1. Introduction

It is widely acknowledged that mathematics is a base for all science and engineering concepts and number theory is the base for mathematics. A vital and challenging task of answering questions in number theory depends heavily on finding an integer's unique factor decomposition. When researchers show involvement, new research approaches emerge. Number theory is largely classified as elementary number theory dealing with divisibility and congruence; analytic number theory supported by complex analysis; algebraic number theory due to the evolution of the study on ring of integers; geometric number theory depending on the perspectives of geometry to explain the pattern of distribution and computational number theory that uses computer algorithms to solve certain mind boggling problems. The universe now witness the rapid use of the concepts of number theory in applied fields such as physics, biology, chemistry, communication, acoustics, electronics, cryptography, computing etc, [1].

2. Prime numbers

A positive integer p which is larger than 1 is called a prime number if \(\forall \ \ n\in N, n|p\Rightarrow n=1 \bigvee n=p\). A number which is not a prime is called a composite number. The Fundamental Theorem of Arithmetic (FTA) says every integer larger than 1 can be expressed as a product of primes in a unique manner apart from their order. Prime numbers are used in cryptography to calculate the public and private keys. Its strength heavily depends on the difficulty of decomposing large integers into their factors. For instance, Diffie-Hellman used prime numbers in his key exchange. He made use of a huge prime number p as a common modulus through which two persons A1 and A2 can communicate in secured way with their undisclosed private keys. That is, if A1 and A2 possess their private key respectively a1 and a2 and publicly share upon a key say b which is smaller than p then one can send a message to the other as: \( A_{1}'s\) message\(=m_1\equiv b^{a_1}\) (mod p)and \( B's\) message\(=m_2\equiv b^{a_2}\) (mod p) then \(m_{2}^{a_2}\) (mod p)\(=m_{1}^{a_2}\) (mod p)\(=b^{a_{1}.a_{2}}\) (modp) \(\equiv x\) is the message shared. The security aspect in this communication depends on the difficulty to know the shared message without knowing \(a_1\) and \(a_2\).

3. Mersenne prime numbers

An integer that is positive and denoted \(M_p = 2^p-1\) is called a Mersenne prime if \(M_p\) is a prime integer. It was already established in the literature that if \(2^p-1\) is a prime then \(p\) is a prime integer as well. But the reverse implication is false. For example, \(\hat{p}= 11\) is a prime integer but \(2^{11}-1 = 2047\) is not a prime integer. These Mersenne primes can be easily represented in binary form without requiring additional space as a p-digit integer can hold upto \(2^p-1\) digits in a binary system. Nishimura and Matsumoto found an efficient pseudorandom generating algorithm by making a good use of Mersenne primes [2].

4. Some results

Proposition 1. Suppose that \(p\) is a prime. Then \(p^2 + p + 1 \neq r^3\) for some \(r \in Z^+\).

Proof. Suppose that \(p^2 + p + 1 \neq r^3\) for some \(r \in Z^+\) then \(r^3-1=p(p+1)\) implies \(p(p+1)=(r-1)(r^2+r-1)\). As \(p\) is a prime, by one of the properties of primes, we have either \(p | r -1\) or \(p | r2 + r + 1\). . If \(p | r -1\) then clearly \(p \leq r -1\). So \(p + 1\) \(r \rightarrow r^3\) \((p + 1)^3 > p^2 + p + 1 = r^3\), a contradiction. Next if \(p | r^2 + r + 1\) implies \(p = 3\) and \(r = 1\), else, \(p = 3\mathcal{L} + 1\) for some \(\mathcal{L} \in Z^+\). Hence \(p^2 + p + 1 \equiv 3 (mod 9)\). But it can be easily seen that a perfect cube cannot be written as \(9t -3\) for some \(t \in Z^+\).

Proposition 2. Let \(p_1\), \(p_2\), \(p_3\) be odd primes and \(p_{1}^2 + p_1+ 1 = (p_{2}^2 + p_2+ 1)( p_{3}^2 + p_3+ 1)\) then either \(p_{2}^2 + p_2+ 1\) or \(p_{3}^2 + p_3+ 1\) is not a prime.

Proof. Suppose that both \(p_{2}^2 + p_2+ 1\) and \(p_{3}^2 + p_3+ 1\) are primes. Then one of (a) \(p_1\equiv p_2(mod p_{2}^2 + p_2+ 1 )\) or \(p_1\equiv p_{2}^2(mod p_{2}^2 + p_2+ 1 )\) or (b) \(p_1\equiv p_3(mod p_{3}^2 + p_3+ 1 )\) or \(p_1\equiv p_{3}^2(mod p_{3}^2 + p_3+ 1 )\) hold good. Without loss of generality assume that a) holds good. As \(p_1 > p_2\) and \(p_1\) is a prime it readily follows that \(p_1 \neq p_{2}^2\). Further, the other choice for \(p_1\) is \(p_{2}^2+p_{2}+1+p_{2}=(p_2+1)^2\). As \(p_2\) is an odd prime \((p_2 + 1)\) is even and hence \((p_2 + 1)^2\) is also even, and contradiction. Finally, \(p_1\) also cannot be equal \(p_{2}^2+p_{2}+1+p_{2}^2=2p_{2}^2+p_2+1\) as \(2p_{2}^2+p_2+1\) is even. From this it follows that \(p_1\geq 2(p_{2}^2+p_{2}+1)+p_2 > 2(p_{2}^2+p_{2}+1)\). On similar lines we can also get \(p_2 \geq 2(p_{3}^2+p_{3}+1). So 3(p_{2}^2+p_{2}+1)(p_{3}^2+p_{3}+1)=p_{1}^2+p_1+1 >p_{1}^2 > 4(p_{2}^2+p_{2}+1)(p_{3}^2+p_{3}+1)\). This is again absurd. Hence one of the two \(p_{2}^2+p_{2}+1\), \(p_{3}^2+p_{3}+1\) is not a prime.

Proposition 3. Let \(p_1\), \(p_2\) be two distinct odd primes and \(p_{1}^2+p_{1}+1\) is also a prime. If \(p_{1}^2+p_{1}+1|p_{2}^2+p_{2}+1\) then \(p_{1}^2+p_{1}+1 < p_{2}/2\).

Proof. It is easy to see that \(p_2=k(p_{1}^2+p_{1}+1)-p_{1}\) or \(p_2=k(p_{1}^2+p_{1}+1)-p_{1}^2\). Also \(p_{2}\) cannot be equal to \(p_{1}^2\). Moreover \(P_{2}\) canot be equal to \((p_{1}^2+p_{1}+1)-p_{1}\) and \((p_{1}^2+p_{1}+1)-p_{1}^2\) as both these terms are even. So we have \(p_{2}\geq p_{1}+2(p_{1}^2+p_{1}+1) > p_{1}^2+p_{1}+1\).

Proposition 4. It is not possible to produce three odd primes \(p_1\), \(p_2\), and \(p_3\) with \(p_{1}^2+p_{1}+1\) and \(p_{2}^2+p_{2}+1\) primes such that \(p_{3}^2+p_{3}+1=3(p_{1}^2+p_{1}+1)(p_{2}^2+p_{2}+1)\).

Proof. Suppose we assume the contrary then by invoking Proposition 3 one can deduce \((p_{1}^2+p_{1}+1)< p_{3}/2\) and \((p_{2}^2+p_{2}+1)< p_{3}/2\). So, \(p_{3}^2+p_{3}+1=3(p_{1}^2+p_{1}+1)(p_{2}^2+p_{2}+1) < 3(p_{3}/2)(p_{3}/2)=\frac{3p_{3}^2}{4}< p_{3}^2+p_{3}+1\) yields a contradiction to conclude the proof.

5. Perfect numbers

We call a positive integer, a perfect number if it equals the sum of its proper divisors. Euclid about three centuries before the Jesus Christ showed that \(2^{p-1} (2^p-1)\) is perfect if \(2^p-1\) is a prime number. After this one has to wait for almost 2000 years to get one Euler to establish all perfect numbers that are even must be of Euclid's form. It is too wonderful to record that even till date we do not know how many such even perfect numbers are there and also nothing is known about the existence or otherwise of an odd perfect number. See [3,4]. Zelinsky [5] showed the following;

5.1. Wonderful result

Suppose that \(n\) is a perfect number that is odd. Let the distinct prime divisors of n be \(\omega(n)\) in number and let the total number of prime divisors of \(n\) by \(\Omega(n)\). If \(gcd(3,n)=1\) then \((302/113)(\omega(n))-(285/113)(\Omega(n))\). If \(n\equiv 0(mod3)\) then \((66/25)(\omega(n))-5 \leq \Omega(n) \).

5.2. Amicable number pairs

The sum of divisors is the function \(\sigma=\sum_{dn}\), where d varies over all positive factors of n including 1 and n. For example \(\sigma(5)=1+5=6\), \(sigma(6)=1+2+2+3+6=12\). We call n a perfect number if \(\sigma(n)=2n\). The case when \(sigma(n)< 2n\), we call n deficient and the case when \(\sigma(n)> 2n\), we call n abundant. We can also call n, perfect if \(sigma(n)=n\) in which case we consider only proper divisors of n that excludes n. Note that \(\sigma(mn)=\sigma(m)\sigma(n)\). This is because if \(d | mn\) then by unique decomposition we can write d uniquely as the product of factor of m and a factor of n. So every term in \(\sigma(mn)\) occurs exactly once in \(\sigma(m)\) \(\sigma (n)\). Also every such product is a factor of \(mn\) so they yield the same sum. Observe that if \(d | n\) then \(\sigma(d) < \sigma (n)\).

