A new mathematical model of multi-faced COVID-19 formulated by fractional derivative chains

It has been reported that there are seven different types of coronaviruses realized by individuals, containing those responsible for the SARS, MERS, and COVID-19 epidemics. Nowadays, numerous designs of COVID-19 are investigated using different operators of fractional calculus. Most of these mathematical models describe only one type of COVID-19 (infected and asymptomatic). In this study, we aim to present an altered growth of two or more types of COVID-19. Our technique is based on the ABC-fractional derivative operator. We investigate a system of coupled differential equations, which contains the dynamics of the diffusion between infected and asymptomatic people. The consequence is accordingly connected with a macroscopic rule for the individuals. In this analysis, we utilize the concept of a fractional chain. This type of chain is a fractional differential–difference equation combining continuous and discrete variables. The existence of solutions is recognized by formulating a matrix theory. The solution of the approximated system is shown to have a minimax point at the origin.

We investigate the growth of two or more coexisting types of COVID-19. The ABCfractional derivative operator formalizes our procedure. We deal with a system of coupled differential equations, which contains the dynamics of the diffusion between infected and asymptomatic populations. The outcome is accordingly associated with a macroscopic rule for the individuals. Moreover, this analysis is formulated with the concept of a fractional chain. This type of chain is a fractional differential-difference equation combining continuous and discrete variables. The existence of solutions is established by formulating a matrix theory. Some numerical results are illustrated in the sequel.
The rest of the paper is organized as follows: Sect. 2 presents the methodology that will be used in our study; Sect. 3 describes the results and discussion of the suggested model; Sect. 4 provides the conclusion and directions for future works.

ABC-definition
The elementary viewpoint and appearances of fractional calculus and its applications are realized in numerous assessments and evaluations. Most studies on the fractional calculus contain kernels. For example, the main difference between the Caputo differential operator, the Caputo-Fabrizio operator [31], and others is that the Caputo differential operator is associated with a power law, the Caputo-Fabrizio differential operator is modified by employing an exponential growth term. Atangana-Baleanu differential operator is formulated by suggesting the generalized Mittag-Leffler function [32]. Definition 2.1 Let α , α ∈ (0, 1) be the Atangana-Baleanu differential operator of order α of a function g having the structure where D(α) denotes a normalization function, while E α indicates the Mittag-Leffler function Associated with α , the ABC integral is realized by Example 2.2 The function g(t) = t κ has the ABC integral In our study, since we focus on the approximated solutions, we assume that D(α) → 1, for all α ∈ (0, 1).

Infected dynamics
We assume that T(t) is the total number of infected individuals, which characterizes the sum of two numbers, the customary infected individuals χ(t) and those involved in the asymptomatic transmission ϒ(t), so that T(t) = χ(t) + ϒ(t). We take into account that χ(t) includes people who were previously sick. Consequently, there are frequency functions, combining χ and ϒ. In this study, we assume that T contains two sets of variables: continuous time variables and multiple discrete variables, namely numbers of infected and asymptomatic. Since COVID-19 has multiple faces, we may assume that T has chain descriptions in both categories of the variables. Two faces of COVID-19 have the description T(m, n, t, s), where (m, n) ∈ N 2 are the discrete variables and (s, t) ∈ R 2 , s ≤ t are the continuous variables. One can extend the functional T into three faces as T(m, n, k, t, s, ), and so on for finite faces, when we have T(m 1 , . . . , m j , t 1 , . . . , t j ), where (m 1 , . . . , m j ) ∈ N j are the discrete variables and (t 1 , . . . , t j ) ∈ R j .

ABC-fractional chain
In general, a chain is an integrable differential-difference equation joining at least one continuous variable and one discrete variable. The first derivative of this chain is used to suggest a system of differential equations. A fractional chain was formulated for the first time by using the Riemann-Liouville differential operator (see [33]). Based on this idea, we improve the fractional chain using a fractional differential operator for several continuous and discrete variables, namely the ABC-fractional differential operator.
In this part, we use the above information to define the ABC-fractional chain. We deal with a two-dimensional functional T. That is, T has two discrete variables, as well as two continuous variables. Similarly, for the extension to higher dimension. Define the ABCfractional chain as follows: where T(m, n, t, s) is a function depending on discrete and continuous variables (m, n) ∈ N 2 (discrete variables) and (t, s) ∈ R 2 (continuous variables), and α t is Atangana-Baleanu differential operator of order α with respect to the continuous variable t. Moreover, we consider the lowest order of (2.1) to be structured by  In this case, we suggest another dynamical system. where and ℘ are fixed constants. Equation (2.9) indicates the KdV-type and pKdV-type equations. Also, (2.1) and (2.2) imply the symmetry of (2.9). Hence, the conclusion is that there exists a function (t, s) such that Thus, we obtain the shifted dynamical system

Shifted dynamic system
System (2.13) indicates the dynamics of multi-face COVID-19, where m is the number sick people on the recent face and n is the number of the previous face, which is not terminated yet. Both systems (2.4) and (2.13) can be generalized into j faces. Moreover, one can generalize the above systems by using the 1D-parametric structure as follows: where ν is an arbitrary integer. Similarly, for the shifted system. From (2.21), we have the system where = T(m + ν, n, t, s), = T(m + ν + 1, n, t, s) and T(m + ν, n + 1, t, s) = φ, T(m + ν + 1, n + 1, t, s) = ψ. In addition, 2D-parametric structure can be realized by considering a new parameter for n to become

Results and discussion
In this section, we investigate the stability of systems (2.4) and (2.13). We have the following results for system (2.4), which can be extended to system (2.13).
The above system can be approximated at the fixed point of α t and α t to obtain the linear system The eigenvalues of this system are λ 1,2 = ± √ a 2 + 4bc, a 2 + 4bc > 0, (3.2) which correspond to the eigenvectors Hence, the critical point is a saddle point (minimax point) satisfying max( , 0 ) = ( 0 , 0 ) = min( 0 , ).  • In view of Theorem 3.1, system (3.7) has a minimum point. Figure 1 shows two important cases, a global minimum and a local minimum. Then the solution can be formulated by the integral system of equations, with the initial condition 0 = 0, 0 = 0, (3.9)

Conclusion
The minimax point theorem is one of the greatest significant consequences of the mathematical analysis theory. It indicates that there is a technique, which together minimizes the maximum loss (sick people) and maximizes the minimum improvement (healthy people). Roughly speaking, there is an approach, which normal people would take supposing the worst-case situation.
Summarizing the above analysis, we have formulated a new mathematical technique based on fractional calculus with the ABC-derivative operator. We formulated a system that satisfies multiple faces of the coronavirus. The total number is suggested as a continuous function of time, which is discrete in the number of faces. We used an approximation method to analyze the system. We recognized that the solution possesses a minimax point. This point indicates the termination of the recent face and realizes a new face of the corona virus.