A new mathematical model for Zika virus transmission

We present a new mathematical model for the transmission of Zika virus between humans as well as between humans and mosquitoes. In this way, we use the fractional-order Caputo derivative. The region of the feasibility of system and equilibrium points are calculated, and the stability of equilibrium point is investigated. We prove the existence of a unique solution for the model by using the fixed point theory. By using the fractional Euler method, we get an approximate solution to the model. Numerical results are presented to investigate the effect of fractional derivative on the behavior of functions and also to compare the integer-order derivative and fractional-order derivative results.

In mathematical models of Zika virus transmission it is assumed that the virus is usually transmitted from mosquitoes to humans, while according to WHO, in addition to the transmission through mosquitoes, Zika virus is transmitted through infected blood as well as through sexual contact with an infected person. In this article, we consider a mathematical model based on both ways of transmitting the virus. Also, according to the good results of fractional-order derivative in the modeling of real-world phenomena in recent years, we use Caputo fractional-order derivative instead of the integer-order derivative in this model.
The structure of the paper is as follows. In Sect. 2 some basic definitions and concepts of fractional calculus are recalled. The transmission model of Zika virus with fractionalorder derivative is presented in Sect. 3, and the equilibrium points and the reproduction number are calculated. The existence and uniqueness of solution for the system are proved in Sect. 4. Numerical method and numerical results are presented in Sect. 5.

Preliminaries
In this section, we recall some basic concepts of fractional differential calculus. Definition 2.1 ([44]) For an integrable function g, the Caputo derivative of fractional order ν ∈ (0, 1) is given by Also, the corresponding fractional integral of order ν with Re(ν) > 0 is given by 45,46]) For g ∈ H 1 (c, d) and d > c, the Caputo-Fabrizio derivative of fractional order ν ∈ (0, 1) for g is given by where t ≥ 0, M(ν) is a normalization function that depends on ν and M(0) = M(1) = 1. If g / ∈ H 1 (c, d) and 0 < ν < 1, this derivative for g ∈ L 1 (-∞, d) is given by Also, the corresponding CF fractional integral is presented by The Laplace transform is one of the important tools in solving differential equations that are defined below for two kinds of fractional derivative.

Definition 2.3 ([44])
The Laplace transform of Caputo fractional differential operator of order ν is given by which can also be obtained in the form

Model formulation
In this section, we provide a mathematical model for the transmission of Zika virus using with the initial conditions S h (0) = S 0h , I h (0) = I 0h , S m (0) = S 0m , I m (0) = I 0m .
The model parameters are: the recruitment rate of human population h , the recruitment rate of mosquito population m , the effective contact rate human to human β 1 , the effective contact rate mosquitoes to human β 2 , the effective contact rate human to mosquitoes μ, the natural death rate of human k 1 , the natural death rate of mosquitoes k 2 .
Model (1) does not include the internal memory effects of the system. To improve the model, we change the first-order time derivative to the Caputo fractional derivative of order ν. With this change, the right-and left-hand sides will not have the same dimension.
To solve this problem, we use an auxiliary parameter θ , having the dimension of sec., to change the fractional operator so that the sides have the same dimension ( [47,48]). According to the explanation presented, the transmission model of Zika virus for t ≥ 0 and ν ∈ (0, 1) is given as follows: where the initial conditions are S h (0) = S 0h , I h (0) = I 0h , S m (0) = S 0m , I m (0) = I 0m .

Nonnegative solution
we show that the closed set is the region of the feasibility of system (2).

Lemma 3.1 The closed set is positively invariant with respect to fractional system (2).
Proof To obtain the fractional derivative of the total population, we add the first two relations in system (2). So where N h (t) = S h (t) + I h (t). Using the Laplace transform, we obtain where N h (0) is the initial human population size, and the terms E ν , E ν,ν in the above equation are represented by the Mittag-Leffler function and its general form defined by , ν > 0.
With some calculations, we get Consequently, the closed set is positively invariant with respect to fractional model (2).