Proposition 5. If \(d^{*}|n\) then \(\frac{\sigma(d^{*})}{d^{*}}\leq \frac{\sigma(n)}{n}\) and equality holds good only if \(d^{*}=n\).

Proof. If \(d|n\) then \(n=\mathcal{L}d\) for some \(\mathcal{L}\), so \(\mathcal{L}=(n/d)|n\). Hence \(\sigma (n)=\sum _{d|n}d=\sum _{d|n}\frac{n}{d}=n\sum_{d|n}\frac{1}{d}\). If \(d^{*}\) is proper factor of \(n\) then \(\sigma(n)/n=\sum _{d|n}(1/d)>\sum_{d'|d^{*}}\frac{1}{d'}=\frac{\sigma (d^{*})}{d^{*}}\).

Proposition 5 implies that \(\sigma (n)=\sum _{d|n}d=\sum _{d|n}\frac{n}{d}=n\sum_{d|n}\frac{1}{d}=2n\). So \(\sum_{d|n}1/d=2\) is \(n\) is perfect. Euclid established that if \(2^n-1\) is a prime then \(2^{n-1}(2^n-1)\) is a perfect number. This is because, the only prime divisors of \(2^{n-1}(2^n-1)\) are \(2^n-1\) and \(2\). So \(\sigma[2^{n-1}(2^n-1)\sigma(2^{n-1})\sigma(2^n-1)=\left(\frac{2^n-1}{2-1}\right)2^n=2\{2^{n-1}(2^n-1)\}\). Similarly, \(2^n-1\) is a prime then \(n\) itself is a prime. This is because, \(y^n-1=(y-1)(y^{n-1}+y^{n-2}+...+y+1)\). If \(n=\mathcal{L}n\) then \(2^n-1=(2^{\mathcal{L}})^{m}-1=(2^{\mathcal{L}}-1)(1+2^{\mathcal{L}}+...+(2^{\mathcal{L}})^{m-1})\). Hence \((2^{\mathcal{L}}-1)|2^n-1\), which is a prime, a contradiction. Its converse is not true. For example, 11 is a prime but \(2^{11}-1=2047=23 \times 89\) is not a prime. Perfect numbers have some interesting properties. We call a number a triangular number if it can be arranged as a triangular lattice.

Proposition 6. If \(n\) is an even perfect number then it is triangular.

Proof. We deem \(\mathcal{L}\) as a triangular number if \(\mathcal{L}=\sum_{j=1}^{t-1}j=1+2+...+(t-1)t/2\) for some t. However note here that \(n^{*}=2^{n-1}(2^n-1)=\frac{1}{2}(2^n)(2^n-1)\). So an even perfect number is triangular.

Proposition 7. \(n^{*}=2^{n-1}(2^n-1)\) is perfect then \(n=1^3+3^3+...+(2^{(n-1)/2}-1)^3\).

Proof. We know that \(\sum_{j=1}^{n}=n^{2}(n+1)^2/4\). Let \(\mathcal{L}=2^{(n-1)/2}\). Then \(n=1^3+3^3+...+(2\mathcal{L}-1)^3=(1^3+2^3+...+(2\mathcal{L})^3)-(2^3+4^3+...+(2\mathcal{L})^3) =[(2\mathcal{L})^2(2\mathcal{L}+1)^2]/4-2^3[(\mathcal{L})^2(\mathcal{L}^2+1)^2]/4 (\mathcal{L})^2(\mathcal{L}^2-1)\). Put \(\mathcal{L}=2^{(n-1)/2}\) to complete the proof.

Proposition 8. All even perfect numbers have its unit digit as 6 or 8.

Proof. Note that if \(p\) is an odd prime then it is either congruent to 1 (mod 4) or congruent to 3 (mod 4). Let it be first congruent to 1 (mod 4). Then \(2^{n-1}(2^n-1)=2^{4\mathcal{L}}(2^{4\mathcal{L}+1}-1)=16^{\mathcal{L}}(2\times 16^{\mathcal{L}}-1)\equiv 6^{\mathcal{L}}(2\times 6^{\mathcal{L}}-1)\equiv 6(12-1) \equiv 6(mod 10) \). In a similar way, \(2^{n-1}(2^n-1)=4\times 16^{\mathcal{L}}(8\times 16^{\mathcal{L}}-1)\equiv 4 \times 6 (8\times 6-1)\equiv 4(8-1)\equiv 8(mod10)\). In both instances we made use of the fact that \(6^{\mathcal{L}}\equiv 6(mod 10)\).

A pair of integers \((r, s)\) with \(r, s \in Z^+\) and \(r < s\) is called amicable if each of \(r\) and \(s\) is the sum of the proper divisors of the other. Euler was the first mathematician to investigate amicable pairs in a systematic manner. He looked at amicable pairs of the form \((rm, rn)\) where \(r\) is a given known factor and m, n are unknowns with \(gcd(m, n) = 1\). By setting \(r = 2^s\), \(s \in Z^+\), \(m = pq\), \(n = t\) with \(p, q, t\) are distinct primes, one can obtain the forms of Thabit period. By taking \(r = 3^2 \times 7 \times 13, m = pq\) and \(n = t,\) Euler got the first amicable pair whose elements are odd, \((3^2 \times 7 \times 13 \times 5 \times 17 = 69615, 3^2 \times 7 \times 13 \times 107 = 87633)\). A pair of positive numbers \((r_1, r_2)\) is called a breeder if \(r_1 + r_2x = \sigma (r_1) = \sigma (r_2) (x + 1)\) have a positive integer solution x. Borho suggested a rule to find an amicable pair by using breeders. Suppose that \((rd, r)\) be a breeder, with integer solution x. If a pair of distinct prime numbers \(p_1, p_2\) exist with \(gcd(r, p_1p_2) = 1\), fulfilling the bilinear equation \((p_1 -x) (p_2 -x) = (x + 1) (x + d)\) and if a third prime \(p_3\) exists with \(gcd(rd, p_3) = 1\) such that \(p_3 = p_1 + p_2 + d\) then \((rdp_3, rp_1p_2)\) is an amicable pair. Another great mathematician Erdos suggested: for a given \(s \in Z^+\) if \(x_j, j = 1, 2, … \)are solutions of \(\sigma(x) = s\) then any pair \((x_i, x_k), i \neq k\) for which \(x_i + x_k = s\) is an amicable pair.

We call a set of three integers \(m_1, m_2, m_3\) form an amicable triple if \(\sigma(m_1) = m_1 + m_2\), \(\sigma(m_2) = m_2 + m_2\) and \(\sigma(m_3) = m_3 + m_1\), and a set of four integers \(m_1, m_2, m_3\) and \(m_4\) is said to form an amicable quadruple if \(\sigma(m1) = m2, \sigma(m2) = m3, \sigma(m3) = m4\) and \(\sigma(m4) = m1\). Fig. 1 shows a python program to find perfect numbers, amicable pairs, amicable triples and amicable quadruples. Accordingly we record that \(6, 28, 496, 8128\) are perfect numbers, \((220, 284), (1184, 1210), (2620, 2924), (5020, 5564)\) are amicable pairs, \((1980, 2016, 2556), (9180, 9504, 11556)\), are amicable triples and \((1236402232, 1369801928, 1603118392, 1412336648)\) is an amicable quadruple.

It is worth to re-publicize the following open questions. 1) Are there an infinite number of amicable pairs 2) Is there an amicable pair whose elements are of different parity 3) Is there an amicable pair whose elements are relatively prime 4) Are there amicable pairs whose elements have smallest but different prime divisors Some of the amicable pairs are (\(1184, 1210), (2620, 2924), (5020, 5564)\) etc. These numbers have the feature that one represents the other. This stands for love, harmony, friendship etc. There numbers have lot of applications in astrology in casting horoscopes and magic.

6. Semiprimes

A semiprime number is a number that can be expressed as a product of two prime numbers [6]. Look at 35, it is easy to see that 5 and 7 are the only factors other than 1 and itself. Determination of prime factors of a large number with millions of digits is a huge task. The difficulty in factoring forms the basis of security in RSA encryption algorithms. The semiprime number serves as a public key to encrypt a message and its prime factors act as private keys to decrypt a message.

