cured in lagoon water when compared to potable water is due to the quickening effects of some of the chemical compounds present in the curing medium.

3.7. Split Tensile Strength of pozzolan blended steel slag aggregate concrete

The results of the split tensile test have been analysed and presented in Figure 5. These figures clearly demonstrate that the composite performs better in potable water than in lagoon water curing when compared with its control specimens. In the case of potable water curing, composite shows higher strength than control having \(2.29N/mm^2\) to \(2.16 N/mm^2\) at 28 days curing age.

In lagoon water curing medium, the split tensile strength for 28 days exposure period is \(2.03N/mm^2\) for control and \(1.89N/mm^2\) for composite. The test reveals that the split tensile strength of both the control and composite is greatly affected by the curing medium and the curing age. From Figure 5, the relationship between relative strength and curing ages or exposure periods for different curing environment was presented. As observed, the strength gain with respect to potable water is 8.07%, 23.04% and 34.16% as the curing age increases for control and 6.98%, 20.93% and 33.14% for composite whereas for lagoon water curing slower strength gains were observed from 5.23% to 18.02% for control while 7.95% to 25.17% for composite at 14 and 28 days respectively. The reason for the lower performance of composite in lagoon water curing is due to expansive materials developed as a result of reactions formed during hydration causing microcracks that weaken the bond between the hydrated products and aggregate particles [28]. Thus, the concrete loses its strength and failure occurs.

3.8. Mathematical models for sawdust ash blended steel slag aggregate concrete using a multivariate interpolation method

Mathematical models were developed from the experimental data to predict some key properties such as density, compression and split tensile strength of sawdust ash steel slag aggregate concrete specimens in different curing medium.

Density prediction model

Potable curing:

\begin{align*} \gamma_{d}=&-0.17978620\times 10^{-4}SC^{3}+0.710408163 \times 10^{-2}SC^{2}-0.016976192 SC + 0.094000000 S\notag\\ & +0.3255580 \times 10^{-4}C^{3}-0.0019591837 C^{2} +0.042690478C + 0.02255999997. \end{align*}

Lagoon curing:

\begin{align*} \gamma_{d}=&-0.1\times 10^{-11}SC^{3}+0.1 \times 10^{-10}SC^{2}-0.001428572 SC + 0.010000001 S\notag\\ & -4.859086 \times 10^{-6}C^{3}+0.0003061224 C^{2} -0.001904762C + 2.5000000000. \end{align*}

Compressive strength prediction model

Portable water:

\begin{align*} F_{cu}=&-0.3401361\times 10^{-4}SC^{3}-0.004285714 SC^{2}+0.19738095 SC -0.26999998 S\notag\\ & +0.71914481 \times 10^{-3}C^{3}-0.040000000 C^{2} +0.96333334C + 20.81999999. \end{align*}

Lagoon water:

\begin{align*} F_{cu}=&9.3292260\times 10^{-4}SC^{3}-0.047653061 SC^{2}+0.74928572 SC -4.74999998 S\notag\\ & -0.31098153 \times 10^{-3}C^{3}+0.01214857 C^{2} +0.19023809C + 24.010000000. \end{align*}

Split tensile strength prediction model

Potable curing:

\begin{align*} F_{y}=&9.718172\times 10^{-6}SC^{3}-0.3061224\times 10^{-3}C^{2}+0.001666666 SC +0.0109999999 S\notag\\ & -0.82604470 \times 10^{-4}C^{3}+0.0048979591 C^{2} -0.049523809C + 1.760000000. \end{align*}

Lagoon water:

\begin{align*} F_{y}=&6.3168124\times 10^{-5}SC^{3}-0.0031632653SC^{2}+0.049047619 SC +0.4109999996 S\notag\\ & -9.718173 \times 10^{-6}C^{3}+0.0006122449 C^{2} +0.003333333C + 1.669999999, \end{align*} where, C is curing age in days. For control, \(S=0\) and for composite, \(S=1\).

Table 5. Validation of developed mathematical models for the density of pozzolan blended steel slag aggregate concrete using MVI method.

Curing Medium	Curing Age	Control			Composite
		Exp.	MVI Models	Perc. Diff.	Exp.	MVI Models	Perc. Diff.
Normal Curing	7	2.470123	2.470000008	0.004998	2.4908642	2.489999996	0.0346995
	14	2.545185	2.559000008	-0.54278	2.56888889	2.569999992	-0.04325
	21	2.58963	2.589000003	0.024313	2.61925926	2.619999984	-0.02828
	28	2.634074	2.630000038	0.154667	2.66469136	2.669999997	-0.19922
Lagnoon Curing	7	2.503704	2.499999997	0.147929	2.21851852	2.520000007	-0.05882
	14	2.511882	2.51999999	-0.32319	2.55111111	2.550000016	0.043553
	21	2.545185	2.549999981	-0.18917	2.59259259	2.590000029	0.099999
	28	2.577778	2.57999997	-0.08621	2.63111111	2.630000043	0.042228

Table 6. Validation of developed mathematical models for the compressive strength of pozzolan blended steel slag aggregate concrete using MVI method.

Curing Medium	Curing Age	Control			Composite
		Exp.	MVI Models	Perc. Diff.	Exp.	MVI Models	Perc. Diff.
Normal Curing	7	25.8518519	25.85000004	0.00716317	26.7400000	26.74000006	-2.2438E-07
	14	28.4444444	28.44000011	0.01562461	30.0000000	30.00000014	-4.6667E-07
	21	30.0740741	30.07000021	0.0135461	31.7400000	31.74000026	-8.1916E-07
	28	32.2222222	32.22000038	0.00689537	33.3700000	33.37000046	-1.3785E-06
Lagnoon Curing	7	25.9300000	25.92999996	1.5426E-07	24.4100000	24.41000003	-1.229E-07
	14	28.3000000	28.29999991	3.1802E-07	27.2592593	27.26000004	-0.00271754
	21	30.5800000	30.57999988	3.9241E-07	29.1851852	29.19000006	-0.01649767
	28	32.1300000	32.12999986	4.3573E-07	31.4800000	31.48000008	-2.5413E-07

Table 7. Validation of developed mathematical models for the split tensile strength of pozzolan blended steel slag aggregate concrete using MVI method.

Curing Medium	Curing Age	Control			Composite
		Exp.	MVI Models	Perc. Diff.	Exp.	MVI Models	Perc. Diff.
Normal Curing	7	1.61	1.610000002	-1.2E-07	1.72	1.719999998	1.16E-07
	14	1.74	1.740000002	-1.1E-07	1.84	1.839999999	5.43E-08
	21	1.981	1.979999999	0.05048	2.08	2.079999996	1.92E-07
	28	2.16	59999996	-2.8E+09	2.29	2.289999999	3.06E-08
Lagnoon Curing	7	1.72	1.719999997	1.74E-07	1.51	1.510000001	-6.6E-08
	14	1.81	1.809999995	2.76E-07	1.63	1.629999998	1.23E-07
	21	1.92	1.919999993	3.65E-07	1.72	29.19000006	2.91E-07
	28	2.03	2.029999991	4.43E-07	1.89	1.88999999	5.29E-07

4. Conclusion

From the results of this investigation, the following conclusions are made:

• 1.   The chemical composition of sawdust ash in this investigation revealed a total of 61.59% combined percentage of \(SiO_2\), \(Al_2O_3\) and \(Fe_2O_3\) and thus classified as class C type pozzolan.
• 2.   The application of sawdust ash and steel slag aggregate in the production of concrete for building structures will result in a reduced overall weight of the concrete structure due to their low specify gravities and bulk densities. Steel slag aggregate was found to possess good resistance to impact, crushing and abrasion.
• 3.   Lower workability was observed for composite concrete when compared with normal concrete due to the water-absorbing properties of sawdust ash and steel slag which leaves fever amounts of water in the mortar available for hydration in the concrete matrix.
• 4.   The composite concrete recorded higher compressive and split tensile strengths when compared to normal concrete as the curing age increases from 7 to 28 days when cured in potable water.
• 5.   The density, compressive and split tensile strengths of specimens (composite and control) cured in lagoon water were lower than specimens cured in potable water.
• 6.   The developed mathematical models using multivariate interpolation method were found to be in good agreement with the experimental data.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Higher order boundary value problems (BVPs) in ordinary differential equations (ODEs) are important tools for modelling different physical phenomena in sciences and engineering [1,2,3]. Although many ordinary differential equations especially linear ODEs have known analytical solutions, searching for numerical solutions is important because they provide reliable approximations to problems that are difficult to solve analytically [4]. In many situations, sound mathematical theories are often required for analysis, however, the closed form solutions may be too complicated, thus approximate solutions may be preferred [5].

Over the years, researchers have developed many numerical methods besides collocation methods for handling higher order BVPs. The least squares solutions of \(8th \) order boundary value problems using the theory of functional connections was developed by [3]. Similarly, [6] considered the numerical solution of \(8th \) order boundary value problems which arise in magnetic fields and cylindrical shells. In the same vein, [7] used the Legendre Galerkin method for the numerical solution of 8th order linear boundary value problems. [8] approximated the solution of some mth order linear boundary value problems where \(2\leq m\leq 9 \) by the use of a numerical method constructed with "Tchebychev" polynomial. The approximation of linear 10th order boundary value problems via polynomial and non-polynomial cubic spline techniques was considered by [9]. [10] applied the optimal homotopy asymptotic method to \(8th \) order initial and boundary value problems. [11] developed a continuous k-step linear multistep method (LMM) that was utilized to generate finite difference methods which were assembled and applied as simultaneous numerical integrators to solve \(4th \) order initial value and boundary value problems directly. [12] proposed a method for the numerical solution of special \(4th \) order BVPs via modified decomposition method.

Collocation methods using splines, polynomials and orthogonal polynomials have been developed and applied for the solution of higher order BVPs. [1] proposed a B-Spline collocation method for approximating higher order linear boundary value problems while a quintic B-spline and sixtic B-spline collocation methods were developed by [13,14] for the treatment of \(8th \) order boundary value problems. Similarly, a Haar wavelet collocation method for approximating \(8th \) order boundary value problems was developed by [15]. Cubic spline collocation tau method for handling 4th order linear ordinary differential equations was constructed by [16]. The Chebyshev polynomial was utilized by [17] to develop a multiple perturbed collocation tau-method which was used for solving \(4th- 6th \) order BVPs. Again, [2] applied the Taylor series polynomials as basis to form a standard collocation method and further developed a perturbed collocation method using Chebyshev polynomials as perturbation terms for approximating \(4th \) order BVPs. All the numerical methods mentioned above provided accurate approximations although with different accuracies.

The ease of implementing collocation methods with polynomial basis which provide accurate results that are comparable with other numerical methods is the motivation of this work. Since few orthogonal polynomials have been applied as trial functions to develop higher order collocation methods for solving BVPs, the Laguerre polynomial of degree \(N=10 \) is utilized as basis function to construct a collocation method which is implemented on 2mth higher order BVPs, \(2\leq m\leq 5 \). The existence and uniqueness of higher order BVPs are not considered in this work, however, this subject matter is comprehensively presented in [18] and [19].

The Laguerre collocation method is presented in Section 2, while the implementation is done in Section 3. Finally, Section 4 deals with the discussion and conclusion.

2. Methods

2.1. Laguerre polynomials

Laguerre polynomials are solutions of the Laguerre differential equation obtained via series solution using the Frobenius method at the centre \(x_{0}=0 \). According to [20], Laguerre polynomials can be generated by the formula which satisfies the following recursive relationship and The first two terms \(L_{0}(x) \) and \(L_{1}(x) \) can be generated from (2) while the rest of terms can be obtained from (3) or (4). The first eleven terms which are polynomials of various degrees generated as explained above are given below

2.2. Higher order linear boundary value problems

Consider the \(nth \) order ordinary differential equation defined on the interval \(a\leq x \leq b \) with the boundary conditions The linear form of (6) is given as where \(p_n (x),\ldots,p_0 (x) \) are coefficients of the unknown function and its derivatives which may either be a constant or function of \(x \), and similarly is \(g(x). \)

This work seeks to obtain the approximate solution to (8) and the boundary conditions as given in (7). However, the boundary value problems considered here have the same order \(n \) with the number of boundary conditions \(k \).

2.3. Derivation of the Laguerre collocation method

We assume that (8) and (7) can be approximated with a linear combination of the Laguerre polynomials provided in (5) as given below where \(N \) is the degree of the polynomial, and \(a_0,a_1,\ldots,a_N \) are constants to be determined. Equation (9) can be written in more compact sigma notation as Equation (10) is differentiated \(n \) number of times corresponding to the order of the boundary value problem given and thereafter substituted in Equation (8) to get Each term of Equation (11) is expanded and the like coefficients \(a_j,j=0,1,...,N \) are collected resulting to \(Q_j (x) \) are polynomials of various degrees while \(p_j^{*} (x) \) are coefficients which may be constants or functions of \(x \) which may not necessarily be polynomials.

