key: cord-0958239-ccq7hvit
authors: Baleanu, Dumitru; Mohammadi, Hakimeh; Rezapour, Shahram
title: A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative
date: 2020-06-18
journal: Adv Differ Equ
DOI: 10.1186/s13662-020-02762-2
sha: d115e8a4d8b1cbe5f3cb06f52e8d70083b171b36
doc_id: 958239
cord_uid: ccq7hvit

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.

very systematic way such as in the approach of Akbari et al. [11] , Baleanu et al. [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] , and Talaee et al. [25] . In this paper, we use the new fractional Caputo-Fabrizio derivative [26] to express the mathematical modeling for simulating the transmission of COVID- 19 . Recently, many works related to the fractional Caputo-Fabrizio derivative have been published (see, for example, [21, 23, 24, [27] [28] [29] [30] ). The Caputo-Fabrizio fractional derivative is also used to study the dynamics of diseases (see, for example, [31] [32] [33] [34] ). Mathematical models are used to simulate the transmission of corona virus (see, for example, [35, 36] ). A mathematical model for the transmission of COVID-19 was presented by Chen et al. [37] . In this work, we investigate this model by using the Caputo-Fabrizio fractional derivative. Now, we recall some fundamental notions. The Caputo fractional derivative of order η for a function f via integrable differentiations is defined by C D η f (t) = 1

where n = [η] + 1. Our second notion is a fractional derivative without singular kernel which was introduced by Caputo and Fabrizio in 2015 [26] . Let b > a, f ∈ H 1 (a, b), and η ∈ (0, 1). The Caputo-Fabrizio derivative of order η for a function f is defined by

where t ≥ 0, M(η) is a normalization function that depends on η and M(0) = M(1) = 1. If f / ∈ H 1 (a, b) and 0 < η < 1, this derivative can be presented for f ∈ L 1 (-∞, b) as

(see [38] ). Let n ≥ 1 and η ∈ (0, 1). The fractional derivatives CF D η+n of order η + n are defined by CF D η+n f (t) := CF D η (D n f (t)) [28] . The Laplace transform of the Caputo- s+η (1-s) , where 0 < η ≤ 1 and M(η) = 1 [38] .

The Riemann-Liouville fractional integral of order η with Re(η) > 0 is defined by I η f (t) = 1 Γ (η) t 0 (ts) η-1 f (s) ds [28] . The fractional integral of Caputo-Fabrizio is defined by CF I η f (t) = 2(1-η) (2-η)M(η) f (t) + 2η (2-η)M(η) t 0 f (s) ds (0 < η < 1) [38] . The Sumudu transform is derived from the classical Fourier integral ( [39] [40] [41] ). Consider the set

The Sumudu transform of a function f ∈ A is defined by

for all t ≥ 0, and the inverse Sumudu transform of F(u) is denoted by f (t) = ST -1 [F(u)] [40] . The Sumudu transform of the Caputo derivative is given by

where (m -1 < η ≤ m) [39] . Let F be a function such that its Caputo-Fabrizio fractional derivation exists. The Sumudu transform of F with Caputo-Fabrizio fractional derivative is defined by ST( CF 0 D η t )(F(t)) = M(η) 1-η+ηu [ST(F(t)) -F(0)] [42] .

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.

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.

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:

Proof From condition (7) and equation (6), we conclude that

In the same way, we can show that

The proof is complete.

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:

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.

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.

This iterative recursive is T-stable in L 1 (a, b) if the following conditions are achieved:

(1 + δω p f 12 (η) -(γ + m)f 13 (η)) < 1,

(1 + γ f 14 (η) + γ f 15 (η)mf 16 (η)) < 1,

(1 + μf 17 (η) + μ f 18 (η)εf 19 (η)) < 1.

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.

In this section, we apply the homotopy analysis transform method (HATM) to implement the fractional model (1) appropriately. Notice that HATM is a well-developed mixture of the standard Laplace transform technique [45] and the homotopy analysis method (HAM) [46] . To solve model (1) by HATM, first we apply the Laplace transform in the following way:

which results in

Then we get

Using the homotopy method, we define N 1 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q) = L Λmφ 1 (t; q)β p φ 1 (t; q)φ 3 (t; q) + κφ 4 (t; q))β w φ 1 (t; q)φ 6 (t; q) , N 2 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q)

= L β p φ 1 (t; q) φ 3 (t; q) + κφ 4 (t; q) + β w φ 1 (t; q)φ 6 (t; q)

-(1δ)ωφ 2 (t; q)δω φ 2 (t; q)mφ 2 (t; q) , N 3 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q)

= L (1δ)ωφ 2 (t; q) -(γ + m)φ 3 (t; q) , N 4 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q) = L δω p φ 2 (t; q)γ + m φ 4 (t; q) , N 5 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q) = L γ φ 3 (t; q) + γ φ 4 (t; q)mφ 5 (t; q) ,

Then the deformation equations become

= qhH(t)N 4 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q) ,

= qhH(t)N 5 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q) ,

(1q)L φ 6 (t; q) -W 0 (t) = qhH(t)N 6 φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q) ,

where q ∈ [0, 1] denotes an embedding parameter; φ i (t; q), i = 0, 1, are unknown functions; S 0 , E 0 , I 0 , A 0 , R 0 , W 0 are initial guesses; L[·] is the Laplace operator; H(t) = 0 is an auxiliary function, and h = 0 is a nonzero auxiliary parameter. Clearly, for q = 0 and q = 1, we have

φ 6 (t; 0) = W 0 (t), φ 6 (t; 1) = W (t).