When analyzing the logic gates, one can analyze both AND gate and OR gate together. This is because both generate almost identical information. For any given state of AND or OR there is a 22% probability that the state could be an error. This occurs when at least two I/O's are defined at the same time. If one of the I/O is defined there is 50% probability that at least one more I/O will be deduced. Given a gate with random inputs and no error there is 62% chance that the gate is fully define. It is wonderful to observe that the probability of error do not get altered while changing from the Full-Adder to Array Multiplier Cell. As the I/O count increases, the ability to generate new information decreases. This is crucial to factorize the semiprime numbers, because as the semiprimes grows, the complexity and I/O count of the array multiplier will grow and result in lesser quantity of information generation. With a fully operating reversible array multiplier, one can test a huge quanta of possibilities for the semiprime number and perform data analysis to quantity the information generated. Python code is involved to generate prime numbers for various binary lengths, from 2 bits to 512 bits. The prime numbers are then multiplied and fed into the output of the reversible array multiplier. After deducing all information, it can be saved as a CSV file. Then one can depict it as a graph to show the information as a function of the size of the semiprime number.

7. 7\(\Pi\) and primes

The most precious gem of India, Srinivasa Ramanujan in one of his shocking revelation formulae have said that: \[1/\pi=(2\sqrt{2}9801)\sum_{n=0}^{\infty}\frac{4n!(1103+26390n)}{(n!)^4(306)^{4n}}.\] The first term in the r.h.s of the above equation gives 7 digits of \(\pi\). That is \(\pi=\frac{9801}{2206\sqrt{2}}\approx 3.14159273...\) (7digits). Note that one can add eight correct digits with each additional term. The rate of convergence is also unbelievably fast. Some more of wonderful expressions for \(\pi\) are \(\pi \approx (3/\sqrt{163}ln(640320))=3.14159273...\) (15 digits) \(\pi \approx (3/\sqrt{67}ln(5280))=3.141592653...\) (9 digits). It is amazing to observe that the first six digits \(3.14159\) is a prime. Infact, the first 38 digits of is also a prime. \(3, 31, 314159, 31415926535897932384626433832795028841\) are all primes. Here the reverse of first three primes viz., \(3, 13, 951413\) are also primes. It was checked up to first \(432\) digits of and only the above four numbers are primes. It is till date unknown whether there is another prime for more digits of . It is strikingly shocking that \(314159\) is a very unusual peculiar prime. The complement number of this prime obtained by replacing each digit of this number by the difference of it with \(10\), viz., \(796951\) is also a prime. If this number is split into three two digit numbers viz., 31, 41 and 59 are again prime numbers. The sum of them \(31 + 41 + 59 = 131\) is also a prime and the sum of the cube of them \(313 + 413 + 593 = 304091\) is also a prime. There are lot of coincidences among and primes. May be much more in depth relations exist between them and they are yet to be found.

8. Prime numbers as a physical system

Marshall and Smith [6] explored the prime numbers from different viewpoint. They treated prime numbers as a physical system and represented it as a differential equation \(f'(x)=-f(xf(x\sqrt{x}))/2x\). \(f(x)\) stands for the "density of primes at x". It predicts the known results regarding the distribution of primes. They have said a deeper relation between number theory and feedback & control, a field in Engineering. They assumed that the density of primes said above is a point density and not average density. They considered two intervals. IA is the interval on R given by \([x, x + dx]\). IB is the square of IA: \([x^2, (x + dx)^2]\). Now approximate the quantity \(f((x + dx)^2) -f(x^2)\). Consider only the effect of primes on IA on the density along IB. Every prime in IA changes the density on IB by a factor of \(1 -1/x\). So every prime subtracts \(f(x^2)/x\) from density. There are \(f(x)\) dx primes in IA. The first approximation of the change in density on IB is \(f((x+dx)^2)-f(x^2)\approx \frac{f(x^2)f(x)dx}{x}\). Another way to calculate the change in density on IB is through derivatives. \((x + dx)^2- x^2 = 2x dx + (dx)^2\). Change in density in IB is roughly evaluated at x2 times the length of IB. So \(f((x+dx)^2)-f(x^2)\approx f'(x^2)x2dx\) By comparing these we get \(f'(x^2)=-f(x^2)f(x)/2x^2\) and by replacing \(x^2\) with \(x\) we get \(f'(x)=-f(x)f(\sqrt{x})/2x.\)

9. Conclusion

We have briefly discussed about the prime numbers and its variations and how their properties and distribution reveal several interesting features and how they are exploited in different application areas of real life scenario. A lot of things about primes are yet to be said and it holds a huge opportunity for researchers to probe and bring out the unknown. We hope to revert more on this elsewhere.

Authorcontributions

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

Conflict of Interests

The authors declare no conflict of interest.

1. Introduction

This study considers the general first order stiff initial value problems of ordinary differential equations of the form

The problem of stiffness in most ordinary differential equations (ODEs) has posed a lot of computational difficulties in many practical application modeled by ODEs. A very important special class of differential equations taken up in the initial value problems termed as stiff differential equations result from the phenomenon with widely differing time scales [2,3]. There is no universally acceptable definition of stiffness. Stiffness is a subtle, difficult and important concept in the numerical solution of ordinary differential equations. It depends on the differential equation, the initial condition and the interval under consideration. The initial value problems with stiff ordinary differential equations occur in many field of engineering science, particularly in the studies of electrical circuits, vibrations, chemical reactions and so on. Stiff differential equations are ubiquitous in astrochemical kinetics, many non-industrial areas like weather prediction and biology. A set of differential equations is `stiff' when an excessively small step is needed to obtain correct integration.

Linear multistep methods (LMMs) are very popular for solving first-order initial value problems (IVPS). LMMs are not self-starting hence, need starting values from single-step methods like Euler's method and Runge-Kutta family of methods. The general k-step method or LMM of step number \(k\) is given by Lambert [4] as follows

where the coefficients \(\alpha_{j}\)'s and \(\beta_{j}\) 's are real constants. The LMM in Equation (2) generates discrete schemes which are used to solve first-order ordinary differential equations.

The techniques for the derivation of continuous LMMs for direct solution of initial value problems in ordinary differential equations have been discussed in literature over the years and these include, among others collocation, interpolation, integration, and interpolation polynomials. Basis functions such as, power series, Chebyshev polynomials, trigonometric functions, monomials, the canonical polynomial of the Lanczos Tau method in a perturbed collocation approach have been employed for this purpose [1,5,6,7,8].

Berhan et al. [1] constructed block procedure with implicit sixth order linear multistep method using Legender polynomial for solving stiff initial value problems. In this study, we constructed implicit linear multistep method in block form of uniform step size for the solution of stiff first order ordinary differential equation using probabilists Hermite polynomial as a base function. The procedure yields four linear multistep schemes which are combined as simultaneous numerical integrators to form block method. The method is found to be consistent and zero-stable and hence convergent. Briefly, the present method is stable, accurate and effective method for solving stiff first order differential equations.

2. Description of the method

2.1. Derivation of the linear multistep methods

In [9,10], some continuous LMM of the type in Equation (3) were developed using the collocation function of the form: Awoyemi et al. [11] proposed a similar function to Equation (3) as to develop LMM for the solution of third-order IVPs. Adeniyi and Alabi [12] used Chebyshev polynomial function of the form: \[y(x)=\sum_{j=0}^{k}\alpha_{j}T_j\left(\frac{x-x_k}{h}\right),\] where \(T_j(x)\) are Chebyshev functions to develop continuous LMM.

In this paper, we applied the Probabilists Hermite polynomial proposed by Koornwinder et al. [13] which is given as

\[y(x)=\sum_{j=0}^{k}\alpha_{j}H_j(x-x_k)\], where \(H_j\) are probabilists Hermite polynomials generated by the recursive relation \[H_{n+1}(x)=xH_n(x)-H_n\prime(x), \,H_{0}=1.\] The first seven probabilists Hermite polynomials are We wish to approximate the exact solution \(y(x)\) to the IVP in Equation (1) by a polynomial of degree \(n\) of the form Hence \[y\prime(x)=f(x,y)=\sum_{j=1}^{k}a_{j}H\overset{^{\prime}}{_j}(x-x_k),~~~ x_k \leq x \leq x_{k+p}.\]

2.2. Derivation of the method for \(k=1\)

Using Equations (5) and (6), we get Differentiating Equation (7) gives Interpolating Equation (7) at \(x=x_k\) and collocating Equation (8) at \(x=x_k\) and \(x_{k+1}\), we get \begin{equation} \label{new1} \begin{cases} y(x_k)=&a_0-a_2, \nonumber \\ y\prime(x_k)=& a_1=f_k, \text{ and }\nonumber\\ y\prime(x_{k+1})=& a_1+2a_2h=f_{k+1}. \end{cases} \end{equation} The system of Equations (9) can be written in matrix form as \begin{equation*} \left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & h & 0 \\ 0 & h & -2h^{2} \end{array} \right)\left(\begin{array}{c} a_0 \\ a_1 \\ a_2 \end{array} \right)=\left(\begin{array}{ccc} y(x_k) \\ hf_k \\ hf_{k+1} \end{array} \right). \end{equation*} Solving the system of equations, we obtain \begin{eqnarray*} a_0&=&\frac{1}{2h}(f_{k+1}-f_k)+y(x_k),\\ a_1&=& f_k, \text{ and }\\ a_2&=&\frac{1}{2h^{2}}(f_{k+1}-f_k). \end{eqnarray*} Substituting \(a_j,\) for \(j=0,1,2\) in Equation (7) yields the continuous method Interpolating Equation (9) at \(x=x_{k+1}\), we obtain the discrete form