2.4. Generating \(N+1 \) systems of linear equations

Since the boundary value problems considered in this work have the same order \(n \) with the number of boundary conditions \(k \), to solve for \(a_j \), \(j=0,1,…,N, k \) number of equations are generated using the boundary conditions, \(\frac{k}{2} \) each at the lower and upper boundaries respectively. The remaining \(N-k+1 \) equations are generated at the collocation points using Equation (12).

2.4.1. Generating \(k \) systems of linear equations using boundary conditions

The \(k=2,…,n \) equations generated using the boundary conditions are given as The \(k=2,\ldots,n \) equations generated using the boundary conditions are given as

2.4.2. Generating \(N-k+1 \) systems of linear equations at the collocation points

First, we state the equation which is used to generate the collocation points Equation (14) is used to get the various collocation points which are substituted into Equation (12) to get the remaining \(N-k+1 \) equations. Thus Equation (12) can be recast as for \(i=1,2,\ldots,N-(k-1) \) and \(a_j,j=0,1,…,N. \)

2.4.3. Representing the system of \(N+1 \) equations in matrix form

Altogether, Equations (13) and (15) give \(N+1 \) system of equations which can be written in matrix form where and where \(\sum_{j=0}^{N}a_jL_j^{*}(a) \) , \(\sum_{j=0}^{N}a_jL_j^{*}(b) \) and \(A^\star \), \(B^\star \) are the left and right hand side of the systems of \(k \) equations generated from the boundary conditions, while \(\sum_{j=0}^N a_j\left(p_j^{\star}(x_i)Q(x_i)\right) \) and \(g(x_i) \) is the left and right hand side of Equation (15) used to get the remaining \(N-K+1 \) system of equations.

3. Results

The Laguerre collocation method developed in Section 2 is applied to approximate some higher order linear boundary value problems. The approximate solutions inform of series solutions are displayed in Tables and compared to the analytical solution at some selected mesh points, while the accuracy is measured using absolute errors. Suppose x represents an independent variable and y the dependent variable, the approximate solution is denoted by \(y_n \), the analytical solution by \(y(x_n ) \) and an absolute error at a mesh point by \(e_n=|y_n-y(x_n )| \) in this work.

Example 1. Consider the \(4th \) order boundary value problem \(y^{(4)}=y+4\exp(x),   \quad 0\leq x \leq 1; \) with the boundary conditions \(y(0)=1, \)   \(y(1)=2\exp(1), \)   \(y^\prime(0)=1, \)   \(y^\prime(1)=3\exp(1) \)   and the analytical solution is given by \(y(x)=(1+x)exp(x). \) The results at the mesh points are given in Table 1.

Table 1. Approximate solution, analytical solution and absolute errors for Example 1.

\(n \)	\(x_n \)	\(y_n \)	\(y(x) \)	\(\left|y_n-y(x_n)\right| \)
\(0 \)	\(0 \)	\(0.9999999995995449000 \)	\(1.0000000000000000000 \)	\(4.0046\times 10^{-10} \)
\(1 \)	\(0.1 \)	\(1.2156880096812333908 \)	\(1.2156880098832123873 \)	\(2.0198\times10^{-10} \)
\(2 \)	\(0.2 \)	\(1.4656833097351525863 \)	\(1.4656833097922038007 \)	\(5.7051\times10^{-11} \)
\(3 \)	\(0.3 \)	\(1.7548164498926186474 \)	\(1.7548164498488040352 \)	\(4.3815\times10^{-11} \)
\(4 \)	\(0.4 \)	\(2.0885545768060973480 \)	\(2.0885545766977784449 \)	\(1.0832\times 10^{-10} \)
\(5 \)	\(0.5 \)	\(2.4730819061942987293 \)	\(2.4730819060501922202 \)	\(1.4411\times10^{-10} \)
\(6 \)	\(0.6 \)	\(2.9153900807828490922  \)	\(2.9153900806248143598 \)	\(1.5803\times10^{-10} \)
\(7 \)	\(0.7 \)	\(3.4233796028562222864 \)	\(3.4233796026998100867 \)	\(1.5641\times10^{-10} \)
\(8 \)	\(0.8 \)	\(4.0059736714316378601 \)	\(4.0059736712864416883 \)	\(1.4519\times10^{-10} \)
\(9 \)	\(0.9 \)	\(4.6732459113274837527 \)	\(4.6732459111982043612 \)	\(1.2928\times10^{-10} \)
\(10 \)	\(1.0 \)	\(5.4365636570322238390 \)	\(5.4365636569180904708 \)	\(1.1413\times10^{-10} \)

Example 2. Consider the \(6th \) order boundary value problem \(y^{(6)}=y-6\exp(x),   \quad 0\leq x \leq 1; \)   with the boundary conditions \(y(0)=1, \)   \(y(1)=0 \),   \(y^{(2)}(0)=-1, \)   \(y^{(2)}(1)=-2\exp(1), \)   \(y^{(4)}(0)=-3, \)   \(y^{(2)}(1)=-3\exp(1), \)   and the analytical solution is given by \(y(x)=(1-x)exp(x). \) The results at the mesh points are given in Table 2.

Table 2. Approximate solution, analytical solution and absolute errors for Example 2.

\(n \)	\(x_n \)	\(y_n \)	\(y(x) \)	\(\left|y_n-y(x_n)\right| \)
\(0 \)	\(0 \)	\(1.0000000000001685000 \)	\(1.00000000000000000000 \)	\(1.6850\times10^{-13} \)
\(1 \)	\(0.1 \)	\(0.9946538258967874881 \)	\(0.99465382626808286232 \)	\(3.7130\times10^{-10} \)
\(2 \)	\(0.2 \)	\(0.9771222054592636538 \)	\(0.97712220652813586712 \)	\(1.0689\times10^{-9} \)
\(3 \)	\(0.3 \)	\(0.9449011631547264200 \)	\(0.94490116530320217280 \)	\(2.1485\times10^{-9} \)
\(4 \)	\(0.4 \)	\(0.8950948151452182868 \)	\(0.89509481858476219068 \)	\(3.4395\times10^{-9} \)
\(5 \)	\(0.5 \)	\(0.8243606307092493746 \)	\(0.82436063535006407340 \)	\(4.6408\times10^{-9} \)
\(6 \)	\(0.6 \)	\(0.7288475147567316736 \)	\(0.72884752015620358996 \)	\(5.3995\times10^{-9} \)
\(7 \)	\(0.7 \)	\(0.60412580684272018001 \)	\(0.60412581224114295648 \)	\(5.3984\times10^{-9} \)
\(8 \)	\(0.8 \)	\(0.4451081812551064650 \)	\(0.44510818569849352092 \)	\(4.4434\times10^{-9} \)
\(9 \)	\(0.9 \)	\(0.2459603085739087748 \)	\(0.24596031111569496638 \)	\(2.5418\times10^{-9} \)
\(10 \)	\(1.0 \)	\(-3.25690\times10^{-14} \)	\(0 \)	\(3.2570\times10^{-14} \)

Example 3. Consider the \(8th \) order boundary value problem \(y^{(8)}-y=8\exp(x),   \quad 0\leq x \leq 1; \)   with the boundary conditions \(y(0)=1, \)   \(y(1)=0, \)   \(y^{\prime}(0)=0, \)   \(y^{\prime}(1)=-2, \)   \(y^{(2)}(0)=-1, \)   \(y^{(2)}(1)=-2\exp(1), \)   \(y^{(3)}(0)=-2, \)   \(y^{(3)}(1)=-3\exp(1), \)   and the analytical solution is given by \(y(x)=(1-x)\exp(x). \) The results at the mesh points are given in Table 3.

Table 3. Approximate Solution, Analytical Solution and Absolute Errors for Example 3.

\(n \)	\(x_n \)	\(y_n \)	\(y(x) \)	\(\left|y_n-y(x_n)\right| \)
\(0 \)	\(0 \)	\(0.999999999999919520 \)	\(1.00000000000000000000 \)	\(8.0480\times10^{-14} \)
\(1 \)	\(0.1 \)	\(0.9946538262678937601 \)	\(0.99465382626808286232 \)	\(1.8910\times10^{-13} \)
\(2 \)	\(0.2 \)	\(0.9771222065321117607 \)	\(0.97712220652813586712 \)	\(3.7959\times10^{-12} \)
\(3 \)	\(0.3 \)	\(0.9449011653172361413 \)	\(0.94490116530320217280 \)	\(1.4034\times10^{-11} \)
\(4 \)	\(0.4 \)	\(0.8950948185965164628 \)	\(0.89509481858476219068 \)	\(1.1754\times10^{-11} \)
\(5 \)	\(0.5 \)	\(0.8243606353404964961 \)	\(0.82436063535006407340 \)	\(9.5676\times10^{-12} \)
\(6 \)	\(0.6 \)	\(0.7288475201290590871 \)	\(0.72884752015620358996 \)	\(2.7415\times10^{-11} \)
\(7 \)	\(0.7 \)	\(0.60412581221979442386 \)	\(0.60412581224114295648 \)	\(2.1348\times10^{-11} \)
\(8 \)	\(0.8 \)	\(0.4451081856932311579 \)	\(0.44510818569849352092 \)	\(2.2624\times10^{-12} \)
\(9 \)	\(0.9 \)	\(0.2459603111159681689 \)	\(0.24596031111569496638 \)	\(2.7320\times10^{-13} \)
\(10 \)	\(1.0 \)	\(4.78512\times10^{-14} \)	\(0 \)	\(4.7851\times10^{-14} \)

Example 4. Consider the \(10th \) order boundary value problem \(y^{(10)}=-\left(1-x\right)\sin(x)+10\cos(x),   \quad 0\leq x \leq 1; \) with the boundary conditions \(y(0)=1, \)   \(y(1)=0, \)   \(y^{(2)}(0)=2, \)   \(y^{(2)}(1)=2\cos(1), \)   \(y^{(4)}(0)=-4, \)   \(y^{(4)}(1)=-4\cos(1), \)   \(y^{(6)}(0)=6, \)   \(y^{(6)}(1)=6\cos(1), \)   \(y^{(8)}(0)=-8, \)   \(y^{(8)}(1)=-8\cos(1) \) and the analytical solution is given by \(y(x)=(x-1)\sin(x). \) The results at the mesh points are given in Table 4.

Table 4. Approximate solution, analytical solution and absolute errors for Example 4.

\(n \)	\(x_n \)	\(y_n \)	\(y(x) \)	\(\left|y_n-y(x_n)\right| \)
\(0 \)	\(0 \)	\(6.8300\times10^{-17} \)	\(0 \)	\(6.8300\times10^{-17} \)
\(1 \)	\(0.1 \)	\(-0.0898511265815970128 \)	\(-0.089850074982145337076 \)	\(1.0516\times10^{-6} \)
\(2 \)	\(0.2 \)	\(-0.1589374674789321263 \)	\(-0.15893546463604897237 \)	\(2.0028\times10^{-6} \)
\(3 \)	\(0.3 \)	\(-0.2068669068902013555 \)	\(-0.20686414466293770258 \)	\(2.7622\times10^{-6} \)
\(4 \)	\(0.4 \)	\(-0.2336542608365488710 \)	\(-0.23365100538519029500 \)	\(3.2555\times10^{-6} \)
\(5 \)	\(0.5 \)	\(-0.2397162019696233407 \)	\(-0.23971276930210150014 \)	\(3.4327\times10^{-6} \)
\(6 \)	\(0.6 \)	\(-0.2258602632950288019 \)	\(-0.22585698935801414288 \)	\(3.2739\times10^{-6} \)
\(7 \)	\(0.7 \)	\(-0.19326809832291728014 \)	\(-0.19326530617130731610 \)	\(2.7922\times10^{-6} \)
\(8 \)	\(0.8 \)	\(-0.1434732509647720103 \)	\(-0.14347121817990455233 \)	\(2.0328\times10^{-6} \)
\(9 \)	\(0.9 \)	\(-0.0783337610764255054 \)	\(-0.078332690962748338846 \)	\(2.1.070\times10^{-6} \)
\(10 \)	\(1.0 \)	\(-4.8700\times10^{-17} \)	\(0 \)	\(4.8700\times10^{-17} \)

4. Discussion and Conclusion

4.1. Discussion

In this section, the numerical experiments carried out with our proposed collocation method as presented in Section 3 are discussed. MAPLE 17 is used to implement all the problems. To reduce round-off errors, the numerical approximations were rounded up to 20 digits.