Thus, increasing q from zero to one varies the solution (φ 1 (t; q), φ 2 (t; q), φ 3 (t; q), φ 4 (t; q), φ 5 (t; q), φ 6 (t; q)) from (S 0 (t), E 0 (t), I 0 (t), A 0 (t), R 0 (t), W 0 (t)) to (S(t), E(t), I(t), A(t), R(t), W (t)). Now, we expand φ i (t; q) (i = 1, 2, 3, 4, 5, 6) in the Taylor series with regard to q.

This procedure yields

If the auxiliary function H(t), the auxiliary parameter h, and the initial guesses are properly chosen, then series (16) converges at q = 1, as proved by Liao [46] . Thus, we get

In addition, we can express the mth order deformation equation by 

where

, 

,

and

Applying the inverse Laplace transform to equation (17), we obtain S n (t) = χ n S n-1 (t) + hHL -1 T 1,n (S n-1 ) , E n (t) = χ n E n-1 (t) + hHL -1 T 2,n (E n-1 ) , I n (t) = χ n I n-1 (t) + hHL -1 T 3,n (I n-1 ) , A n (t) = χ n A n-1 (t) + hHL -1 T 4,n (A n-1 ) , R n (t) = χ n R n-1 (t) + hHL -1 T 5,n (R n-1 ) , W n (t) = χ n W n-1 (t) + hHL -1 T 6,n (W n-1 ) .

Solving these equations for different values of n = 1, 2, 3, . . . , we derive

I 1 (t) = -hHρ 1-η (1 + α(t -1))((1δ)ωE 0 (t) -(γ + m)I 0 (t)) = -hHM 3 ρ 1-η (1 + α(t -1)),

= -hHM 4 ρ 1-η (1 + α(t -1)), R 1 (t) = -hHρ 1-η (1 + α(t -1))(γ I 0 (t) + γ A 0 (t) -mR 0 (t)) = -hHM 5 ρ 1-η (1 + α(t -1)), W 1 (t) = -hHρ 1-η (1 + α(t -1))(μI 0 (t) + μ A 0 (t) -εW 0 (t))

Finally, the solutions of system (1) are obtained as follows:

In the following, we discuss the convergence of HATM by presenting and proving the following theorem. 

, and W (t), respectively, where {S n (t), E n (t), I n (t), A n (t), R n (t), W n (t)} ∈ L(R + ) are produced by the mth order deformation (17) . Also, assume that ∞ n=0 ( CF D α t S n (t)), ∞ n=0 ( CF D α E n (t)), ∞ n=0 ( CF D α I n (t)), ∞ n=0 ( CF D α A n (t)), ∞ n=0 ( CF D α R n (t)), ∞ n=0 ( CF D α W n (t)) are convergent. Then S(t), E(t), I(t), A(t), R(t), W (t) are the exact solutions of system (15) .

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 + .

Since Laplace is a linear operator, we have Thus, from (19) and (20) Since h = 0, H = 0, this yields ∞ n=1 T 1,n ( S n-1 (t)) = 0. Similarly, we can prove β p S n-1 (t) I n-1 (t) + κA n-1 (t)β w S n-1 (t)W n-1 (t)

Therefore S(t) is the exact solution of system (15) . Similarly, we can prove that E(t), I(t), A(t), R(t), and W (t) are the exact solutions of system (15) , and the proof is complete.

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.

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-

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Research of the third author was supported by Azarbaijan Shahid Madani University. Also, research of the second author was supported by Miandoab Branch of Islamic Azad University. The authors are thankful to dear referees for the valuable comments which improved the final version of this work.

Not available.

Plots of the results of Caputo derivative and Caputo-Fabrizio derivative for S, E with η = 0.96 of COVID-19 is 14 days. Of course, the new Chinese study, which has yet to be peerreviewed, suggests that the incubation period for the virus could be as long as 24 days.Because the information is changing and due to the lack of complete information on many parameters related to the transmission of this virus, we had to consider some of the coefficients hypothetically. In this simulation, according to the news, we have chosen the parameters as β p = 0.0025, β w = 0.001, κ = 0.05, δ = 0.25, ω = 0.071, ω = 0.1, γ = 0.047, γ = 0.1, μ = 0.003, μ = 0.001, ε = 0.033, and the initial values are S 0 = 35, I 0 = 25, R 0 = 0, E 0 = 25, A 0 = 10, W 0 = 5.In Figures 1-3 , we show the three-term solution of homotopy analysis transform method (HATM) with the auxiliary parameter h = -1 and the auxiliary function H = 1 corresponding to proposed model (1) for different values of η and modification parameter ρ = 0.99. Figures 1 and 2 show that the number of susceptible and exposed people increases first with a birth rate of 1.146. And then, with COVID-19 infection, the population of these two groups declines, and the population of the symptomatic and asymptomatic infected people increases. Figure 3 shows that the population of the out-group, 

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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The authors declare that they have no competing interests.

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The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript. 

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Received: 8 March 2020 Accepted: 9 June 2020