2.3. Derivation of the method for \(k=2\)

Using Equations (5) and (6), we get Differentiating Equation (11) gives Interpolating Equation (11) at \(x=x_k\) and collocating Equation (12) at \(x=x_k,x_{k+1}\), and \(x_{k+2}\), we get The matrix form of system of Equations (13) is \begin{equation*} \left( \begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & h & 0 & -3h \\ 0 & h & 2h^{2} & 3h^{2}-3h\\ 0 & h & 4h^{2} & 12h^{3}-3h \end{array} \right)\left(\begin{array}{c} a_0 \\ a_1 \\ a_2 \\ a_3 \end{array} \right)=\left(\begin{array}{cccc} y(x_k) \\ hf_k \\ hf_{k+1}\\ hf_{k+2} \end{array} \right) \end{equation*} Solving the system of equations, we obtain \begin{eqnarray*} a_0 &=& \frac{1}{12h}(-5h^{2}f_k+8h^{2}f_{k+1}-h^{2}f_{k+2}-12y_{k+1}h+9f_k-12f_{k+1}+3f_{k+2}),\\ a_1&=&\frac{1}{h^{2}}(2h^{2}f_k+f_k-2f_{k+1}+f_{k+2}),\\ a_2&=&\frac{1}{4h}(3f_k-4k_{k+1}+f_{k+2}), \text{ and } \\ a_3 &=& \frac{1}{6h}(f_k-2f_{k+1}+f_{k+2}). \end{eqnarray*} Substituting \(a_j,\) for \(j=0,1,2,3\) in Equation (11) yields Interpolating Equation (14) at \(x=x_{k+2}\), we obtain

2.4. Derivation of the method for \(k=3\)

Using Equations (5) and (6), we get Differentiating Equation (15) gives Interpolating Equation (16) at \(x=x_{k+1}\) and collocating Equation (17) at \(x=x_k,x_{k+1},x_{k+2}\), and \(x_{k+3}\), we get The matrix form of system of Equations (18) is \begin{equation*} \left( \begin{array}{ccccc} 1 & h & h^{2}-1 & h^{3}-3h & h^{4}-6h^{2}+3\\ 0 & h & 0 & -3h & 0 \\ 0 & h & 2h^{2} & 3h^{3}-3h & 4h^{4}-12h^{2}\\ 0 & h & 4h^{2} & 12h^{3}-3h & 32h^{4}-24h^{2}\\ 0 & h & 6h^{2} & 27h^{3}-3h & 108h^{4}-36h^{2} \end{array} \right)\left(\begin{array}{c} a_0 \\ a_1 \\ a_2 \\ a_3\\ a_4 \end{array} \right)=\left(\begin{array}{ccccc} y_{k+1} \\ hf_k \\ hf_{k+1}\\ hf_{k+2}\\ hf_{k+3} \end{array} \right).\label{21} \end{equation*} Solving the system of equations, we have \begin{eqnarray} a_0 &=& \frac{1}{24h^{3}}(9h^{4f_k}+19h^{4}f_{k+1}-5h^{4}f_{k+2}+h^{4}f_{k+3}-24h^{3}y_{k+1}+22h^{2}f_k-36h^{2}f_{k+1}\nonumber\\&&{}+18h^{2}f_{k+2}-4h^{2}f_{k+3}+3f_k-9f_{k+1}+9f_{k+2}-3f_{k+3}) \nonumber\\ a_1 &=& \frac{1}{2h^{2}}(2h^{2}f_k+2f_k-5f^{k+1}+4f_{k+2}-f_{k+3})\nonumber\\ a_2 &=& \frac{1}{12h^{3}}(-11h^{2}f_k-18h^{2}f_{k+1}+9h^{2}f_{k+2}-2h^{2}f_{k+3}+3f_{k}-9f_{k+1}+9f_{k+2}-3f_{k+3})\nonumber\\ a_3 &=& \frac{1}{24h^{2}}(2f_k-5f_{k+1}+4f_{k+2}-f_{k+3})\nonumber\\ a_4&=& \frac{1}{24h^{3}}(-f_k-3f_{k+1}+3f_{k+2}-f_{k+3})\nonumber \end{eqnarray} Substituting \(a_j,\) for \(j=0,1,2,3,4\) in Equation (16) yields Interpolating Equation (19) at \(x=x_{k+3}\), we obtain the discrete form

2.5. Derivation of the method for \(k=4\)

Using Equations (5) and (6), we get Differentiating Equation (21) gives Interpolating Equation (21) at \(x=x_{k+2}\) and collocating Equation (22) at \(x=x_k,x_{k+1},x_{k+2},x_{k+3}\), and \(x_{k+4}\), we get The matrix form of system of Equations (23) is \begin{eqnarray*} \left( \begin{array}{cccccc} 1 & 2h & 4h^{2}-1 & 8h^{3}-6h & 16h^{4}-24h^{2}+3 & 32h^{5}-80h^{3}+30h\\ 0 & h & 0 & -3h & 0 & 15h \\ 0 & h & 2h^{2} & 3h^{3}-3h & 4h^{4}-12h^{2} & 5h^{5}-30h^{3}+15h \\ 0 & h & 4h^{2} & 12h^{3}-3h & 32h^{4}-24h^{2} & 80h^{5}-120h^{3}+15h\\ 0 & h & 6h^{2} & 27h^{3}-3h & 108h^{4}-36h^{2} & 405h^{5}-270h^{3}+15h\\ 0 & h & 8h^{2} & 48h^{3}-3h & 256h^{4}-48h^{2} & 1280h^{5}-480h^{3}+15h \end{array} \right)\left(\begin{array}{c} a_0 \\ a_1 \\ a_2\\ a_3 \\ a_4\\ a_5 \end{array} \right)=\left(\begin{array}{cccccc} y_{k+2} \\ hf_k \\ hf_{k+1}\\ hf_{k+2}\\ hf_{k+3}\\ hf_{k+4} \end{array} \right).\end{eqnarray*} Solving the system of equations, we have \begin{eqnarray*} a_0 &=& \frac{-1}{720h^{3}}(232h^{4}f_k+992h^{4}f_{k+1}+192h^{4}f_{k+2}+32h^{4}f_{k+3}-8h^{4}f_{k+4}-720h^{3}y_{k+2}\\ &&+750h^{2}f_k-1440h^{2}f_{k+1}+1080h^{2}f_{n+2}-480h^{2}f_{k+3}+90h^{2}f_{k+4}+225f_k-810f_{k+1}\\ &&+1080f_{k+2}-630f_{k+3}+135f_{k+4}),\\ a_1&=& \frac{1}{24h^{4}}(24h^{4}f_k+35h^{2}f_k-104h^{2}f_{k+1}114h^{2}f_{k+2}-56hf_{k+3}+11h^{2}f_{k+4}+3f_k- 12f_{k+1}\\&&+18f_{k+2}-12f_{k+3}+3f_{k+4}),\\ a_2&=&-\frac{1}{24h^{3}}(25h^{2}f_k-48h^{2}f_{k+1}+36h^{2}f_{k+2}-16h^{2}f_{k+3}3h^{2}f_{k+4}+15f_k-54f_{k+1}+72_{k+2}\\ &&-42f_{k+3}+9f_{k+4}),\\ a_3&=&\frac{1}{72h^{4}}(35h^{2}f_k-104h^{2}f_{k+1}+114h^{2}f_{k+2}-56h^{2}f_{k+2}+11h^{2}f_{k+4}+6f_k- 28f_{k+1}+36f_{k+2}\\&&-24f_{k+3}+6f_{k+4}),\\ a_4&=&-\frac{1}{48h^{3}}(5f_k-18f_{k+1}+24f_{k+2}-14f_{k+3}+3f_{k+4}), \text{ and }\\ a_5&=&\frac{1}{120h^{4}}(f_k-4f_{k+1}+6f_{k+2}-4f_{k+3}+f_{k+4}). \end{eqnarray*} Substituting \(a_j,\) for \(j=0,1,2,3,4,5\) in Equation (21) yields \( y(x)= \frac{-1}{720h^{3}}(232h^{4}f_k+992h^{4}f_{k+1}+192h^{4}f_{k+2}+32h^{4}f_{k+3}-8h^{4}f_{k+4}- 720h^{3}y_{k+2}+750h^{2}f_k-1440h^{2}f_{k+1}+1080h^{2}f_{n+2}-480h^{2}f_{k+3}+ 90h^{2}f_{k+4}-810f_{k+1}+1080f_{k+2}-630f_{k+3}+135f_{k+4})+\frac{1}{24h^{4}}(24h^{4} f_k+ 35h^{2}f_k-104h^{2}f_{k+1}114h^{2}f_{k+2}-56hf_{k+3}+11h^{2}f_{k+4}+3f_k- 12f_{k+1}+ 18f_{k+2}-12f_{k+3}+3f_{k+4})(x-x_k)-\frac{1}{24h^{3}}(25h^{2}f_k-16h^{2} f_{k+3}+ 3h^{2}f_{k+4}+15f_k-54f_{k+1}-104h^{2}f_{k+1}+114h^{2}f_{k+2}+ 72f_{k+2} -42f_{k+3} +9f_{k+4})[(x-x_k)^{2}-1]+\frac{1}{72h^{4}}(35h^{2}f_k-56h^{2}f_{k+2}-28f_{k+1} +36f_{k+2}-24f_{k+3}+6f_{k+4})+\frac{1}{48h^{3}}(5f_k-18f_{k+1}+24f_{k+2}-14f_{k+3} +3f_{k+4})[(x-x_k)^{4}-6(x-x_k)^{2}+3]+\frac{1}{120h^{4}}(f_k-4f_{k+1}+6f_{k+2} -4f_{k+3}+f_{k+4})-10(x-x_k)^{3}+15(x-x_k)].\) Interpolating Equation (24) at \(x=x_{k+4}\), we obtain