The Laguerre polynomial of degree \(10 \) was used to develop an orthogonal collocation method for solving higher order boundary value problems in ordinary differential equations. Four test problems on \(4th \), \(6th \), \(8th \) and \(10th \) order boundary value problems were used to verify the efficiency and accuracy of the proposed method via absolute errors. The numerical results are displayed in Tables 1-4. The results from Tables 1-3 which are BVPs of order \(4 \), \(6 \) and \(8 \) respectively are highly accurate, while the result in Table 4 which is a BVP of order \(10 \) is fairly accurate when compared to the other problems. In general, our proposed collocation method provides an accurate numerical method for approximating higher order BVPs. However, we observed from Tables 1-4 that the accuracy of the numerical results increased at the boundaries as the order and number of boundary conditions also increased. On the other hand, the accuracy at the interior mesh points were less accurate to that at the boundary points as the order and number of boundary conditions increased. This may be as a result of the increase in the number of boundary conditions and a corresponding decrease in the number of collocation points. This may be the reason for the poor accuracy of Example 4 which has only one collocation point and \(10 \) boundary conditions. To develop a collocation method that may handle such higher order BVPs, it is advisable to consider many basis terms in order to get higher order polynomials so as to have many collocation points which may be equal or more than the equations obtained at the boundaries.

4.2. Conclusion

The Laguerre polynomial which is an orthogonal polynomial was used as a basis function to develop a collocation method. The proposed method was easier to develop and implement as compared to other functions which are used as basis for developing other collocation methods. The method is also accurate and comparable to many other collocation methods in literature. The collocation method can be extended to solve higher order BVPs by considering higher order Laguerre polynomials. Other orthogonal polynomials may similarly be used to develop collocation methods for handling higher order BVPs.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

In recent years, some extensions of special functions have been considered by many authors [1,2,3,4,5,6,7,8]. In 1994, Chaudhry and Zubair [3] have introduced the following extension of Gamma function

In 1997, Chaudhry et al., [1] presented the following extension of Euler's Beta function It is clearly seem that \(\Gamma_0(x)=\Gamma(x)\) and \(B_0(x,y)=B(x,y),\) where \(\Gamma(x)\) and \(B(x,y)\) are the classical Gamma and Beta functions (see [9]).

Afterwards, Chaudhry et al., [2] used \(B_p(x,y)\) to defined the Gauss hypergeometric and confluent hypergeometric functions as

\begin{align*} F_p(a,b;c;z)&=\sum_{n=0}^{\infty}(a)_n\frac{B_p(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},&\left(Re(p)\geq0;\ \ |z|< 1;\ \ Re(c)>Re(b)>0\right),\\ \Phi_p(b;c;z)&=\sum_{n=0}^{\infty}\frac{B_p(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},&\left(Re(p)\geq0;\ \ Re(c)>Re(b)>0\right), \end{align*} respectively, where \((a)_n\) denotes the Pchhammer symbol [9] defined as \((a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}\) and \((a)_0=1.\) Also, \(F_p(a,b;c;z)\) and \(\Phi_p(b;c;z)\) are defined as [2] \[\left(Re(p)\geq0;\ \ |\arg(1-z)|< \pi;\ \ Re(c)>Re(b)>0\right),\] \begin{align*} \Phi_p(b;c;z)=&\frac{1}{B(b,c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}e^{zt}\exp\left[\frac{-p}{t(1-t)}\right]\,dt,~~~~~~~ \left(Re(p)\geq0;\ \ Re(c)>Re(b)>0\right). \end{align*} Note that \begin{align*} F_0(a,b;c;z)&={}_2F_1(a,b;c;z),\\ \Phi_0(b;c;z)&=\Phi(b;c;z)={}_1F_1(b;c;z). \end{align*} In 2011, Özergin [10] (see also Özergin et al., [6]) introduced a further extension of the Gamma and Beta functions as and \[(Re(p)\geq0;\ \ Re(\alpha)>0;\ \ Re(\beta)>0;\ \ Re(x)>0;\ \ Re(y)>0), \] respectively. Clearly, we have \( \Gamma_p^{(\alpha,\alpha)}(x)=\Gamma_{p}(x),\) \(\Gamma_0^{(\alpha,\alpha)}(x)=\Gamma(x), \) \(B_p^{(\alpha,\alpha)}(x,y)=B_p(x,y)\) and \(B_0^{(\alpha,\beta)}(x,y)=B(x,y).\) In [6], \(B_p^{(\alpha,\beta)}(x,y)\) is used to defined \(F_p^{(\alpha,\beta)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta)}(b;c;z)\) as \begin{align*} F_p^{(\alpha,\beta)}(a,b;c;z)=&\sum_{n=0}^\infty(a)_n\frac{B_p^{(\alpha,\beta)}(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},\end{align*} \[(Re(p)\geq0;\ \ |z|< 1;\ \ Re(c)>Re(b)>0;\ \ Re(\alpha)>0;\ \ Re(\beta)>0), \] and \begin{align*} \Phi_p^{(\alpha,\beta)}(b;c;z)=&\sum_{n=0}^\infty\frac{B_p^{(\alpha,\beta)}(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},\;\;\;\;(Re(p)\geq0;\ \ Re(c)>Re(b)>0;\ \ Re(\alpha)>0;\ \ Re(\beta)>0), \end{align*} respectively. We observe that \begin{align*} F_p^{(\beta,\beta)}(a,b;c;z)&=F_p(a,b;c;z),\\ F_0^{(\alpha,\beta)}(a,b;c;z)&={}_2F_1(a,b;c;z), \end{align*} and \begin{align*} \Phi_p^{(\beta,\beta)}(b;c;z)&=\Phi_p(b;c;z),\\ \Phi_0^{(\alpha,\beta)}(b;c;z)&=\Phi(b;c;z)={}_1F_1(b;c;z). \end{align*} Very recently, Shadab et al., [11] introduced another form of the extended Beta function as where \(E_\alpha(z)\) is the classical Mittag-Leffler function given in [12].

The above extended Beta function (6) is used to defined \(F_{p,\alpha}(a,b;c;z)\) and \(\Phi_{p,\alpha}(b;c;z)\) as [11]

\begin{align*} F_{p,\alpha}(a,b;c;z)=&\sum_{n=0}^\infty(a)_n\frac{B^p_\alpha(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},& (\alpha\in R^+;\ \ p\in R^+_0;\ \ |z|< 1;\ \ Re(c)>Re(b)>0), \end{align*} and \begin{align*} \Phi_{p,\alpha}(b;c;z)&=\sum_{n=0}^\infty\frac{B^p_\alpha(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},& (\alpha\in R^+;\ \ p\in R^+_0;\ \ Re(c)>Re(b)>0). \end{align*} Also, very recently, Al-Gonah and Mohammed [13], extended the Gamma and Beta functions as [13] where \(E_{\alpha,\beta}^\gamma(z)\) denotes the generalized Mittag-Leffler function defined as [14] \begin{align*} E_{\alpha,\beta}^\gamma(z)&=\sum_{n=0}^\infty\frac{(\gamma)_n}{\Gamma(\alpha n+\beta)}\frac{z^n}{n!},& (z,\alpha,\beta,\gamma\in\mathbb{C};\ \ Re(\alpha)>0;\ \ Re(\beta)>0;\ \ Re(\gamma)>0). \end{align*} It clear that \begin{align*} \Gamma(\beta)E_{1,\beta}^\gamma(z)&={}_1F_1(\gamma;\beta;z),\\ E_{\alpha,1}^1(z)&=E_\alpha(z),\\ E_{1,1}^1(z)&=e^z. \end{align*} From (7) and (8), we note that \begin{align*} \Gamma_p^{(1,\beta,\gamma)}(x)&=\frac{1}{\Gamma(\beta)}\Gamma_p^{(\gamma,\beta)}(x),\\ B_p^{(1,\beta,\gamma)}(x,y)&=\frac{1}{\Gamma(\beta)}B_p^{(\gamma,\beta)}(x,y),\\ \Gamma_p^{(1,1,1)}(x)&=\Gamma_p(x),\\ B_p^{(1,1,1)}(x,y)&=B_p(x,y), \end{align*} where \(\Gamma_p^{(\gamma,\beta)}(x), B_p^{(\gamma,\beta)}(x,y), \Gamma_p(x)\) and \(B_p(x,y)\) denote the various forms of generalized Gamma and Beta functions given in (4), (5), (1) and (2) respectively. Also, we not that \begin{equation*} B_p^{(\alpha,1,1)}(x,y)=B^p_\alpha(x,y), \end{equation*} where \(B^p_\alpha(x,y)\) denotes the new extended Beta function given in (6).

2. A new forms of hypergeometric functions

In this section, we use the new extended Beta function (8) to introduce a new forms of extended Gauss hypergeometric and confluent hypergeometric functions as follows: \[(Re(p)\geq0;\ \ \vert z\vert< 1;\ \ Re(c)>Re(b)>0;\ \ Re(\alpha)>0;\ \ Re(\beta)>0;\ \ Re(\gamma)>0), \] and \[(Re(p)\geq0;\ \ Re(c)>Re(b)>0;\ \ Re(\alpha)>0;\ \ Re(\beta)>0;\ \ Re(\gamma)>0), \] respectively. Also, we observe that and Some properties of the above functions are given in the form of the following theorems:

Theorem 1. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), the following functional relations hold;

Proof. Using the following known relation [13]

in (9), we get \begin{eqnarray*}F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=&\sum_{n=0}^\infty(a)_n\left[\frac{B_p^{(\alpha,\beta,\gamma)}(b+n+1,c-b)+B_p^{(\alpha,\beta,\gamma)} (b+n,c-b+1)}{B(b,c-b)}\right]\frac{z^n}{n!},\\ &=&\sum_{n=0}^\infty(a)_n\frac{B_p^{(\alpha,\beta,\gamma)}(b+n+1,c-b)}{B(b,c-b)}\frac{z^n}{n!}+\sum_{n=0}^\infty(a)_n\frac{B_p^{(\alpha,\beta,\gamma)} (b+n,c-b+1)}{B(b,c-b)}\frac{z^n}{n!},\\ &=&\frac{B(b+1,c-b)}{B(b,c-b)}\sum_{n=0}^\infty(a)_n\frac{B_p^{(\alpha,\beta,\gamma)}(b+n+1,c-b)}{B(b+1,c-b)}\frac{z^n}{n!}\\ &&+\frac{B(b,c-b+1)}{B(b,c-b)}\sum_{n=0}^\infty(a)_n\frac{B_p^{(\alpha,\beta,\gamma)}(b+n,c-b+1)}{B(b,c-b+1)}\frac{z^n}{n!},\end{eqnarray*} which on using the relation [9] \( B(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}, \) and (9) yields the desired result.

Similarly, using (21) in (10) and following the same procedure leading to result (19), we get the desired result (20) and thus the proof of Theorem 1 is completed.

Using [13]

\begin{align*} (1+\alpha\,\gamma-\beta)\,B_p^{(\alpha,\beta,\gamma)}(x,y)&=\alpha\,\gamma\,B_p^{(\alpha,\beta,\gamma+1)}(x,y)-B_p^{(\alpha,\beta-1,\gamma)}(x,y),\\ p\,B_p^{(\alpha,\beta,\gamma)}(x-1,y-1)&=B_p^{(\alpha,\beta-\alpha,\gamma-1)}(x,y)-B_p^{(\alpha,\beta-\alpha,\gamma)}(x,y), \end{align*} together in (9) and (10) respectively and proceeding on the same lines of proof of Theorem 1, we get the following functional relations:

Theorem 2. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), the following functional relations hold; \begin{align*} (1+\alpha\,\gamma-\beta)\,F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\alpha\,\gamma\,F_p^{(\alpha,\beta,\gamma+1)}(a,b;c;z)-F_p^{(\alpha,\beta-1,\gamma)}(a,b;c;z),\\ (1+\alpha\,\gamma-\beta)\,\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)&=\alpha\,\gamma\,\Phi_p^{(\alpha,\beta,\gamma+1)}(b;c;z)-\Phi_p^{(\alpha,\beta-1,\gamma)}(b;c;z), \end{align*} \[ (\alpha,\beta,\gamma\in\mathbb{C};\ \ Re(\alpha)>0;\ \ Re(\gamma)>0;\ \ Re(\beta)>1), \] \begin{align*} p\,c\,(c+1)\,F_p^{(\alpha,\beta,\gamma)}(a,b;c;z) &=b\,(c-b)\left\{F_p^{(\alpha,\beta-\alpha,\gamma-1)}(a,b+1;c+2;z)-F_p^{(\alpha,\beta-\alpha,\gamma)}(a,b+1;c+2;z)\right\},\\ p\,c\,(c+1)\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z) &=b\,(c-b)\left\{\Phi_p^{(\alpha,\beta-\alpha,\gamma-1)}(b+1;c+2;z)-\Phi_p^{(\alpha,\beta-\alpha,\gamma)}(b+1;c+2;z)\right\}, \end{align*} \[ (Re(p)\geq0;\ \ Re(c)>Re(b)>0;\ \ Re(\beta)>Re(\alpha)>0; \ \ Re(\gamma)>1). \] Using [13] \begin{align*}B_p^{(\alpha,\beta,\gamma)}(x,y)&=\sum_{n=0}^\infty B_p^{(\alpha,\beta,\gamma)}(x+n,y+1),\\ B_p^{(\alpha,\beta,\gamma)}(x,1-y)&=\sum_{n=0}^\infty\frac{(y)_n}{n!} B_p^{(\alpha,\beta,\gamma)}(x+n,1),\\ B_p^{(\alpha,\beta,\gamma)}(x,y)&=\sum_{n=0}^k\left(\begin{array}{c}k\\n\end{array}\right) B_p^{(\alpha,\beta,\gamma)}(x+n,y+k-n),~~~~~(k\in\mathbb{N}),\end{align*} together in (9) and (10) respectively and proceeding on the same lines of proof of Theorem 1, we get the following summation relations:

Theorem 3. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), the following summation relations hold; \begin{align*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=(c-b)\sum_{k=0}^\infty\frac{(b)_k}{(c)_{k+1}}F_p^{(\alpha,\beta,\gamma)}(a,b+k;c+k+1;z),\\ \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)&=(c-b)\sum_{k=0}^\infty\frac{(b)_k}{(c)_{k+1}}\Phi_p^{(\alpha,\beta,\gamma)}(b+k;c+k+1;z),\\ F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\sum_{k=0}^\infty\frac{(b-c+1)_kB(b+k,1)}{k!\,B(b,c-b)}F_p^{(\alpha,\beta,\gamma)}(a,b+k;b+k+1;z),\\ \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)&=\sum_{k=0}^\infty\frac{(b-c+1)_kB(b+k,1)}{k!\,B(b,c-b)}\Phi_p^{(\alpha,\beta,\gamma)}(b+k;b+k+1;z),\\ F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\sum_{n=0}^k\left(\begin{array}{c}k\\n\end{array}\right)\frac{B(b+n,c-b-n+k)}{B(b,c-b)}F_p^{(\alpha,\beta,\gamma)}(a,b+n;c+k;z),\\ \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)&=\sum_{n=0}^k\left(\begin{array}{c}k\\n\end{array}\right)\frac{B(b+n,c-b-n+k)}{B(b,c-b)}\Phi_p^{(\alpha,\beta,\gamma)}(b+n;c+k;z). \end{align*} Using the relations [9] \(B(b,c-b)=\frac{(c)_k}{(b)_k}B(b+k,c-b),\) \((a)_{n+k}=(a)_n(a+n)_k,\) and \(\frac{d^k}{dz^k}z^n=\frac{n!}{(n-k)!}z^{n-k},\) we get the following result

Theorem 4. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), the following differentiation formulas hold \begin{align*} \frac{d^k}{dz^k}\left\{F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\right\}=\frac{(a)_k(b)_k}{(c)_k}F_p^{(\alpha,\beta,\gamma)}(a+k,b+k;c+k;z), \end{align*} \begin{align*} \frac{d^k}{dz^k}\left\{\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\right\}=\frac{(b)_k}{(c)_k}\Phi_p^{(\alpha,\beta,\gamma)}(b+k;c+k;z). \end{align*}

3. Integral representations

In this section, some integral representations for \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\) are given in the form of the following theorems:

Theorem 5. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\), the following integral representations hold;

Proof. Using

in (9) respectively, we get the desired results (22)-(29).

Patting \(u=\tanh^2t\) in assertion (22) of Theorem 5, we get the following result:

Corollary 1. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\), the following integral representation holds; \begin{align*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\frac{2}{B(b,c-b)}\int_0^\infty \frac{(\sinh t)^{2b-1}(\cosh t)^{2a-2c+1}}{(\cosh^2t-z\sinh^2t)^a} E_{\alpha,\beta}^\gamma\left(-p\cosh^2t\coth^2t\right)\,dt. \end{align*}

Theorem 6. For \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), the following integral representations hold;

Proof. Using (30)-(37) in (10), respectively, we get the desired results (38)-(45).

Patting \(u=\tanh^2t\) in assertion (38) of Theorem 6, we get the following result:

Corollary 2. For \(Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), the following integral representation holds; \begin{equation*} \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)=\frac{2e^z}{B(b,c-b)}\int_0^\infty(\sinh u)^{2b-1}(\cosh u)^{1-2c}\tanh^2u E_{\alpha,\beta}^\gamma\left(-p\cosh^2u\coth^2u\right)\,du. \end{equation*}

Theorem 7. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have

and

Proof. Since

\begin{align} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\sum_{n=0}^\infty(a)_n\int_0^1\frac{B_{pu}^{(\alpha,\alpha,\gamma)}(b+n,c-b)\left(1-u^\frac{1}{\alpha}\right) ^{\beta-\alpha-1}\,du}{\alpha~\Gamma(\beta-\alpha)B(b,c-b)}\frac{z^n}{n!}. \notag \end{align} Rearranging the integration and summation, we get \begin{equation*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)=\frac{1}{\alpha~\Gamma(\beta-\alpha)}\int_0^1\left[\sum_{n=0}^\infty(a)_n\frac{B_{pu}^{(\alpha,\alpha,\gamma)}(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!}\right]\left(1-u^\frac{1}{\alpha}\right)^{\beta-\alpha-1}\,du, \end{equation*} which on using (9), gives the desired result (46).

Similarly, using (48) in (10) and following the same procedure leading to result (46), we obtain (47) and thus the proof of Theorem 7 is completed.

Putting \(\mu=u^{\frac{1}{\alpha}}\) in assertions (46) and (47) of Theorem 7, we get the following results:

Corollary 3. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have \begin{equation*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)=\frac{1}{\Gamma(\beta-\alpha)}\int_0^1F_{p\mu^\alpha}^{(\alpha,\alpha,\gamma)}(a,b;c;z)\,\mu^{\alpha-1}(1-\mu)^{\beta-\alpha-1}\,d\mu,\end{equation*} and \begin{equation*} \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)=\frac{1}{\Gamma(\beta-\alpha)}\int_0^1\Phi_{p\mu^\alpha}^{(\alpha,\alpha,\gamma)}(b;c;z)\,\mu^{\alpha-1}(1-\mu)^{\beta-\alpha-1}\,d\mu. \end{equation*} Now, using the relations [13] \begin{equation*} B_p^{(\alpha,\beta,\gamma)}(x,y)=\frac{1}{\Gamma(\alpha)}\int_0^1B_{p(1-u)^\alpha}^{(\alpha,\beta-\alpha,\gamma)}(x,y)\,u^{\alpha-1}(1-u)^{\beta-\alpha-1}\,du, \end{equation*} \begin{equation*} (\alpha,\beta,\gamma\in\mathbb{C};\ \ Re(\beta)>Re(\alpha)>0,\ \ Re(\gamma)>0), \end{equation*} \begin{equation*} B_p^{(\alpha,\beta,\gamma)}(x,y)=\frac{1}{B(\gamma,l-\gamma)}\int_0^1B_{pu}^{(\alpha,\beta,\,l)}(x,y)\,u^{\gamma-1}(1-u)^{l-\gamma-1}\,du, \end{equation*} \begin{equation*} (\alpha,\beta,\gamma,l\in\mathbb{C};\ \ Re(l)>Re(\gamma)>0,\ \ Re(\alpha)>0,\ \ Re(\beta)>0), \end{equation*} \begin{equation*} B_p^{(\alpha,\beta+l,\gamma)}(x,y)=\frac{1}{\Gamma(l)}\int_0^1B_{pu^\alpha}^{(\alpha,\beta,\gamma)}(x,y)\,u^{\beta-1}(1-u)^{l-1}\,du, \end{equation*} \begin{equation*} (\alpha,\beta,\gamma,l\in\mathbb{C};\ \ Re(\alpha)>0,\ \ Re(\beta)>0,\ \ Re(\gamma)>0,Re(l)>0) \end{equation*} and following the same procedure leading to the results in the above theorem, we get the following results:

Theorem 8. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have \begin{align*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\frac{1}{\Gamma(\alpha)}\int_0^1F_{p(1-u)^\alpha}^{(\alpha,\beta-\alpha,\gamma)}(a,b;c;z)\,u^{\alpha-1}(1-u)^{\beta-\alpha-1}\,du, \\ \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)&=\frac{1}{\Gamma(\alpha)}\int_0^1\Phi_{p(1-u)^\alpha}^{(\alpha,\beta-\alpha,\gamma)}(b;c;z)\,u^{\alpha-1}(1-u)^{\beta-\alpha-1}\,du, \end{align*} \[(Re(\beta)>Re(\alpha)>0;\ \ Re(\gamma)>0;\ \ Re(c)>Re(b)>0).\] \begin{align*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\frac{1}{B(\gamma,l-\gamma)}\int_0^1F_{pu}^{(\alpha,\beta,\,l)}(a,b;c;z)\,u^{\gamma-1}(1-u)^{l-\gamma-1}\,du, \\ \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)&=\frac{1}{B(\gamma,l-\gamma)}\int_0^1\Phi_{pu}^{(\alpha,\beta,\,l)}(b;c;z)\,u^{\gamma-1}(1-u)^{l-\gamma-1}\,du, \end{align*} \[(Re(l)>Re(\gamma)>0;\ \ Re(\alpha)>0;\ \ Re(\beta)>0;\ \ Re(c)>Re(b)>0).\] \begin{align*} F_p^{(\alpha,\beta+l,\gamma)}(a,b;c;z)&=\frac{1}{\Gamma(l)}\int_0^1F_{pu^\alpha}^{(\alpha,\beta,\gamma)}(a,b;c;z)\,u^{\beta-1}(1-u)^{l-1}\,du, \\ \Phi_p^{(\alpha,\beta+l,\gamma)}(b;c;z)&=\frac{1}{\Gamma(l)}\int_0^1\Phi_{pu^\alpha}^{(\alpha,\beta,\gamma)}(b;c;z)\,u^{\beta-1}(1-u)^{l-1}\,du, \end{align*} \[(Re(\alpha)>0; \ \ Re(\beta)>0;\ \ Re(\gamma)>0;\ \ Re(l)>0;\ \ Re(c)>Re(b)>0).\]

Theorem 9. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have \begin{eqnarray*} F_p^{(k,\beta,\gamma)}(a,b;c;z)& =\frac{1}{\Gamma(\beta)B(b,c-b)}\int_0^1u^{b-1}(1-u)^{c-b-1}(1-zu)^{-a}{}_1F_{k} \left[\begin{array}{ccc}\gamma\hspace{1.0in};& \\ & \frac{-p}{k^ku(1-u)}\\ \frac{\beta}{k},\frac{\beta+1}{k},\ldots,\frac{\beta+k-1}{k};& \\\end{array}\right]\,du, \\ \Phi_p^{(k,\beta,\gamma)}(b;c;z)& = \frac{1}{\Gamma(\beta)B(b,c-b)}\int_0^1u^{b-1}(1-u)^{c-b-1}e^{zu}{}_1F_{k} \left[\begin{array}{ccc}\gamma\hspace{1.0in};& \\ & \frac{-p}{k^ku(1-u)}\\ \frac{\beta}{k},\frac{\beta+1}{k},\ldots,\frac{\beta+k-1}{k};& \\\end{array}\right]\,du. \end{eqnarray*}

Proof. Using the known relation [13]

in (9) (for \(\alpha=k),\) we get \begin{align*} F_p^{(k,\beta,\gamma)}(a,b;c;z)=&\frac{1}{{\Gamma(\beta)B(b,c-b)}}\sum_{n=0}^\infty(a)_n\int_0^1u^{b+n-1}(1-u)^{c-b-1}\notag\\& \times{}_1F_{k}\left[\begin{array}{ccc}\gamma\hspace{1.0in};&\\&\frac{-p}{k^ku(1-u)}\\ \frac{\beta}{k},\frac{\beta+1}{k},\ldots,\frac{\beta+k-1}{k};&\\\end{array}\right]\frac{z^n}{n!}\,du. \end{align*} Rearranging the integration and summation, we get \begin{align*} F_p^{(k,\beta,\gamma)}(a,b;c;z)=&\frac{1}{{\Gamma(\beta)B(b,c-b)}}\int_0^1u^{b-1}(1-u)^{c-b-1}\notag\\& \times{}_1F_{k}\left[\begin{array}{ccc}\gamma\hspace{1.1in};&\\&\frac{-p}{k^ku(1-u)}\\ \frac{\beta}{k},\frac{\beta+1}{k},\ldots,\frac{\beta+k-1}{k};&\\\end{array}\right]\sum_{n=0}^\infty(a)_n\frac{(zu)^n}{n!}\,du, \end{align*} which on using [9] \begin{equation*}\label{3.48} (1-t)^{-\alpha}=\sum_{n=0}^\infty\frac{(\alpha)_n}{n!}t^n,~~~~~~\left(|t|< 1\right), \end{equation*} in the right hand side yields the desired result (49).

