A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative

We present a fractional-order model for the COVID-19 transmission with Caputo–Fabrizio derivative. Using the homotopy analysis transform method (HATM), which combines the method of homotopy analysis and Laplace transform, we solve the problem and give approximate solution in convergent series. We prove the existence of a unique solution and the stability of the iteration approach by using fixed point theory. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative.


A mathematical model for the transmission of COVID-19 with Caputo-Fabrizio fractional derivative
Chen and colleagues have proposed a transmission network model to simulate possible transmission from the source of infection (possibly bats) to human infection [37]. They assumed that the virus was transmitted among the bats' population, and then transmitted to an unknown host (probably wild animals). Then hosts were hunted and sent to the seafood market, which was defined as the reservoir or the virus. People exposed to the market got the risks of the infection. In the presented model, people were divided into five groups: susceptible people (S), exposed people (E), symptomatic infected people (I), asymptomatic infected people (A), and removed people (R) including recovered and dead people. COVID-19 in the reservoir was denoted as (W). This model was presented as follows: where Λ = n × N , N refer to the total number of people and n is the birth rate, m: the death rate of people, β p : the transmission rate from I to S, κ: the multiple of the transmissible of A to that of I, β w : the transmission rate from W to S, δ: the proportion of asymptomatic infection rate of people We moderate the system by substituting the time derivative by the Caputo-Fabrizio fractional derivative in the Caputo sense [26]. With this change, the right-and left-hand sides will not have the same dimension. To solve this problem, we use an auxiliary parameter ρ, having the dimension of sec., to change the fractional operator so that the sides have the same dimension [43]. According to the explanation presented, the COVID-19 transmission fractional model for t ≥ 0 and η ∈ (0, 1) is given as follows: where the initial conditions are In the next section we investigate the existence and uniqueness of the solution for system (1) by fixed point theorem.

Existence of a unique solution
In this section, we show that the system has a unique solution. For this purpose, employing the fractional integral operator due to Nieto and Losada [38] on the system (1), we obtain Using the definition of Caputo-Fabrizio fractional integral [38], we obtain For convenience, we consider Proof Consider functions S(t) and S 1 (t), then Let λ 1 = m + β p l 1 + β w l 2 , where l 1 = I(t) and l 2 = W (t) are bounded functions, then we have Thus, the Lipschitz condition is fulfilled for P 1 . In addition, if 0 < m + β p l 1 + β w l 2 ≤ 1, then P 1 is a contraction.
Similarly, P 2 , P 3 , P 4 , P 5 , P 6 satisfy the Lipschitz condition as follows: On consideration of P 1 , P 2 , P 3 , P 4 , P 5 , P 6 , we can write equation (2) as follows: Thus, consider the following recursive formula: Given the above equations, one can write According to H 1n 's definition and using the triangular inequality, we have Thus we get It can be shown that similar results are obtained for H in , i = 2, 3, 4, 5, 6, as follows: According to the above result, we show that system (1) has a solution.

Theorem 3.2
The fractional COVID-19 model (1) has a system of solutions if there exist t i , i = 1, 2, 3, 4, 5, 6, such that We have shown that kernels H in , i = 1, 2, 3, 4, 5, 6, satisfy the Lipschitz condition. By using the recursive method and the results of (4) and (5), we obtain Thus, functions (3) exist and are smooth. We claim that the above functions are the solutions of system (1). To prove this claim, we assume We have By repeating this process, we obtain By taking limit on recent equation as n tends to infinity, we obtain G 1n (t) → 0. By the same way, we get G in (t) → 0, i = 2, 3, 4, 5, 6, and this completes the proof.
To prove the uniqueness of solution, we assume that system (1) has another solution such as S 1 , E 1 , According to the Lipschitz condition of S, we get Thus (1) is unique if the following condition holds:

Theorem 3.3 The solution of COVID-19 fractional model
Proof From condition (7) and equation (6), we conclude that In the same way, we can show that The proof is complete.

Stability analysis by fixed point theory
Using the Sumudu transform, we obtain a special solution to the COVID-19 model and then prove the stability of the iterative method using fixed point theory. At first, we apply the Sumudu transform on both sides of equations in model (1), then We conclude from the Sumudu transform definition of the Caputo-Fabrizio derivative the following: If we rearrange the above inequalities, then We obtain The approximate solution of system (1) is as follows:

Stability analysis of iteration method
Consider the Banach space (G, · ), a self-map T on G, and the recursive method q n+1 = φ(T, q n ). Assume that Υ (T) is the fixed point set of T which Υ (T) = ∅ and lim n→∞ q n = q ∈ Υ (T). Suppose that {t n } ⊂ Υ and r n = t n+1φ(T, t n ) . If lim n→∞ r n = 0 implies that lim n→∞ t n = q, then the recursive procedure q n+1 = φ(T, q n ) is T-stable. Suppose that our sequence {t n } has an upper boundary. If Picard's iteration q n+1 = Tq n is satisfied in all these conditions, then q n+1 = Tq n is T-stable.

Theorem 4.1 ([44]) Let (G, · ) be a Banach space and T be a self-map of G satisfying
for all x, y ∈ G where B ≥ 0 and 0 ≤ b < 1. Suppose that T is Picard T -stable.
According to (8), the fractional model of COVID-19 (1) is connected with the subsequent iterative formula. Now consider the following theorem.
Proof To prove that T has a fixed point, we compute the following inequalities for (i, j) ∈ N × N : By applying norm on both sides, we obtain Since the solutions have the same roles, we can consider From equations (9) and (10), we get From equations (11) and (12), we get Similarly, we will obtain where Thus the T-self mapping has a fixed point. Also, we show that T satisfies the conditions in Theorem 4.1. Consider that (13), (14) hold, we assume (1 + δω p f 12 (η) -(γ + m)f 13 (η)), So, all the conditions of Theorem 4.1 are satisfied and the proof is complete.

Convergency of HATM for FDEs
In the following, we discuss the convergence of HATM by presenting and proving the following theorem.
Proof By assuming that ∞ n=0 S n (t) is uniformly convergent to S(t), we can clearly state lim n→∞ S n (t) = 0, for all t ∈ R + .

Numerical results
In this section, we present a numerical simulation for the transmission model of COVID-19 (1) by using the homotopy analysis transform method (HATM). To this end, we assume that the total population is N = 100, and since the birth rate for China in 2020 is about 11.46 births per 1000 people, then Λ = n × N = 1.146. According to the news released by the World Health Organization, the death rate is 3.4 percent and the incubation period i.e., the recovered and the dead, also increases with time. The amount of virus in the reservoir also decreases first and then increases as people enter the reservoir from the two infected groups. We put the Caputo fractional derivative in model (1) instead of the Caputo-Fabrizio fractional derivative and solved the new model similarly and obtained the results of the two derivatives for η = 0.96. Then, in Figs. 4-6, we compared these results for system (1). We observe that the difference between the results of these two derivatives increases with time.

Conclusion
In this paper, we investigate a model of the COVID-19 transmission in different groups of people using the Caputo-Fabrizio fractional derivative. Using the fixed point theorem, we prove a unique solution for the system. The resulting differential system is solved using the homotopy analysis transform method (HATM), and we obtain approximate solutions in convergent series. With the numerical results, we present a simulation for COVID-19, which shows the rapid transmission of the virus to different groups of people. We com-