2.6. The proposed block method

The proposed block procedure with implicit linear multistep method is given by

3. Analysis of the method

3.1. Order and error constant

It is convenient at this point to introduce the so called characteristic polynomials \[\rho(z)=\sum_{j=0}^{k}\alpha_{j}z^{j}\] and \[\sigma(z)=\sum_{j=0}^{k}\beta_{j}z^{j}\] for the linear multistep methods given in Equation (2) obtained by using the substitutions \(y_{n+j}=z^{j}\) and \(f_{n+j}=\lambda\) \(z^{j}\) where \(z\) is a variable and \(j=0,1,2,3,\cdots,k\). Moreover, following Henric [14], the approach adopted in Fatunla [15], Lambert [16], and Suli and Mayer [17], they define the local truncation error associated with Equation (26) by the difference operator where \(y(x)\) is the exact solution. Assuming \(y(x)\) is smooth and expanding Equation (27) in Taylor series give us and According to Lambert [16], any linear multistep method of the form Equation (2) is of order p if \(c_0=c_1=c_2=\) \(\ldots\) \(c_p=0\) and \(c_{p+1}\neq0.\) In this case the number \(\frac{c_{p+1}}{\sigma(1)}\) is called the error constant of the method. Thus, the order of Equation (26) is \((2~3~4~5)^{T}\) with error constant \((-0.833333~-0.83333~-0.011~-0.011)^{T}\).

3.2. Zero stability of the method

Definition 1. [18] A block method is zero-stable provided that the root \( z_{j},j=1(1)k \) of the first characteristics polynomial satisfies \(|z_{j}|\leq1\) and for those root with \(|z_{j}|\) the multiplicity must not exceed two.

The characteristic polynomials of Equations (10),(15),(20) and (25) are \(z-1=0, z^{2}-z=0, z^{3}-z^2=0\) and \(z^4-z^3=0\) respectively. Hence, they are all zero stable according to Definition 1.

3.3. Consistency of the method

Definition 2. [16] A linear multistep method is said to be consistent if it has order at least one.

Using Definition 2, the linear method is said to be consistent if it has an order greater than or equal to one. Therefore, the block method (26) is consistent, since the orders of each method is greater than one.

3.4. Convergence of the method

Theorem 3.[19] A necessary and sufficient condition for a linear multistep method to be convergent is that it be consistent and zero-stable.

The proposed method satisfies the two conditions stated in Definition 1 and Definition 2. Hence, according to Theorem 1 the scheme in Equation (26) is convergent.

4. Numerical examples

The mode of implementation of our method is by combining the schemes Equation (26) as a block for solving Equation (1). It is a simultaneous integrator without requiring the starting values. To assess the performance of the proposed block method, we consider two stiff first order initial value problems in ODEs. The maximum absolute errors of the proposed method is compared with that of Runge Kutta order 4 (RK4) and Berhan et al. [1]. All calculations are carried out with the aid of MATLAB software.

Example 1.[18] Consider the first order stiff ordinary differential equation \[y\prime=-1000(y-x^{3})+3x^{2},~~y(0)=0,~~x\in[0,1]\]. The exact solution is \(y(x)=x^{3}\). Maximum Absolute errors of RK4 and the present method is given in Table 1

Table 1. Maximum Absolute errors of RK4 and the present Method for Example 1.

h	RK4	Present method
\(10^{-1}\)	\(2.81614e+60\)	\(1.78054e-04\)
\(10^{-2}\)	\(1.07457e+239\)	\(3.67265e-07\)
\(10^{-3}\)	\(9.98899e-08\)	\(5.00000e-10\)
\(10^{-4}\)	\(6.5319e-12\)	\(5.00033e-12\)
\(10^{-5}\)	\(2.88657e-15\)	\(5.11812e-14\)

Example 2.[20] Consider the first order stiff ordinary differential equation \[y\prime=-2100\Bigl(y-cos(x)\Bigr)-sin(x),~~~y(0)\in[0,1].\] The exact solution is \(y(x)=cos(x).\) Maximum Absolute error of Berhan et al. [1] and the present method in Table 2.

Table 2. Maximum Absolute errors of Berhan \textit{ et al.} \cite{1} and the present Method for Example 2.

h	Berhan \textit{et al.} [8]	Present method
\(10^{-1}\)	\(1.22516e-5\)	\(4.06068e-06\)
\(10^{-2}\)	\(9.67880e-8\)	\(3.78971e-8\)
\(10^{-3}\)	\(6.46040e-11\)	\(3.3317e-11\)
\(10^{-4}\)	\(3.33844e-13\)	\(3.33844e-13\)
\(10^{-5}\)	\(4.10783e-15\)	\(4.10782e-15\)

5. Concluding remarks

This paper presented a block procedure with the linear multistep method based on probabilists' Hermite polynomials for solving first order IVPs in ODEs. A collocation approach along with interpolation at some grid points which produces a family block scheme with maximum order five has been proposed for the numerical solution of stiff problems in ODEs. The method is tested and found to be consistent, zero stable and convergent. We implement the method on two numerical examples, and the numerical evidence shows that the method is accurate and effective for stiff problems.

Acknowledgments

The authors would like to thank the College of Natural Sciences, Jimma University for funding this research work.

Author Contributions

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

Conflict of Interests

The authors declare no conflict of interest.

1. Introduction

Not only for the urgency of detecting SARS-COV-\(2\) epidemic spreading patterns among humans since December \(2019\) and early \(2020\) [1], due to the increasing number of infections and deaths [1], the importance of a mathematical epidemiologic framework remains for the public policies and healthcare infrastructure needs in order to estimate the disease patterns of transmission that is related to the potential damage within a pandemic scenario quickly affecting economics and survival of human individuals [2,3,4,5,6].

As COVID-\(19\) epidemics continuously keep a growing pattern of infection since its beginning until July \(2020\), theoretical analysis was briefly performed to track the main transmission pattern of the virus reproductive behavior during the period from January to March, \(2020\). Latest empirical data retrieved from researches about COVID-\(19\) transmission patterns were used to identify modeling patterns of SARS-COV-\(2\) transmission based on SIR model equation and its derivatives for this period and how those models failed to find a pattern of transmission involving individual's actions or groups behaviors.

SARS-COV-\(2\) was noted as expressing different patterns of transmission among humans [2,3,4]. This feature is being investigated not only by clinical trials/data [2,3,4,5,6,7], statistical tools [2,4,6,7,8] and medical interviews with the patients [2,3,5,6,7], but also through the mathematical point of view concerning the maximum possible rate of infection of the virus and human daily life [2,9,10] that keeps the dissemination with an increasing margin of probability/statistical outcomes.

This mini review also focused on the investigation of a specific theoretical framework understood as the high biological convergence phenomena (author conceptualization) between SARS-COV-\(2\) and high random forms (nonlinear properties of biological interactions) this virus transmission patterns presents associated with human behavior from an strictly empirical point of view. Biological high convergence was defined as an uncontrolled biological association of rapid dissemination between organisms, presenting distinct possible causes of transmission and mainly important to say, expressing an emergent phenomena of transmission originated from the host individual to the groups behavior since early global outbreak.

Using the new definition of biological convergence phenomena, ecological consequences in global terms were briefly described as the limitations of SIR models. Also an important point was addressed and stressed to the individual level of infection, rather than community transmission since biological behavior of the infected organism represents the key aspect for the disease convergence towards high exponential growth curve. The urgency for mathematical modeling of the virus transmission needs to be improved in many senses due to the presence of nonlinear behavior of the phenomena and exponential cumulative daily new cases behavior.

2. Empirical evidences

Based on researches from January to March, \(2020\), at that time it was pointed that the transmission nature of COVID-\(19\) was caused by near human's proximity and interactions within a set of empirical given variables constituted by the most basic forms of social interactions such as cough, sneeze, hand shake, clothes, cups, general touching and general objects sharing behaviors [1,4,11]. This set of variables are the type of interactions that don't follow a specific pattern of expression among each of the samples (individual) [9] due to psychological nature of the event in terms of how hosts will behave in personal and communitarian behavior. In this sense, the pre-assumed forms of transmission at that time presented a limitation concerning the continuous form of observation (partially unpredictable) of human behavior. And being this feature a higher domain of probabilities distributions assuming distinct patterns of occurrence for symptoms (pre-asymptomatic, symptomatic, asymptomatic) [4,6,7] and mainly transmission [9,10,11], researches based on traditional SIR models on this period encountered important limitations regarding predictive analysis and SARS-COV-\(2\) spreading patterns definition.