Similarly, using (49) in (10) and following the same procedure leading to result (49), we obtain result (49) and thus the proof of Theorem 9 is completed.

4. Some transformation formulas

First, the Mellin transform representation for the new extended Gauss hypergeometric and confluent hypergeometric functions are obtained in the form of the following theorem:

Theorem 10. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have

Proof. Using the definition of Mellin transform, we get \begin{equation*}\label{4.3} \mathcal{M}\left\{F_p^{(\alpha,\beta,\gamma)}(a,b;c;z);s\right\}=\int_0^\infty p^{s-1}F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\,dp. \end{equation*} Now using relation (22) in the right hand side, we get \begin{equation*}\label{4.4} \mathcal{M}\left\{F_p^{(\alpha,\beta,\gamma)}(a,b;c;z);s\right\}=\frac{1}{B(b,c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\left[\int_0^\infty p^{s-1}E_{\alpha,\beta}^\gamma\left(\frac{-p}{t(1-t)}\right)\,dp\right]\,dt. \end{equation*} Now using the one-to-one transformation (except possibly at the boundaries and maps the region onto itself) \(~u=\frac{p}{t(1-t)},\,w=t\) in the above equation and considering that the Jacobian of the transformation is \(J=w(1-w),\) we get \begin{align*} &\mathcal{M}\left\{F_p^{(\alpha,\beta,\gamma)}(a,b;c;z);s\right\}\notag\\ &=\frac{B(b+s,c+s-b)}{B(b,c-b)}\frac{1}{B(b+s,c+s-b)}\int_0^1w^{b+s-1}(1-w)^{c+s-b-1}(1-zw)^{-a}\,dw \times\int_0^\infty u^{s-1}E_{\alpha,\beta}^\gamma(-u)\,du, \end{align*} which on using relations (3) and (7) (for \(p=0\)) in the right hand side yields the desired result (50).

Similarly, following the same procedure leading to result, we obtain result (51) and thus the proof of Theorem 10 is completed.

Theorem 11. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\) and \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have \begin{align*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)&=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma^{(\alpha,\beta,\gamma)}_0(s)B(b+s,c+s-b)}{B(b,c-b)}{}_2F_1(a,b+s;c+2s;z)p^{-s}\,ds,\\ \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)&=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma^{(\alpha,\beta,\gamma)}_0(s)B(b+s,c+s-b)}{B(b,c-b)}{}_1F_1(b+s;c+2s;z)p^{-s}\,ds. \end{align*}

Proof. Taking Mellin inversion of Theorem 10, we get the desired results.

Next, we prove some other transformation formulas in the form of the following theorems:

Theorem 12. For \(F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)\), we have

Proof. By writing \begin{equation*}\label{4.11} [1-z(1-t)]^{-a}=(1-z)^{-a}\left(1+\frac{z}{1-z}t\right)^{-a}, \end{equation*} and replacing \(u\rightarrow 1-t\) in (22), we get \begin{equation*} F_p^{(\alpha,\beta,\gamma)}(a,b;c;z)=\frac{(1-z)^{-a}}{B(b,c-b)}\int_0^1(1-t)^{b-1}t^{c-b-1}\left(1-\frac{z}{z-1}t\right)^{-a} \times E_{\alpha,\beta}^\gamma\left(\frac{-p}{t(1-t)}\right)\,dt, \end{equation*} which on using (22), gives the desired result (52).

Similarly, using the same steps above and following the same procedure leading to result (52), we get the desired result (53).

Now from (52) and (53), we obtain

\begin{equation*} (1-z)^{-a}F_p^{(\alpha,\beta,\gamma)}\left(a,c-b;c;\frac{z}{z-1}\right)=(1-z)^{-b}F_p^{(\alpha,\beta,\gamma)}\left(c-a,b;c;\frac{z}{z-1}\right), \end{equation*} which on putting \(a=c-a\) and \(z=\frac{z}{z-1},\) we get the desired result (54) and thus the proof of Theorem 12 is completed.

Theorem 13. For \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have \begin{equation*}\label{4.13} \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)=\exp(z)\Phi_p^{(\alpha,\beta,\gamma)}(c-b;c;-z).\end{equation*}

Proof. Putting \(u=1-t\) in (38), we obtain \begin{equation*} \Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)=\frac{1}{B(b,c-b)}\int_0^1(1-t)^{b-1}t^{c-b-1}e^{z(1-t)} E_{\alpha,\beta}^\gamma\left(\frac{-p}{t(1-t)}\right)\,dt,\end{equation*} which on using some simplification and using (38), yields the desired result.

Theorem 14. For \(\Phi_p^{(\alpha,\beta,\gamma)}(b;c;z)\), we have the following relation

\[(Re(\alpha)>0;\ \ Re(\beta)>0;\ \ Re(\gamma)>0;\ \ Re(p)>0;\ \ Re(a)>0).\]

Proof. From Theorem 13 for \(z=-z\), we obtain

Multiplying both sides by \(z^{a-1}\) and integrating the resultant equation with respect to \(z\) from \(z=0\) to \(z=\infty,\) we get \begin{align*} \int_0^\infty z^{a-1}\Phi_p^{(\alpha,\beta,\gamma)}(b;c;-z)\,dz&=\int_0^\infty z^{a-1}e^{-z}\Phi_p^{(\alpha,\beta,\gamma)}(c-b;c;z)\,dz,\\ &=\int_0^\infty z^{a-1}e^{-z}\sum_{n=0}^\infty\frac{B_p^{(\alpha,\beta,\gamma)}(c-b+n,b)}{B(b,c-b)}\frac{z^n}{n!}\,dz. \end{align*} Interchanging the order of integration and summation, we get \begin{equation*}\label{4.18} \int_0^\infty z^{a-1}\Phi_p^{(\alpha,\beta,\gamma)}(b;c;-z)\,dz=\sum_{n=0}^\infty\frac{B_p^{(\alpha,\beta,\gamma)}(c-b+n,b)}{B(b,c-b)n!}\int_0^\infty z^{n+a-1}e^{-z}\,dz, \end{equation*} which on using definition of Euler Gamma function [9] and then using (9), yields the desired result.

5. Concluding remarks

In this paper, the authors established new extension forms of the hypergeometric functions with the help of the new definition of extended Beta function given in [13]. Also, various properties of this extended functions are obtained. The authors conclude that if we let \(\alpha=\beta=\gamma=1\) throughout in the paper and use the relations (13) and (17), then some known and new results due to the work of Chaudhry et al., [2] will be obtained. Also, if we let \(\alpha=1\) throughout in the paper and use the relations (11) and (15) then some known results due to work of Özergin et al., [6] will be obtained. Further if we let \(\beta=\gamma=1\) throughout in the paper and use the relations (12) and (16), then some known results due to work of Shadab et al., [11] will be obtained.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Transmission of television signals ever since the inception of television broadcast [1] has been on the Analogue Television (ATV) technology until a couple of years ago that Digital Terrestrial Television (DTTV) technology was proposed in a bid to maximize frequency spectrum and have good quality of television services [2]. With DTTV technology, television signals are transmitted by multiplex transmitters (transmission of many channels in a single frequency range) and are being radiated from the transmitting antenna whereas in ATV technology, the signals are transmitted through a single channel by an analogue transmitter [3]. Reception of DTTV signals is always influenced by a number of factors like; the height of transmitting and receiving antenna's, the output power of the transmitter [4], the terrain between the transmitter and the receiver, the effect of meteorological parameters [5] and foliage, amongst others.

All forms of radiation in free space are governed by the inverse square law. For this law, if the distance from the transmitter is doubled, the power density of the radiated wave is reduced to a quarter of its former value at the new location. Power density is the radiated power per unit area, hence a measure of radiofrequency intensity and received signal strength. In radiation, power density is directly proportional to the transmitted power and is inversely proportional to the square of the distance from the source [6,7,8]. Therefore;

where, \(\mathfrak{p}_{d}\) is the power density at a distance \(r(m)\) from the transmitter and \(\mathfrak{p}_{t}\) is the transmitted power, all the powers measured in watts.

Globally, almost all countries have switched onto DTTV in accordance with the dates of 17 June 2015 (for Ultra High Frequency) and 17 June 2020 (for Very High Frequency) as set by ITU [9] for its member states, including Uganda.

Uganda being in ITU region 1, her DTTV broadcasting service has a primary allocation of 470-790 MHz frequency band and the Digital Switch Over (DSO) was done on June 2015, within 60 Km radius of Kampala Metropolitan [10].

The motivation for this research was because Kampala Metropolitan, Uganda, was the first region to undergo full DSO and as a result, there was need to know the quality of service, the propagation pattern and coverage of DTTV signals with distance from the transmitter at various climatic zones.

Radiofrequency intensity is a measure of the DTTV received signal (received power). The higher the value of RF intensity, the higher is the received power from the DTTV transmitter and the better is the quality of the received TV signal. The purpose of this work is to study the variation of radiofrequency intensity for different distances from the transmitter. Related studies by different researchers from different countries (Nigeria, Korea, India, and Spain) have been carried out as explained by [11,12,13,14]. In these studies, the quality of DTTV signals reception at different distances and locations have been analyzed.

In Uganda, since little is known about radiofrequency intensities from DTTV transmitter and their relation to TV signals reception, this study is important in order to give a clear sense about the DTTV situation. With this study, we will be able to know the signal attenuation in DTTV transmission. In ATV technology, signal attenuation may not lead into total loss of the signal, which is not the case with DTTV technology. In DTTV technology, TV reception levels below \(-116 dBm\), will lead to total loss of signals [13,15,16].

2. Materials and methods

In order to investigate the variation of radiofrequency intensities from the broadcasting transmitter in Kampala, measurements were done in an outdoor setting for a period of two months. The measurements first considered all the DTTV- UHF frequency band of \(470-862 MHz\) and there after a \(470-790 MHz\) frequency band as assigned to Uganda by the ITU [9] and as implemented by UCC [10] was selected.

2.1. Measurement location

The two month measurements were conducted during day hours at seven different locations which were marked on different eight concentric circles, that's, CC1, CC2, CC3, CC4, CC5, CC6, CC7 and CC8. The radius of these concentric circles determined the horizontal distances between the measured points, that's receiver antenna and the transmitter. The radii of CC1, CC2, CC3, CC4, CC5, CC6, CC7 and CC8 were; 1 Km, 2 Km, 3 Km, 4 Km, 5 Km,6 Km,7 Km and 8 Km respectively, from the transmitter. A Google map of one of the measurement location at Latitude: 0.348340, Longitude: 32.599286 is shown in Figure 1.

Each measurement location chosen in this research had at least the minimum achievable Direct Line of-Sight (DLoS) with the DTTV-Transmitter and the minimum distance of the receiving antenna from the ground. The locations are in urban setting hence allowing signal measurements which have encountered less distortion in form of; absorption, diffraction, reflection, refraction, scattering and directional characteristics of both the transmitter and the receiver antennas [17].

2.2. Measurement setup

The Setup measurement at every measurement location was made up of a calibrated Aaronia Spectran HF-6065V4 spectrum analyzer, an Aaronia AG HyperLOG 4025 Antenna, a \(T430s\) Lenovo Laptop, connected to the spectrum analyzer via a USB cable, and the MCS software. The MSC software is specifically designed to run on Aaronia spectrum analyzers and can easily suit a given measurement since it is easy to configure. Points on each concentric circle as shown in Figure 2, where randomly selected and determined the Measurement Locations (MLos).

A full over view of the UHF frequency band of 470-862 MHz at different seven location measurements were taken. An average of two times measurements at every measurement location were taken for every 5 minutes for a period of 10 minutes before taking the actual average measurements of the frequency band variation of radiofrequency intensities. The radiofrequency intensity measurements were measured in form of Reference Signal Received Power (RSRP).

In Kampala Uganda, the DVB-Transmitter broadcasts all the television signals within the frequency range of \(470-790 MHz\), with the \((470-694) MHz\) as the sub \(700 MHz\) frequency band and \((694-790) MHz\) as the \(700 MHz\) frequency band.

The configurations of the parameters for the spectrum analyzer on the MCS software during the measurements are as in Table 1.

Table 1. Parameter configuration for the spectrum analyzer.

\textbf{Parameter}	\textbf{Value}
UHF Frequency range	470 MHz-862 MHz
DVB-T broadcasting band	Sub 700 MHz band
	700 MHz band
Resolution Band Width	100 KHz
Video Band Width	100 KHz
Sweep time	5ms
Detection type	RMS
Sample points	100
Attenuation factor	Auto
Reference level	-10
Unit	dBm

3. Results and discussion

Measurements of all the UHF frequency range for DTTV has been made for the eight concentric circles whose GPS coordinates for every measurement locations on each circle are as seen in Table 2.