This feature of analysis was noted during January to March in the forms of transmission patterns based on traditional methods as standard [2,6,10,11,12,13,14]. These traditional approaches gave a false observation of the event and other unsuspected factors for transmission and modeling patterns that were not inferred [9,10,15,16,17] as mathematical counterproof predictions. In this sense, the high chances of transmission, statistical fluctuations, distinct SIR (mathematical modeling of infectious diseases) patterns formation of the epidemiological data observed for that period and until now demands more empirical and theoretical investigations concerning how the virus gets transmitted and how hosts behavior present an imperative convergence about virus dissemination. Several branches of knowledge are required to understand how lethal and damaging can be biological infections that are in a nonlinear scenario of epidemic spreading [9,10,13,14,17].

Based on the random nature of transmission patterns, the COVID-\(19\) seen to continue the spreading of infection, standing beyond the predefined and known measures of epidemic prevention until now (July, \(2020\)) and therefore demanding alternative scientific and mathematical hypothesis and further probabilistic and statistical frameworks definitions.

2.1. Virus reproductive emergent behavior event

Many diseases share in common the same forms of infection as COVID- \(19\) [2], however, not only the causes of transmission are important issues to be considered, as the chemical and biological properties of transmission forms, but the human emergent behavior events [18,19,20,21] since it is in the main cause of virus community transmission patterns. All these mentioned parameters share a high convergent solutions (affinity) if observed with a nonlinear time series analysis [22,23,24,25,26], thus, possibly presenting high asymptotic stability for the dissemination network [24]. This was proved to be true by July \(2020\) where many researches pointed to the importance of preventive methods towards individual and community behavior [27]. In this sense, further researches in this field of knowledge are required in order to clarify time, space and coupling interactions of the virus towards human society's organizations.

The mathematical patterns of transmission might express an indeterminate pattern in its causation due to the number of variables that causes the transmission and the random variable's variance defined by each of the individual behavior's hosts infected by the virus [9,15]. At this point the concept of high biological convergence phenomena meets its definition and meaning, where not many diseases or organisms interaction find very closed interaction regarding one organism basic forms of expression being it the most effective way of spreading patterns of other organism reproduction. Not defined in science as a traditional occurrence, the previous models of epidemic infection [2,17,25] can be too discrete (SIR tendency to deterministic approaches) in terms of knowing the true nature of event causation and random expression, being this latter feature a continuous indefinite observation of the virus spreading pattern, that promotes an unresolvable distance between the established mathematical predictions during January to March and the empirical evidences found until July, \(2020.\)

The mathematical modeling adopted by several researches that gives a glance of discretization methods, to obtain virus behavior through its hosts, might not present a new novel considering high degree of randomness in the individual scale of virus spreading, since it does not consider human behavior characteristics as the main role of the virus reproductive behavior. In this way, unpredictable spread of the infections might be noticed beyond the traditional methods boundaries [17].

It is suggested that the modeling patterns of transmission and infection in the spatial life course epidemiology need to be observed in the time-varying unresolved empirical data [17] through the human emergent behavior phenomenon [18], to track a high order non autonomous function of the virus infection towards human emergent behavior under the view of local limitations, just to mention a few, public health infrastructure/efficacy, public health policies and biological resistance/drug responsiveness. A theoretical approach for this suggestion was showed to be important during pandemics evolution due to fails in many researches to predict virus quantitative aspects of transmission per population ratio, thus generating high degree of uncertainly for future predictions based on SIR traditional approaches.

3. Evidences synthesis

These epidemiological factors, that is, forms of infection, biological- chemical affinities and human emergent behavior modeling patterns ensures to the preventive epidemic framework the need to consider any given number of infected individuals as potentially dangerous for the pandemic start and continuity (posterior waves of infection). This statement leads to the conclusion that no minimum range of infected individuals are parameters to consider the local epidemics as controlled as it is notably pre assumed in SIR models formulation, therefore, a post critical epidemics event should be treated as an alert phase, since new probabilistic outcomes are expected to generate the same non autonomous phase space of the origin where uncertainty prevails.

To conclude the previous paragraph statements, the analyzed mathematical framework of this mini review, points that the transmission patterns resemble the own host emergent behavior [18,19,20,21], and for this reason it assumes the infodemics domains of analysis in a multidimensional form, where the heteroskedasticity aspect, in which SIR models can not handle this feature by its main variables compartments. Therefore, the virus reproduction need to be quantified, as it is designed theoretically in this review, with a high order organization of the host individual/group/community behavior [17] in the outside scope of the strictly SIR models basis or at least a consistent modification of the basic compartments. The importance of this approach remains mainly in the public policy actions and media information campaign where these confounding variables stands beyond pre assumed forms of observation of empirical data concerning deterministic approaches of new daily infections. It means that more specific definitions, strategies and monitoring of human behavior are needed to estimate disease spreading patterns under the SIR models view point.

These statements lead also to the evidence observed [26] that not only \(140\) thousand (number of infected individuals during January to March), but even \(20\) contaminated individual hosts are at risk of keeping propagating the disease [24] due to convergence of epidemic factors and stability aspect of the biological atypical organism's interaction (biological convergence phenomena). Also being this interaction possibly modeled by an emergent phenomenon [28,29] of the host's behavior thus being this new mathematical model directly proportional to the virus reproductive formed patterns that is being observed since pandemic outbreak and therefore, outside the common mathematical SIR models approaches already performed by nowadays science.

The human emergent behavior phenomenon shares with the COVID-\(19\) virus, the intrinsic relation between the nonlinear expression of forms of transmission plus human forms of socializing, resulting into a high convergence of infection with results pointing in the direction and the view that human most basic form of socializing is also a tool for the self-spreading pattern of transmission and infection [30] of COVID-\(19\), being this virus reproductive behavior pattern the own human emergent phenomena behavior [18,19] itself in a very closed interaction.

Note that this review also take into account the digital behavior (infodemics) [31] that can also present high influential effect at virus transmission. Future researches need be carried out to integrate infodemics and life course epidemiology with a mathematical overview of the pandemic COVID-\(19\), and also this represents nowadays a very important tool for policy making in order to prevent fake news and false science information to take place within human behavior at individual or community scale.

This natural phenomenon of extreme convergence (host emergent phenomena behavior and virus reproductive patterns formation) might give us a new glance of biological organizations, towards rare extreme conditions where nature and/or human society can share uncontrolled biological association of rapid dissemination. This observation gives us also a view of about humanity capacity of organizing, preventing and determining extreme convergence phenomena with rapid preventive and corrective response towards new high convergent pathogens [25].

The more random forms of transmission, associated with human behavior, the more effective are the infection and spread of SARS-COV-\(2\) virus among individuals. This feature is not observed in other type of epidemic diseases [14]. Non convergence solutions patterns for transmission of the virus need to be adopted through human behavior dimension by society as a whole, and this means not only the government through policy, health infrastructure and medical treatment, but all individuals behavior in a single manner [30].

4. Concluding remarks

COVID-\(19\) dissemination was theoretically investigated during the period of January to March \(2020\) and it showed that virus reproductive patterns were wrongly investigated by that time, mainly because the virus spreading patterns are much closed with the human behavior in terms of expressing high randomness or emergent patterns formation. New empirical data need to be investigated regarding human behavior as the main nonlinear property of analysis to sustain SIR model predictive equations efficacy. Also new epidemiologic mathematical methods need to be further advanced towards new variables that influence virus reproductive patterns such as agent-based models and complex networks.

Also, considering these theoretical review findings, and extended with other branches of knowledge, this review brings the attention to adoption of new strategies to be used for policy makers and citizens in general, beyond the mathematical predictions already implemented worldwide as a counterproof of mathematical uncertainty existent in the actual SIR models used for predictive analysis.

The review also points out for the individual scale of prevention very well pointed in Lopes and McKay [32] that in its turns leads to the group and community dimension of human emergent behavior. This approach can be easily transferred by analogy to an infodemics analysis. This novel, since pandemic outbreak until now constituted the main center of the COVID-\(19\) spreading pattern analysis for a local epidemic or pandemic simulation and new methods of control.

conflictofinterests

''The author declares no conflict of interest.''

1. Introduction

Linear and non-linear fractional equations play a major role in various fields such as fluid mechanics, optical fibers, solid state physics, chemical physics, probability and statistics, geochemistry, engineering, acoustics and so on [1,2,3,4,5]. This is due to the fact that fractional derivatives can hold the history of the variable under consideration. In fact, several numerical and analytic techniques for solving FDEs have been presented in the literature and they have their own advantages and limitations.

As it is well known, Riccati differential equations concerned with applications in pattern formation in dynamic games, linear systems with Markovian jumps, river flows, econometric models and control theory [6,7,8]. Many studies have been conducted on solutions of the Riccati differential equations. Some of them, the approximate solution of ordinary Riccati differential equation obtained from homotopy perturbation method (HPM) [9,10,11], homotopy analysis method (HAM) [12], and variational iteration method proposed by He [13]. The He's homotopy perturbation method proposed by He [14,15,16], the variational iteration method [17], Adomian decomposition method (ADM) [18], the Laplace- Adomain- Pade Method [19].

The Variation of Parameters (VPM) is the other name is Variation of constant in Mathematics. The linear conventional differential equations that are inhomogeneous are unraveled by this technique. Duhamel's rule is another name for this strategy. It was named after Jean-Marie Duhamel \((1797 - 1872) \), by whom this VPM was applied for the absolute first time to tackle the inhomogeneous equation.

By and large to discover arrangement through coordinating components or dubious coefficient with bounty substantially less endeavor, we settle first order inhomogeneous linear differential equations, while heuristics are influenced by the once systems. All the inhomogeneous linear differential equations are not worked by the heuristics which include expecting. The proposed technique is liberated from adjust blunder, discretization, linearization, Adomian's polynomial and use just beginning condition, which are simpler to actualize and diminish the computational work.

Right now, expand the use of the VPM so as to determine investigative estimated answers for non-linear fractional Riccati differential equation:

\begin{equation*} D^{\alpha}_{\ast} u(t)=A_{1}(t)+A_{2}(t) u(t)+A_{3}(t) u^{2}(t)\,\,\,\,\,\, t\epsilon R, 0 \,< \,\alpha \leq 1, \end{equation*} subject to the initial conditions \begin{equation*} u^{k}(0)=a_{k},\,\,\,\,\,\,\,\, k=1,2,3.....n-1, \end{equation*} where \( \alpha \) is fractional derivative order, n is an integer, A1 (t), A2 (t) and A3 (t) are known real functions and \(a_{k} \) is a constant. The objective of this paper is to broaden the use of the variation of Parameters technique (VPM) to comprehend fractional nonlinear Riccati differential equations with Riemann-Liouville fractional integral.

2. Basic definitions

Here, some fundamental definitions and properties of the fractional calculus hypothesis which can be found in [5,20,21].

1. Definition The Riemann-Liouville fractional primitive of order is of a function \(h:(0,b]\rightarrow R \) of order \( \alpha\epsilon R^{+} \) is defined by \begin{equation*} j^{\alpha}_{0}u(s)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha - 1} u(s)ds, \end{equation*} provided the right side is point wise defined on \((0, b] \) where \( \Gamma \) is the gamma function.

2. Definition The Riemann-Liouville fractional order derivative of function \(u(s) \) is given as

where \( m-1 \leq \alpha\, <\, m \,\epsilon\, Z^{+} \).

3. Definition The modified Riemann-Liouville derivative is defined as \begin{equation*} D^{\alpha}_{\ast,s} u(s)=\frac{1}{\Gamma(m-\alpha)}\frac{d^m}{ds^{m}}\int^{0}_{t}(t-s)^{m-\alpha} (u(s)-u(0))ds, \end{equation*}where \(x\, \epsilon\, [0,1] \), \( m-1 \leq \alpha < m \) and where \( m\geq 1 \).

3.Variation of parameters method

To emanate the main concept of the VPM [22,23] we assume the general equation

\begin{equation*} L(w)+N(w)+R(w)=f(x), \,\,\,\,\,\,\,\,\,\, a \leq x \leq b, \end{equation*} where \(L,\) \(N\) are linear and nonlinear operators. \(R\) is a linear differential operator but \(L\) has the highest order than \(R\), \(f(x)\) is a source term in the given domain \([a,b].\) By utilizing the VPM, we have the pursuing solution of the equation \begin{equation*} w(x)=\sum^{k-1}_{\iota=0}\frac{c_{\iota+1}x^{\iota}}{\iota\,!}+\int_{0}^{x}\lambda(x,\tau)(-N(w)(\tau)-R(w)(\tau)+f(\tau))d\tau, \end{equation*} where \(k \) represent the order of given differential equation (DE) and \(C_{\iota} \) where \( \iota=1,2,3, \) are unknowns. So \begin{equation*} w(x)=\sum^{k-1}_{\iota=0}\frac{c_{\iota+1}x^{\iota}}{\iota\,!}. \end{equation*}

For homogeneous solution which is taken by

\begin{equation*} L(w)= 0. \end{equation*}

The another part which is obtained from Equation (1) by using VPM is

\begin{equation*} \int_{0}^{x}\lambda(x,\tau)(-N(w)(\tau)-R(w)(\tau)+f(\tau))d\tau. \end{equation*}

Here \( \lambda(x,\tau) \) is a Lagrange multiplier that expel the progressive use of integrals in the iterative scheme and it depending on the order of equation. Commonly the pursue definition is utilized to find the value of the multiplier \( \lambda(x,\tau) \) from

\begin{equation*} \lambda(x,\tau)= \sum^{k}_{\iota=1}\frac{(-1)^{\iota-1}\tau^{\iota-1}x^{k-1}}{(\iota-1)\,! (k-1)\,!}=\frac{(x-\tau)^{k-1}}{(k-1)\,!}, \end{equation*} here \(k\) is the order of the given DE and it varies for different values of \(k.\) We have the pursue conditions: \begin{eqnarray*} &&k=1, \,\,\,\,\ \lambda(x,\tau)=1,\\ &&k=2, \,\,\,\, \lambda(x,\tau)=(x-\tau),\\ &&k=3, \,\,\,\,\ \lambda(x,\tau)=\frac{x^{2}}{2\,!}+\frac{\tau^{2}}{2\,!}-\tau x. \end{eqnarray*}

Therefore, we utilize the pursue iterative scheme for solving equation

\begin{equation*} w_{n+1}=w_{0}+\int_{0}^{x}\lambda(x,\tau)(-N(w)(\tau)-R(w)(\tau)+f(\tau))d\tau. \end{equation*}

We can get the initial guess \( w_{0}(x) \) by using initial conditions. We are taking better approximation by using fixed value of initial guess in each iteration. Here we are settling fractional riccati differential condition by utilizing Reimann-Liouville. For the arrangement method we are consolidating VPM with fractional integral then the iterative plan for fractional equations is

\begin{equation*} w_{n+1}=w_{0}+\frac{1}{\Gamma (\alpha)} \int_{0}^{x}\lambda(x,\tau)^{x-\tau}(-N(w)(\tau)-R(w)(\tau)+f(\tau))d(\tau). \end{equation*}

4. Solution of fractional Riccati differential equation

The point of this area is to contemplate the arrangements and the impacts of the fractional order derivative to Riccati's conditions. In perspective on the nearness of a shut structure arrangement and on other applied numerical procedures to Riccati's conditions, three models are considered to check the capability of the proposed method.

4.1. Problem

We first consider the following fractional Riccati equation:

\begin{equation*} D^{\alpha}_{\tau}u(t)-2u(t)+u^{2}(t)=1,\,\,\,\,\,\,\, 0 < \alpha \leq 1, t\geq 0, \end{equation*} with homogeneous initial condition \(u(0)=0 \). The exact solution of the above problem is \begin{equation*} u(t)= 1+\sqrt{2}\tanh \left(\sqrt{2}t + \frac{1}{2}\log\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}\right)\right). \end{equation*}

We construct the following iterative scheme for the above problem:

\begin{equation*} u_{n+1}=u_{0}+\frac{1}{\Gamma (\alpha)} \int_{0}^{t}(t-\tau)^{\alpha-1}(2u(t)+u^{2}(t)+1)d\tau. \end{equation*}

Taking initial condition \(u(0)=0 \), the following results for \(\alpha=1 \) are produced:

\begin{align*} u_{1}(t)&=t,\\ u_{2}(t)&=t^{2}-\frac{1}{3}t^{3}+t,\\ u_{3}(t)&=t-\frac{1}{63}t^{7}+\frac{1}{9}t^{6}-\frac{1}{15}t^{5}-\frac{2}{3}t^{4}+\frac{1}{3}t^{3}+t^{2},\end{align*} \begin{align*} u_{4}(t)&=t-\frac{1}{59535}t^{15}+\frac{1}{3969}t^{14}-\frac{41}{36855}t^{13}-\frac{1}{1890}t^{12}+\frac{27}{1925}t^{11}\\ &\;\;\;-\frac{62}{4725}t^{10}-\frac{62}{945}t^{9}+\frac{17}{420}t^{8}+\frac{71}{315}t^{7}+\frac{4}{45}t^{6}-\frac{3}{5}t^{5}-\frac{1}{3}t^{4} +\frac{1}{3}t^{3}+t^{2}. \end{align*}

Table 1. Comparison of numerical results of problem 4.1 for \(\alpha\)=1.

t	Exact	VPM	HPM	OHM	Error of VPM
0.0	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
0.2	0.2419744004	0.2419499764	0.2419648204	0.2443164581	\(2.4424\times 10^{-5}\)
0.4	0.5678068604	0.5673979034	0.5681149562	0.5702708053	\(4.089\times 10^{-4}\)
0.6	0.9535582813	0.9525886597	0.9582588343	0.9535657064	\(9.696\times 10^{-4}\)
0.8	1.346354258	1.345789984	1.365239549	1.3475927635	\(5.64\times 10^{-4}\)
1.0	1.689488974	1.688651308	1.723809524	1.6907027573	\(8.37\times 10^{-4}\)

The approximate solutions \(u_{4}(t)\) for different values of \( 0 \,< \,\alpha\leq 1 \) continuously approaches to the exact solution when \( \alpha=1 \). Thus, we anticipate a veracious solution for various values of \( \alpha \). For additional examination Table 1 and Table 2 show a correlation between our methodology and other existing numerical and diagnostic strategies for \( \alpha=1 \). It tends to be found from the table that the outcomes got by the 3-term VPM gives estimated arrangement contrast well overall and different strategies, particularly when draws near to 1. Correlation shows that our strategy gives minimal blunder as contrast with other. Figure 1 is plotted for approximate solution of VPM and exact solution of Problem 4.1. In Figure 2, we have shown the graphic representation of of approximate solution of Problem 4.1 for \(\alpha\) = 0.7, 0.8, 0.9, and 1.