Table 2. GPS coordinates for the different measurements location on the different concentric circles.

Measurement Location (MLo)	\(\mathbf{GPS}\)  (\mathbf{Coordinates}\)	\(\mathbf{CC1}\)	\(\mathbf{CC2}\)	\(\mathbf{CC3}\)	\(\mathbf{CC4}\)	\(\mathbf{CC5}\)	\(\mathbf{CC6}\)	\(\mathbf{CC7}\)	\(\mathbf{CC8}\)
1	Latitude	0.3332 06	0.32894 99999	0.3299 1	0.32592 16666	0.3482 4833	0.3294 266	0.3309 2667	0.2689 25
	Longitude	32.587 644	32.5830 11667	32.573 885	32.5654 5466	32.562 78166	32.552 29	32.545 64167	32.566 18
	Altitude	1203	1175	1201	1188	1173	1242	1271	1169
2	Latitude	03272 016	0.31827	0.3384 85	031893 1666	0.3355 267	0.3382 5	0.2948 15	00.2728
	Longitude	32.596 766	32.5903 04999	32.5741	32.57049833	32.561 0267	32.555 02	32.552 28	32.617 5
	Altitude	1212	1172	1187	1165	1194	1213	1192	1159
3	Latitude	03282 15	033480 0000	0.3281	0.32166 63333	0.3234	0.3654 63	0.3608 95	0.3634
	Longitude	32.593 256	32.6014 8	32.556 4	32.615865	32.556 4	32.603058	32.561 648	32.663 4
	Altitude	1215	1178	1206	1163	1206	1198	1195	1178
4	Latitude	0.3369 98	0.34070 16666	0.338033	0.34417333	0.3320 315	0.566 6	0.3672 95	0.3802
	Longitude	32.857303	32.5888 2333	32.6052317	32.61251	32.620 1275	32.61553	32.559543	32.557 1
	Altitude	1222	1220	1182	1196	1176	1226	1194	1221
5	Latitude	0.3378 1	0.33747 6666	0.3263 89	0.34834 0	0.3485 786	0.3123 45	0.3622 68	0.3653
	Longitude	32.672 39	32.5838 5	32.577 222	32.599286	32.619 7121	32.636113	32.62246	32.624 9
	Altitude	1167	1212	1236	1229	1195	1164	1198	1220
6	Latitude	0.3369 98	0.33641	0.3437 7333	0.36100 8333	0.3526 773	0.3608 95	0.3539	0.378564
	Longitude	32.587 303	32.5826 81666	32.585 6083	32599863	32.608 873	32.56 648	32.636 8	32.845 64
	Altitude	1202	1207	1185	1200	1230	1195	1194	32.8451155
7	Latitude	0.33702833	0.32189 83333	0.32368	0.33491 6	0.3626862	0.3021 66	0.3043 56	0.2954
	longitude	32.589 6266	32.6043 3833	3257712	32.5663 01	32.601 4431	32.552 638	32.643 552	32.542 4
	Altitude	1255	1162	1181	1249	1197	1261	1162	1164

In general, as seen in Figure 3, experimental results show that all Measurement Locations (MLo) on the different distances from the transmitter have a good reception of digital television signals.

Theoretically, radiofrequency intensities at a constant distance from the broadcasting transmitter must be constant. However, this is not the case as seen in Figures 3-7 since the graphs obtained show that radiofrequency intensities vary with location for the same distance though measurements were taken at the least possible DLoS with the transmitter at every measurement location. This means that in some locations there are more obstacles yielding to low radiofrequency intensities while in some there a few obstacles leading to high values of radiofrequency intensities. This cause may also be due to multipath effects giving graphs of varying radiofrequency intensities

This same trend was also observed on the different MLos of the; 2 km, 4 km, 6 km and the 8 km distances. From Table 2, every concentric circle has different altitudes for the different Measurement locations indicating that some MLos are in the valley while as others are on the hill; hence differences in RF Intensities for the same CC.

Another observation made here is that there is a general decrease in the radiofrequency intensities with increasing frequency for the 470 MHz -694 MHz frequency band while as for the 694 MHz -790 MHz frequency band, there is a slight general increase in the radiofrequency intensities as the frequency increases, for every measurement location of every frequency value. This also means that more TV stations are located within the sub 700 MHz frequency band than in the 700 MHz frequency band.

When a single frequency value of 542 MHz from the sub 700 MHz frequency band was considered, for each distance with its MLos as seen in Figures 8 below, the RF intensity for every MLo for a particular concentric distance from the transmitter is not constant [18]. This is because different MLos on the same concentric circle had different altitudes, hence giving different values of RF intensities on the same concentric circle. Specifically, MLos on CC2, CC4 and CC6, recorded higher values than those ones on CC1, CC3 and CC5, yet CC1, CC3 and CC5 are closer to the transmitter than CC2, CC3 and CC6, respectively. This same difference was also observed on the 740 MHz frequency selected from the 700 MH frequency band. These observations do not conform to the inverse square law and the mathematical expression as shown in Equation (1).

From Figure 9, it is noted that there is total non-observance of the inverse square law. In (a), it is expected that RF intensities at all MLos on the 1km distance would be greater than the rest of the RF intensities for all other distances, but it was not so. This also applies in (b) for the 2 km RF intensities at all the MLos on the 2 km in comparison to other distances from the transmitter [4]. The same was observed for the 740 MHz frequency selected from the 700 MHz frequency band.

4. Conclusion

In this paper, radiofrequency intensities from the Digital Terrestrial Television (DTTV) Broadcasting Transmitter in Kampala Metropolitan have been measured and analyzed. The measurements generally show that there is a variation of radiofrequency intensities for a constant distance from the DVB-T Transmitter for any point around it. The measurements have shown that the sub 700 MHz frequency band of 470 MHz-694 MHz is more utilized than the 700 MHz frequency band of 694MHz-790 MHz; though both bands are still less utilized as being expected by ITU and UCC. The study has also revealed that in Kampala Metropolitan, there is good reception of DTTV signals at all the Measurement Locations where DTTV RF intensities measurements where carried from. The measurements do not obey the inverse square law which governs all forms of radiation in free space. This is evidenced on comparison of the RF Intensity measurements of CC1 with CC2, CC3 with CC4, CC5 with CC6 and CC7 with CC8. This is due to different Measurement Locations on the same CC having different altitude values. The study has come at the rightful time when there is an increase in the number of DVB-television stations in Kampala Metropolitan, Uganda.

Acknowledgments

The authors are indebted to Dr. Akisophel Kisolo for providing the transport means that were used while collecting the data from the different measurement locations and extend their thanks to Dr. Gertrude Ayugi for providing the Spectrum Analyzer that was used in this study.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

In 1882, Cebyšev [1] gave the following inequality

where \(f,g:\left[ a,b\right] \rightarrow \mathbb{R}\) are absolutely continuous function, whose first derivatives \(f^{\prime }\) and \(g^{\prime }\) are bounded and and \(\left\Vert .\right\Vert _{\infty }\) denotes the norm in \(L_{\infty }% \left[ a,b\right] \) defined as \(\left\Vert f\right\Vert _{\infty }=\underset{% t\in \left[ a,b\right] }{ess\sup }\left\vert f\left( t\right) \right\vert \).

During the past few years, many researchers have given considerable attention to the inequality (1). Various generalizations, extensions and variants have been appeared in the literature [2,3,4,5,6].

Recently, Guezane-Lakoud and Aissaoui [2] gave the analogue of the functional (2) for functions of two variables and established the following Cebyšev type inequalities for functions whose mixed derivatives are bounded as follows;

and where Motivated by the existing results, in this paper we establish some new Cebyšev type inequalities for functions whose mixed derivatives are co-ordinates quasi-convex and co-ordinates \((\alpha ,QC)\) and \((s,QC)\) -convex.

2. Preliminaries

Throughout this paper, we denote by \(\Delta \), the bidimensional interval in \(% [0,\infty )^{2}\), \(\Delta =:[a,b]\times \lbrack c,d]\) with \(a< b\) and \(c< d\), \(% k=:\left( b-a\right) (d-c)\) and \(\frac{\partial ^{2}f}{\partial \lambda \partial w}\) by \(f_{\lambda w}.\)

Definition 1. [7] A function \(f:\Delta \rightarrow \mathbb{R}\) is said to be convex on the co-ordinates on \(\Delta \) if \begin{align*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq &\lambda wf(x,y)+\lambda \left( 1-w\right) f(x,v)+\left( 1-\lambda \right) wf(t,y)+\left( 1-\lambda \right) \left( 1-w\right) f(t,v) \end{align*} holds for all \(\lambda ,w\in \lbrack 0,1]\) and \((x,y),(x,v),(t,y),(t,v)\in \Delta \).

Definition 2. [8] A function \(f:\Delta \rightarrow \mathbb{R}\) is said to be quasi-convex on the co-ordinates on \(\Delta \) if \begin{equation*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq \max \left\{ f(x,y)+f(x,v)+f(t,y)+f(t,v)\right\} \end{equation*} holds for all \(\lambda ,w\in \lbrack 0,1]\) and \((x,y),(x,v),(t,y),(t,v)\in \Delta \).

Definition 3. [9] For some \(\alpha \in \left( 0,1\right] \), a function \(f:\Delta \rightarrow \mathbb{R}\) is said to be \(\left( \alpha ,QC\right) \)-convex on the co-ordinates on \(\Delta \), if \begin{align*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq &\lambda ^{\alpha }\max \left\{ f(x,y)+f(x,v)\right\} +\left( 1-\lambda ^{\alpha }\right) \max \left\{ f(t,y)+f(t,v)\right\} \end{align*} holds for all \(\lambda ,w\in \lbrack 0,1]\) and \((x,y),(x,v),(t,y),(t,v)\in \Delta \).

Definition 4. [10] For some \(s\in \left[ -1,1\right] \), a function \(f:\Delta \rightarrow \left[ 0,\infty \right) \) is said to be \(\left( s,QC\right) \)-convex on co-ordinates on \(\Delta \), if \begin{align*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq &\lambda ^{s}\max \left\{ f(x,y)+f(x,v)\right\} +\left( 1-\lambda \right) ^{s}\max \left\{ f(t,y)+f(t,v)\right\} \end{align*} holds for all \(\lambda \in \left( 0,1\right),\ w\in \lbrack 0,1]\) and \(% (x,y),(x,v),(t,y),(t,v)\in \Delta \).

Lemma 1. [11] Let f : \(\Delta \ \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta \) in \(\mathbb{R}^{2}\). If \(f_{\lambda w}\in L_{1}\left( \Delta \ \right) \) then for any \(\left( x,y\right) \in \Delta \), we have the equality;

3. Main result

Theorem 1. Let \(f,g\) \(:\Delta \ \rightarrow \mathbb{R}\) be partially differentiable functions such that their second derivatives \(f_{\lambda w}\) and \(% g_{\lambda w}\) are integrable on \(\Delta \). If \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \) are co-ordinated quasi-convex on \(\Delta \), then

where \(T(f,g)\) is defined as in (5), \(M=\underset{x,t\in \left[ a,b\right] ,y,v\in \left[ c,d\right] }{\max }\) \(% \left[ \left\vert f_{\lambda w}\left( x,y\right) \right\vert +\left\vert f_{\lambda w}\left( x,v\right) \right\vert +\left\vert f_{\lambda w}\left( t,y\right) \right\vert +\left\vert f_{\lambda w}\left( t,v\right) \right\vert \right] \), and \(N=\underset{x,t\in \left[ a,b\right] ,y,v\in \left[ c,d\right] }{\max }% \left[ \left\vert g_{\lambda w}\left( x,y\right) \right\vert +\left\vert g_{\lambda w}\left( x,v\right) \right\vert +\left\vert g_{\lambda w}\left( t,y\right) \right\vert +\left\vert g_{\lambda w}\left( t,v\right) \right\vert \right] \), and \(k=\left( b-a\right) \left( d-c\right) \).

Proof. From Lemma 1, we have

and Multiplying (8) by (9), and then integrating the resulting equality with respect to \(x\) and \(y\) over \(\Delta \), using modulus and Fubini's Theorem, and multiplying the result by \(\frac{1}{k}\), we get Since \(\left\vert f_{\lambda \alpha }\right\vert \) and \(\left\vert g_{\lambda \alpha }\right\vert \) are co-ordinated quasi-convex, we deduce where we have used the fact that The proof is completed.

Theorem 2. Under the assumptions of Theorem 1, we have

where \(T(f,g)\) is defined as in (5), \(M,\) \(N,\) and \(k\) are as in Theorem 1.