Table 2. Comparison of numerical results of problem 4.1 for different values of \(\alpha\).

t	\(\alpha=0.7\)	\(\alpha=0.8\)	\(\alpha=0.9\)
0.0	0.0000000	0.0000000	0.0000000
0.2	0.3784548231	0.3336603491	0.2867654164
0.4	0.7083751809	0.6756546260	0.6251501607
0.6	1.022225529	1.025455098	0.9976478322
0.8	1.304871418	1.352852736	1.363764296
1.0	1.542084537	1.628041223	1.677804452

4.2. Problem

We consider the fractional Riccati equation,

\begin{equation*} D^{\alpha}_{\tau}u(t)+u^{2}(t)=1,\,\,\,\,\,\,\, 0 < \alpha \leq 1, t\geq 0, \end{equation*} with homogeneous initial condition \(u(0)=0 \). The exact solution of the above problem is \begin{equation*} u(t)=\frac{e^{2t}-1}{e^{2t}+1}. \end{equation*}

We are constructing the following iterative scheme of the above problem is

\begin{equation*} u_{n+1}=u_{0}+\frac{1}{\Gamma (\alpha)} \int_{0}^{t}(t-\tau)^{\alpha-1}(-u^{2}(t)+1)d\tau. \end{equation*}

Taking initial condition \(u(0)=0 \), the following results for \( \alpha=1 \) are produced:

\begin{align*} u_{1}(t)&=t,\\ u_{2}(t)&=t-\frac{1}{3}t^{3},\\ u_{3}(t)&=t-\frac{1}{63}t^{7}+\frac{2}{15}t^{5}-\frac{1}{3}t^{3},\\ u_{4}(t)&=t-\frac{1}{59535}t^{15}+\frac{4}{12285}t^{13}-\frac{134}{51975}t^{11}+\frac{38}{2835}t^{9}-\frac{17}{315}t^{7}+\frac{2}{15}t^{5}-\frac{1}{3}t^{3}. \end{align*}

Table 3 and Table 4 show a comparison between our approach and other existing numerical and analytic methods for \(\alpha= 1\). It can be deduced from the tables that the results obtained by the 3-term VPM gives approximate solution compare very well with other methods especially when \(\alpha\) gets close to 1. Comparison shows that our proposed technique gives the least error as compare to other. Figure 3 is plotted for approximate solution of VPM and exact solution of Problem 4.2. In Figure 4, we have shown the graphic representation of approximate solution of Problem 4.2 for \(\alpha\) = 0.7, 0.8, 0.9, and 1.

Table 3. Comparison of numerical results of problem 4.2 for \(\alpha=1\).

t	Exact	VPM	HPM	OHM	Error of VPM
0.0	0.0000000	0.0000000	0.0000000	0.0000000	0.0000000
0.2	0.1973753203	0.1973753160	0.1973753092	0.1974023559	\(4\times 10^{-9}\)
0.4	0.3799489622	0.3799469862	0.3799435784	0.3800652965	\(1.976\times 10^{-6}\)
0.6	0.5370495670	0.5369833784	0.5368572343	0.5371479432	\(6.61886\times 10^{-5}\)
0.8	0.6640367702	0.6633009217	0.6617060368	0.6640492005	\(7.3584\times 10^{-4}\)
1.0	0.7615941560	0.7571662667	0.7460317460	0.7616344154	\(4.427\times 10^{-3}\)

Table 4. Comparison of numerical results of problem 4.2 for different values of \(\alpha\).

t	\(\alpha=0.7\)	\(\alpha=0.8\)	\(\alpha=0.9\)
0.0	0.0000000	0.0000000	0.0000000
0.2	0.2695421965	0.2490582911	0.2242652315
0.4	0.4163894367	0.4130176413	0.4003438466
0.6	0.5223419886	0.5373149545	0.5419937918
0.8	0.6012630211	0.6311970814	0.6517859873
1.0	0.6595756101	0.6995721576	0.7318890064

4.3. Problem

We consider the following fractional Riccati equation,

\begin{equation*} D^{\alpha}_{\tau}u(t)+u(t)-u^{2}(t)=1,\,\,\,\,\,\,\, 0 < \alpha \leq 1, t\geq 0, \end{equation*} with initial condition \( u(0)=\frac{1}{2} \). The exact solution of the above problem is \begin{equation*} u(t)=\frac{e^{-t}}{e^{t}+1}. \end{equation*}

We are constructing the following iterative scheme of the above problem is

\begin{equation*} u_{n+1}=u_{0}+\frac{1}{\Gamma (\alpha)} \int_{0}^{t}(t-\tau)^{\alpha-1}(u^{2}(t)+u(t))d\tau. \end{equation*}

Taking initial condition \(u(0)=0 \), the following results for \( \alpha=1 \) are produced:

\begin{align*} u_{1}(t)&=\frac{1}{2}-\frac{1}{4}t,\\ u_{2}(t)&=\frac{1}{2}-\frac{1}{4}t+\frac{1}{48}t^{3},\\ u_{3}(t)&=\frac{1}{2}-\frac{1}{4}t+\frac{1}{48}t^{3}+\frac{1}{16128}t^{7}-\frac{1}{480}t^{5},\\ u_{4}(t)&=\frac{1}{2}-\frac{1}{4}t+\frac{1}{3901685760}t^{15}-\frac{1}{50319360}t^{13}+\frac{67}{106444800}t^{11} -\frac{19}{1451520}t^{9}+\frac{17}{80640}t^{7}-\frac{1}{480}t^{5}+\frac{1}{48}t^{3}. \end{align*}

Table 5. Comparison of numerical results of problem 4.3 for different values of \(\alpha\).

t	Exact	\(\alpha=0.7\)	\(\alpha=0.8\)	\(\alpha=0.9\)	\(\alpha=1.0\)	Error
0.0	0.5000000000	0.5000000000	0.5000000000	0.5000000000	0.5000000000	0.00000000
0.2	0.4501660027	0.4309494485	0.4365457215	0.4431444975	0.4501660027	0.00000000
0.4	0.4013123399	0.3893257986	0.3909330219	0.3951259688	0.4013123420	\(-2.1\times 10^{-9}\)
0.6	0.3543436938	0.3553599726	0.3517080337	0.3515020203	0.3543437718	\(-7.80\times 10^{-8}\)
0.8	0.3100255189	0.3262525572	0.3171054591	0.3117623165	0.3100265069	\(-9.880\times 10^{-7}\)
1.0	0.2689414214	0.3007743319	0.2863305011	0.2757427983	0.2689483336	\(-6.912\times 10^{-6}\)

Table 5 represents the different values of \(\alpha\). We see that VPM for \(\alpha=1\) gives the best approximate results with least computational work. Figure 5 is plotted for approximate solution of VPM and exact solution of Problem 4.3. In Figure 6, we have shown the graphic representation of approximate solution of Problem 4.3 for \(\alpha=\) 0.7, 0.8, 0.9, and 1.

In above all problems, it seems that \(u_{4}(t)\) for equal to 1 is in high agreement with the exact solution. Furthermore, \(u_{4}(t)\) for different values of continuously communicates until \(\alpha=1\) is reached. Thus, a convenient solution is expected for various values of \(\alpha\).

5. Discussion of Results:

We have adequately applied the variation of Parameters procedure (VPM) for the numerical outcomes of fractional order riccati's differential condition. We got pleasing results due to great intermingling VPM.

6. Concluding Remarks:

In this work, we utilized the Variation of Parameters Method (VPM) to study the solution of the fractional Riccati equation. We succeed to the fact that VPM is very practical and effective technique for finding the analytic results and in addition numerical results for extensive classes of linear and nonlinear fractional differential equations. It provides the solutions in very less iteration and that converge very rapidly in real physical problems. Three existing models were tested to noticeable the legitimacy of the chosen technique and the attained outcomes were outstanding and compatible with other methods. Furthermore, we noticed that the attained approximate results for various values of alpha continuously communicate until the first order derivative is reached. This is another indicator that our results are likely to be valid.

acknowledgments

We would like to thank the reviewers for carefully reading the manuscript and making several helpful comments to increase the quality of the paper.

authorcontributions

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

conflictofinterests

The authors declare no conflict of interest.