Proof. From Lemma 1, (8) and (9) are valid. Let \(G(x,y)=\frac{1}{2k}g(x,y)\) and \(F(x,y)=\frac{1}{2k}f(x,y)\). Multiplying \(G(x,y)\) by \(F(x,y)\), then integrating the resultant equalities with respect to \(x\) and \(y\) over \(% \Delta \), and by using the modulus, we get

Since \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \) are co-ordinated quasi-convex, (14) implies By a simple computation, we easily obtain Substituting (16) in (15), we get the desired result.

Theorem 3. Let \(f,g\) \(:\Delta \ \rightarrow \mathbb{R}\) be partially differentiable functions, such that their second derivatives \(f_{\lambda w}\) and \(% g_{\lambda w}\) are integrable on \(\Delta \). If \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \) are co-ordinated \(% \alpha \)-quasi-convex on \(\Delta \), for some \(\alpha \in \left( 0,1\right] \) , then

where \(T(f,g)\) is defined as in (5), \(M,N\), and \(k\) are as in Theorem 1.

Proof. Clearly the inequalities (8)-(10) are valid, using the co-ordinated \(% \alpha \)-quasi-convexity of \(\left\vert f_{\lambda w}\right\vert \) and \(% \left\vert g_{\lambda w}\right\vert \), (10) gives

Using (12) in (18), we obtain the desired result.

Theorem 4. Under the assumptions of Theorem 3, we have

where \(T(f,g)\) is defined as in (5) and \(M,N,\) and \(k\) are as in Theorem 3.

Proof. By the same argument given in Theorem 2, we easly obtain the inequality (14), using the \(\alpha \)-quasi-convexity on the co-ordinates of \(% \left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \), we get

Substituting (16) in (20), we get the desired result.

Theorem 5. Let \(f,g\) \(:\Delta \ \rightarrow \mathbb{R}\) be partially differentiable functions such that their second derivatives \(f_{\lambda w}\) and \(% g_{\lambda w}\) are integrable on \(\Delta \), and let \(s\in \left( -1,1\right] \) fixed. If \(\left\vert f_{\lambda \alpha }\right\vert \) and \(\left\vert g_{\lambda \alpha }\right\vert \) are co-ordinated \(s\)-quasi-convex on \(% \Delta \), then

where \(T(f,g)\) is defined as in (5) and \(M,N,\) and \(k\) are as in Theorem 1.

Proof. Clearly inequalities (8)-(10) are satisfied. Using second definition of the co-ordinated \(s\)-quasi-convex of \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \), (10) gives;

Substituting (12) in (22), we get the desired result.

Theorem 6. Under the assumptions of Theorem 5, we have

where \(T(f,g)\) is defined as in (5) and \(M,N\), and \(k\) are as in Theorem 1.

Proof. By the same argument given in Theorem 2, we easily obtain the inequality (14), using the second definition of \(s\)-quasi-convexity on the co-ordinates of \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \), we get

Substituting (16) in (24), we get the desired result.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interests

The authors declare no conflict of interest.

1. Introduction

Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) is the virus strain of COVID-19 which emerged from Wuhan city of China and has spread to Nigeria which is the most populous country in Africa with an estimated population of about 200 million people [1]. The first case of the novel coronavirus was discovered in Nigeria on February 27, 2020 as reported by the Nigeria Centre for Disease Control (NCDC). There are thirty six states in Nigeria and one Federal Capital Territory (FCT). Oyo state is one of the thirty six states with an estimated population of about 7,840,864 as at the year 2016 [2] and its first COVID-19 case was discovered on March 22, 2020. Since the discovery of its first case, there has been increase in the spread of the COVID-19 among the people. This spread has continued despite the proactive measures taken by the policymakers such as creation of isolation centres for the purpose of separating infectious individuals from other susceptible individuals by putting them in quarantine.

Other government interventions to curb the spread of infections include announcement of dusk to dawn curfew ranging between 7 PM and 6 AM local time, banning of interstate travel from and into Oyo state, except vehicles carrying food items and other essential commodities as well as closure of all schools and markets except those selling perishable food items [3,4]. Ban on all social gatherings including religious services, burials, birthday parties and weddings was also imposed on March 23, 2020 [5]. However, as part of strategies to open the economy of the state, the initially imposed curfew was relaxed to range between 8 PM and 5 AM and schools within the state were ordered to open on July 6, 2020 to students at the final year of their studies only [6].

Some researches on COVID-19 epidemic have been conducted which are country or location specific [7,8,9,10,11]. Others are based on the peculiar nature of the novel coronavirus itself, its incubation period, mode of transmission, risk, estimation of its parameters for dynamic modelling, persistence on inanimate surfaces and inactivation with biocidal agents [12,13,14,15]. While considering Nigeria in their study, the research of [16] on Nigeria centred on Lagos, Kano and FCT, they did not investigate infection transmission dynamics of COVID -19 in Oyo state. Also, the study conducted by [17] examined the impact of non-pharmaceutical control measures on the population dynamics of Lagos state, Nigeria using a formulated model. However, their work solely centred on Lagos state in Nigeria being the state having the highest number of detected COVID-19 cases in the country; they also did not consider Oyo state. None of the researches to date in literature has considered the dynamics of the spread of COVID-19 in Oyo state, Nigeria using dual latency compartments approach.

The motivation to embark on this study stems from the fact that Oyo state is in the third position among states in Nigeria with the highest number of COVID-19 cases according to the data obtained from NCDC on September 13, 2020. This could probably be due to its close proximity to Lagos state which is regarded as the epicentre of the COVID-19 in Nigeria. It was also observed that Lagos and Oyo states have been sending large samples for laboratory test for COVID-19. While Lagos has sent 104,532 samples, Oyo state has sent 21,154 samples for laboratory test for COVID-19 as at September 8, 2020; hence they are among the first five states which have sent large samples to laboratories for COVID-19 test as part of their determination to curb the spread of the novel coronavirus. Thus, it will be beneficial to investigate the impact of the non-pharmaceutical interventions on the transmission dynamics of COVID-19 in Oyo state, Nigeria.

In this study, we develop a mathematical model of dual latency compartments to examine effects of non-pharmaceutical control measures on the spread of COVID-19 in Oyo state, Nigeria. The choice of non-pharmaceutical control measures such as the use of face masks, social distancing and the use of mass media is imperative since there has not been any approved vaccine against COVID-19 in Nigeria.

The remaining part of this study is organised as follows: Section 2 centres on methodology, Section 3 deals with qualitative analysis of the developed model, while Section 4 presents results and discussion and finally conclusion comes up in Section 5.

2. Methodology

2.1. Epidemic data

The Nigeria Centre for Disease Control has the responsibility of reporting epidemic data of infectious diseases to the nation. The first case of the novel coronavirus patient was recorded for Oyo state on March 22, 2020, hence NCDC alerted the state of impending COVID-19 outbreak in the state. The Oyo state government then swung into action by implementing various non-pharmaceutical control strategies such as setting up of isolation or quarantine centres, mass media sensitisation and so on. With collaboration between NCDC and Oyo state government, laboratories were set up for testing and reporting of cases of COVID-19. Figure 1 shows the cumulative number of daily epidemic data reported by NCDC from March 22 to September 13, 2020.

2.2. Model formulation

The model formulation is based on community transmission of the virus among human population. An assumption of dual latency compartments is made based on the work of who reported the possibility of transmission of novel coronavirus during incubation period and in pre-symptomatic cases. The human community is divided into six compartments: Susceptible community \(S_c (t)\), Exposed not yet infectious community \(E_1 (t)\) (First latency stage), Exposed infectious community \(E_2 (t)\) (Second latency stage), Infectious (not detected, not quarantined symptomatic and asymptomatic) community \(I_1 (t)\), Infectious (detected, quarantined symptomatic and asymptomatic) community \(I_2 (t)\) and Recovered \(R(t)\). The total human population is denoted by \(N(t)\) so that \(N(t) = S_c (t) + E_1 (t) + E_2 (t) + I_1 (t) + I_2 (t) + R(t)\). The flow diagram of the model is given in Figure 2.

Since this is an epidemic model, the system governing the evolution of the homogeneously distributed community of people for compartments described in Figure 2 is based on short time scale of few months [17]. Thus, an assumption that births and natural deaths do not play a significant role within the specified period is made hence births and natural deaths are not included except deaths from the disease by removal from infectious compartments which is captured in compartment \(R(t)\) [18,19]. The model is governed by the following nonlinear ordinary differential equations;

where \(m_{I{}_1} (t)\), \(m_{I{}_2} (t)\), \(m_{E{}_2} (t)\) are non-pharmaceutical measures taken to reduce the spread of the coronavirus and are defined by with \(\sigma _d ,\sigma _f ,\sigma _m \) representing social distancing, use of face masks and mass media sensitisation measures respectively such that \(0 \leqslant \sigma _d \leqslant 1,0 \leqslant \sigma _f \leqslant 1,0 \leqslant \sigma _m \leqslant 1\). Substitution of Equation (2) into Equation (1) gives; Detailed definitions of the state vectors \(S(t),\ E_1 (t),\ E_2 (t),\ I_1 (t),\ I_2 (t),\ R(t)\) of Equations (3) together with associated parameters are shown in Tables 1 and 2.

Table 1. Parameters for numerical simulation.

State variable	Description
S	Susceptible community
\(E_1 \)	Exposed not yet infectious community (First latency stage)
\(E_2 \)	Exposed and infectious community (not detected, second latency stage)
\(I_1 \)	Infectious (not detected, not quarantined symptomatic and asymptomatic) community
\(I_2 \)	Infectious (detected, quarantined symptomatic and asymptomatic) community
R	Recovered

\begin{table}[h] \centering \caption{{Parameters used in (\ref{eq3}) and their descriptions}} \label{t2}

Table 2. Parameters used in (3) and their descriptions.

State variable	Description
\(\beta _{E_2 } ,\beta _{I_1 } ,\beta _{I2} \)	Transmission rate among exposed infectious, infectious (undetected, not quarantined)
	and infectious (detected, quarantined) communities.
\(\sigma _f ,\sigma _d ,\sigma _m \)	Proportion of people that use face masks, practice social distancing and
	transmission rate of awareness through mass media influence
\(f\)	Fraction of exposed, infectious community  that becomes infectious
	(detected, quarantined) community \(I_2 \).
\(1 - f\)	Fraction of exposed, infectious community  that becomes infectious
	(undetected, not quarantined) community \(I_1 \).
\(\delta \)	Rate of progression of exposed, infectious community  to
	fully infectious communities \(I_1 ,I_2 \)
\(\gamma _1 ,\gamma _2 \)	Recovery rates from infectious (undetected, not quarantined) community \(I_1 \) and
	infectious (detected, quarantined) community \(I_2 \)
k	Rate of progression from exposed non infectious
	community to exposed infectious community.

3. Qualitative analysis of the model (3)

In this section, qualitative analysis of (3) is presented.

3.1. Disease free equilibrium and reproduction number

The disease free equilibrium (DFE) of (3) is given by \[ \eta _o = (S^* ,E_1^* ,E_2^* ,I_1^* ,I_2^* ,R^* ) = (N(0),0,0,0,0,0), \] which is obtained when the right hand side of (3) is set equal to zero. The infectious compartments of (3) are \(E_1 ,E_2 ,I_1 \) and \(I_2 \), hence using the next generation matrix approach [20], the matrix \(F\) of new infections and transition matrix \(V\) representing transfer or number of ways individuals can move between compartments at DFE with N= S=S(0)=\(S_o \) are \[ V = \left[ {\begin{array}{*{20}c} k & 0 & 0 & 0 \\ { - k} & {f\delta + (1 - f)\delta } & 0 & 0 \\ 0 & { - (1 - f)\delta } & {\gamma _1 } & 0 \\ 0 & { - f\delta } & 0 & {\gamma _2 } \\ \end{array} } \right],\] and \[V^{ - 1} = \left[ {\begin{array}{*{20}c} {\frac{1} {k}} & 0 & 0 & 0 \\ {\frac{1} {\delta }} & {\frac{1} {\delta }} & 0 & 0 \\ { - \frac{{ - 1 + f}} {{\gamma _1 }}} & { - \frac{{ - 1 + f}} {{\gamma _1 }}} & {\frac{1} {{\gamma _1 }}} & 0 \\ {\frac{f} {{\gamma _2 }}} & {\frac{f} {{\gamma _2 }}} & 0 & {\frac{1} {{\gamma _2 }}} \\ \end{array} } \right]. \] The control reproduction number \(R_C \) is given by \(R_C = \rho (FV^{ - 1} )\) and using Maple software gives \begin{align*}\label{eq4} R_C =& - \frac{1} {{\delta \gamma _1 \gamma _2 }} f\gamma _1 \beta I_2 \delta \sigma _d \sigma _f \sigma _m - f\gamma _2 \beta I_1 \delta \sigma _d \sigma _f \sigma _m - f\gamma _1 \beta I_2 \delta \sigma _d \sigma _f - f\gamma _1 \beta I_2 \delta \sigma _d \sigma _m - f\gamma _1 \beta I_2 \delta \sigma _f \sigma _m + f\gamma _2 \beta I_1 \delta \sigma _d \sigma _f \notag\\&+ f\gamma _2 \beta I_1 \sigma _d \sigma _m + f\gamma _2 \beta I_1 \delta \sigma _f \sigma _m + \gamma _1 \gamma _2 \beta E_2 \sigma _d \sigma _f \sigma _m + \gamma _2 \beta I_1 \delta \sigma _d \sigma _f \sigma _m + f\gamma _1 \beta I_2 \delta \sigma _d + f\gamma _1 \beta I_2 \delta \sigma _f + f\gamma _1 \beta I_2 \delta \sigma _m\notag\\& - f\gamma _2 \beta I_1 \delta \sigma _d - f\gamma _2 \beta I_1 \delta \sigma _f - f\gamma _2 \beta I_1 \delta \sigma _m - \gamma _1 \gamma _2 \beta E_2 \sigma _d \sigma _f - \gamma _1 \gamma _2 \beta E_2 \sigma _d \sigma _m - \gamma _1 \gamma _2 \beta E_2 \sigma _f \sigma _m - \gamma _2 \beta I_1 \delta \sigma _d \sigma _f \notag\\&- \gamma _2 \beta I_1 \delta \sigma _d \sigma _m - \gamma _2 \beta I_1 \sigma _f \sigma _m - f\gamma _1 \beta I_2 \delta + f\gamma _2 \beta I_1 \delta + \gamma _1 \gamma _2 \beta E_2 \sigma _d + \gamma _1 \gamma _2 \beta E_2 \sigma _f + \gamma _1 \gamma _2 \beta E_2 \sigma _m + \gamma _2 \beta I_1 \delta \sigma _d\notag\\& + \gamma _2 \beta I_1 \delta \sigma _f + \gamma _2 \beta I_1 \delta \sigma _m - \gamma _1 \gamma _2 \beta E_2 - \gamma _2 \beta I_1 \delta. \end{align*}

3.2. Parameter estimation and data fitting

While the parameters that represent the characteristics of the virus itself are chosen from literature, those that represent the circumstantial effects are estimated from data as well as from literature with moderate assumptions. Several non-pharmaceutical intervention strategies such as use of face masks \(\sigma _f \), social distancing \(\sigma _d \) and use of mass media \(\sigma _m \) are quantified in this study through parameter estimation process using data obtained from NCDC. The incubation period of the SARS-CoV-2 range between 3 and 14 days, we chose average incubation period of 5.2 days [21] hence \(f\) is chosen as \(\frac{1}{{5.2}}\) [22], the period for which individuals at first latency compartment \(E_1 \) begin being infectious and proceed to the second latency compartment \(E_2 \) is assumed to be average of 2 days hence \(k = \frac{1}{2}\).

The recovery period of infected individuals ranges between 3 and 30 days, in this study we made an assumption of average recovery period of 15 days for detected, quarantined symptomatic and asymptomatic community so that \(\gamma _2 = \frac{1}{{15}}\) and average recovery period of 20 days for undetected, not quarantined symptomatic and asymptomatic community so that \(\gamma _1 = \frac{1}{{20}}\). The former is based on the assumption that the detected, quarantined community receive special care such as being treated with certain antibiotics and some multivitamins which have potency to boost immune system of infected individuals towards faster recovery than the latter case, hence \(\gamma _1 = 0.05\), \(\gamma _2 = 0.067\).

The parameter for the use of face masks \(\sigma _f \) and variations of its values for this study is centred on and around the values obtained for it in [23] The values associated with the use of face masks \(\sigma _f \) is assumed very low due to large observable population of people not adhering strictly to its use and also due to its improper use or low quality stuffs available among the population. Other non-pharmaceutical control strategies such as social distancing \(\sigma _d \), use of mass media \(\sigma _m \) and disease transmission rates \(\beta _{I1} ,\beta _{I2} \), \(\beta _{E2} \) and \(\delta \) as well as some initial conditions of the model (3) are fitted to the cumulative daily COVID-19 cases from NCDC data using nonlinear least squares technique with the aid of Maple software. Since the total population of Oyo state is estimated as N(t)=7,840,864 and the first COVID-19 case was announced on March 22, 2020, the initial conditions are set as S(0) = 7,838,864, \(R(0) = 0, I_2 (0) = 1\) with \(E_1 (0),E_2 (0),I_1 (0)\) estimated from cumulative daily COVID-19 data obtained from NCDC. The values of the parameters used for this study are shown in Table 3. It should be emphasized that the reported number of COVID-19 cases in Oyo state came from very small number of laboratory tests carried out, thus there is likelihood of NCDC reporting below the actual COVID-19 cases within the population. Also, the choice of high initial parameter value in nonlinear least squares method for mass media is based on the assumption of aggressive broadcast and awareness campaign of the Oyo state government with consequent inducement of the fear of the disease [24].

Table 3. Values of parameters used in Equation (3).

Parameter	Default values	Ref
\(\beta _{I_1 } \)	0.15	Data Fitted
\(\beta _{I2} \)	0.4	Data Fitted
\(\beta _{E2} \)	0.3	Data Fitted
\(\sigma _f  = \sigma _d \)	0.1	Eikenberry \emph{et al.,} [23]  / Data Fitted
\(\sigma _m \)	0.6	Data Fitted
\( f \)	0.1923	Rothan and Byrareddy [21]
\(\delta \)	0.35	Data Fitted
\(\gamma _1 \)	0.05	Assumed
\(\gamma _2 \)	0.067	Chen et al., [22]
\( k \)	0.5	Data Fitted
\( E1(0) \)	40	Data Fitted
\( E2(0) \)	70	Data Fitted
\( I1(0) \)	50	Data Fitted
\( I2(0) \)	1	Data Fitted

4. Results and discussion

In this section, we carry out numerical simulations of the model (3) whose equations are solved numerically using Maple software. The target of the numerical simulations is towards investigating the effects of non-pharmaceutical control strategies such as the use of face masks, social distancing and mass media effects on the transmission of COVID-19 epidemic in Oyo state, Nigeria. The results from Figure 3 compare the actual data of cumulative daily COVID-19 infectious individuals (red) with simulations of infectious, undetected, not quarantined individuals \(I_1 (t)\) (green) and infectious, detected, quarantined individuals \(I_2 (t)\) (blue) in Oyo state, Nigeria.

The graphical display of Figure 3 indicates that the model (3) is well fitted to the actual data and that the peak of the COVID-19 epidemic will be attained in about 350 days for parameters \(\sigma _f = 0.01,\ \sigma _d = 0.1,\ \sigma _m = 0.6\). This means that when level of compliance of Oyo state residents with the use of face masks and social distancing is very low compared with high usage of mass media, there is a tendency to attain the peak of the epidemic in about 350 days after about 491,891 people and 89,388 people have become infectious both from the group of infectious, undetected, not quarantined individuals \(I_1 (t)\) (green) and the group of infectious, detected, quarantined individuals \(I_2 (t)\)(blue) respectively in Oyo state, Nigeria.

Figure 4 indicates that with \(\sigma _f = 0.05,\sigma _d = 0.1,\sigma _m = 0.6\), there is a shift of the peak of COVID-19 infections towards the right between 350 days and 400 days with about 418,638 and 75,807 infectious individuals in compartments \(I_1 (t)\) and \(I_2 (t)\) at peak of the epidemic. This means that with a slight increase from 0.01 to 0.05 representing about \(5% \) increase in the use of face masks with low social distancing but high mass media effect, there could be reduction in the population of infectious individuals to 418,638 and 75,807 against 491,891 and 89,388 people observed at the peak in Figure 3 for both groups of infectious, quarantined people and those who are infectious but not quarantined. It should be noted that in the nonlinear least squares method, the value of \(\sigma _f \) is fitted with low initial value of parameter for face masks usage due to observable low compliance of people towards its use and due to low enforcement of its usage by law enforcement agencies. Evidence of the contributions of face masks usage is further displayed in Figure 5 with the value of \(\sigma _f \) set to 0.1 indicating a further \(5% \) increase from the value used in Figure 4. Some further shift in the peak of the infection rate of the COVID-19 cases is observed but with reduction in the number of individuals that might have acquired fully infectious status. This means that with \(\sigma _f = 0.1\), about 328,440 and 59,217 individuals from fully infectious compartments \(I_1 ,I_2 \) would have contracted the disease and become infectious at about 347 days starting from the date of its first incidence which is March 22, 2020.

Mass media effect on the population of people in Oyo state is also investigated as shown in Figures 6,7 and 8. Results from Figure 6 show possibility of attaining the peak of the epidemic between 300 days and 350 days after about 533,530 and 97,147 have contracted the coronavirus for both groups of infectious undetected, not quarantined individuals and infectious detected, quarantined individuals for parameter values \(\sigma _m = 0.55,\ \sigma _f = 0.1,\ \sigma _d = 0.1\). However, a slight increase to about \( 60% \) in the mass media effects as shown in Figure 7 suggests a decrease to about 328,440 and 59,217 at the peak of the COVID-19 disease incidence when compared with Figure 6.

Further increase in the value of the parameter for mass media to about \( 62% \) of the population as shown in Figure 8 also indicate further shift in the peak to the right with possibility of population of infected individuals dropping to about 250,146 and 44,940 individuals respectively for infectious undetected, not quarantined individuals and infectious detected, quarantined individuals respectively.

Though not indicated here, investigation on social distancing for values of \(\sigma _d = 0.01,\ \sigma _d = 0.05,\ \sigma _d = 0.1\) also showed a shift in the peak of the COVID-19 disease incidence to the right similar to what is shown in Figures 3,4 and 5. It should be noted that the values of parameters used for social distancing are based on similar assumptions for the use of face masks as the model (3) is fitted to data obtained from NCDC. The low values of parameters for the simulation of the social distancing are based on the assumption that quite a large observable number of people especially in the market places and some other areas of social gatherings do not observe the prescribed social distancing. Thus, while fitting the model of this study to data, initial value representing social distancing for fitting the model was set low in the nonlinear least squares method.

Although Figures 9 and 10 only show effects of social distancing for various values \(\sigma _d = 0.01,\ \sigma _d = 0.05,\ \sigma _d = 0.1\), similar effects of shift in peaks of incidence rates of COVID-19 are observed in the variations of the values of social distancing for the first and second latency compartments of exposed individuals. This means that enforcement of these non-pharmaceutical measures at higher values could significantly reduce the spread of COVID-19 epidemic for the exposed individuals in the latency compartments in Oyo state, Nigeria.

Generally, control reproduction number \(R_C \) below the value of 1 is required for the elimination of the pandemic but using values of Table 3 gives \(R_C = 1.4\) . The contour plots shown in Figures 11, 12 and 13 also indicate that the control reproduction number is greater than 1 at the baseline values \(\sigma _f = 0.1,\sigma _d = 0.1,\sigma _m = 0.6\) of face masks, social distancing and mass media. Figure 11 shows that about \( 50% \) in the use of face masks and about \( 45% \) social distancing with a baseline value of 0.6 of mass media could guarantee control reproduction number lower than 1. Figure 12 indicates that at baseline value 0.1 of social distancing as in Table 3, about \( 45% \) use of face masks and about \( 80% \) use of mass media, there can also be control reproduction number less than 1. Furthermore, for about \( 55% \) social distancing and about \( 80% \) of mass media, there is possibility of control reproduction number being below 1 for baseline value 0.1 of the use of face masks as shown in Figure 13.

Results in Figure 14 shows \(SE_1 E_2 I_1 I_2 R\) model which indicate that the population of susceptible individuals \(S(t)\) reduces and the population of the recovered \(R(t)\) increases before attaining equilibrium state.

5. Conclusion

In this research, we investigate effects of non-pharmaceutical control measures on the spread of COVID-19 epidemic in Oyo state, Nigeria using a mathematical model of dual latency compartments. The three main control measures examined are the use of face masks, social distancing and the use of mass media. After fitting the data obtained from NCDC to the model (3), it was observed that mass media contributed most significantly to the reduction in the spread of the COVID-19 disease in the state. This suggests that possible increase in the use of mass media could probably have a long range impact towards informing large population of people in Oyo state about the danger inherent in the neglect of safety measures such as use of face masks and social distancing. Although policy makers in the state do not want to significantly disrupt economic activities and other activities such as schooling, there is need to intensify efforts in the area relating to the use of mass media coupled with enforcement of the use of face masks and social distancing pending when approved vaccine will be available. Particularly, it is imperative to use mass media relative to programs which specifically explain how to correctly use face masks and the need to continue using it in public places.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.