Equilibrium points and reproduction number
To determine the equilibrium points of fractional order system (2), we solve the following equations: By solving the above algebraic equations, we obtain two equilibrium points of system (2). The disease-free equilibrium point is obtained as E 0 = ( h k 1 , 0, m k 2 ). In addition, if R 0 > 1, then system (2) has a positive endemic equilibrium point , Also, R 0 is the basic reproduction number and is obtained using the next generation method [49]. To find R 0 , we first consider the system as follows: At E 0 , the Jacobian matrix for F and V is obtained as follows: FV -1 is the next generation matrix for the system (2), and the basic reproduction number is obtained from R 0 = ρ(FV -1 ), where ρ(FV -1 ) is the eigenvalue of matrix FV -1 . We get This basic reproduction number R 0 is an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents.

Stability of equilibrium point
To investigate the stability of the equilibrium point, we first consider the Jacobian matrix of system (2) as follows: At E 0 , the Jacobian matrix of system (2) is (2) is locally asymptotically stable.
Proof At the disease-free equilibrium point E 0 , the characteristic equation of the Jacobian matrix is det(λI -J(E 0 )) = 0. Then we obtain . By simplifying the above equations, the eigenvalues of characteristic equation are obtained as λ 1 = -k 1 , λ 2 = -k 2 and the roots of the equation If R 0 < 1, since all of the parameters are positive, then Also, from R 0 < 1 we have Since B > 0, C > 0, applying the Routh-Hurwitz criteria, we obtain that E 0 is locally asymptotically stable.

Existence and uniqueness of solution
To show that the system has a unique solution, we write system (2) as follows: By applying integral on both sides of the above equations, we have We show that the kernels W i , i = 1, 2, 3, 4, satisfy the Lipschitz condition and contraction.

Theorem 4.1 The kernel W 1 satisfies the Lipschitz condition and contraction if the following inequality holds:
Proof For S h and S 1h , we have Suppose that M 1 = β 1 u 1 + β 2 u 2 + k 1 , where I h (t) ≤ u 1 , I m ≤ u 2 are bounded functions, then Thus, for W 1 , the Lipschitz condition is obtained, and if 0 ≤ β 1 u 1 + β 2 u 2 + k 1 < 1 then W 1 is a contraction.
Similarly, we can prove that W i , i = 2, 3, 4, satisfies the Lipschitz condition as follows: According to system (3), consider the following recursive forms: with the initial conditions S 0h (t) = S h (0), I 0h (t) = I h (0), S 0m (t) = S m (0), and I 0m (t) = I m (0). We take the norm of the first equation in the above system, then By Lipschitz condition (4), we have In a similar way, we obtained Then we can obtain We prove the existence of a solution in the next theorem.

Theorem 4.2 The fractional model of Zika virus (2) has a solution if there exists t 1 such that
Proof From the recursive technique and Eq. (5) and Eq. (6), we conclude that Then the system has a solution, and also it is continuous. Now we show that the above functions construct a solution for model (2). We assume that By repeating the method, we obtain At t 1 , we get Taking limit on recent equation as n approaches ∞, we obtain B 1n (t) → 0. In the same way, we can show that B in (t) → 0, i = 2, 3, 4. This completes the proof.
In the following, we show that system (2) has a unique solution. We suppose that the system has another solution such as S 1h (t), I 1h (t), S 1m (t), and I 1m (t), then we have By taking the norm from this equation, we obtain It follows from Lipschitz condition (4) that Then

Theorem 4.3 The solution of the transmission model of Zika virus is unique if the following condition holds:
1 - Proof Suppose that condition (7) holds Then S h (t) -S 1h (t) = 0. So, we obtain S h (t) = S 1h (t). Similarly, we can show the same equality for I h , S m , I m .

Numerical results
Using the fractional Euler method for Caputo derivative, we present the approximate solutions for the transmission model of Zika virus [50]. We present simulations for investigating the dynamics of the system.

Numerical method
We consider system (2) in the compact form as follows: where u = (S h , I h , S m , I m ) ∈ R 4 + , u 0 = (S 0h , I 0h , S 0m , I 0m ) is the initial vector, and p(t) ∈ R is a continuous vector function satisfying the Lipschitz condition p u 1 (t)p u 2 (t) ≤ r u 1 (t)u 2 (t) , r > 0.
The stability analysis of the obtained scheme has been proved in Theorem (3.1) in [50].
Thus, the solution of system (2) is written as follows:

Simulation
In this section, using numerical results, we investigate the behavior of the answers of the transmission model of Zika virus obtained from system (2). The numerical values of the model parameters are considered as h = 1.2, m = 0.3, k 1 = 0.004, k 2 = 0.0014, β 1 = 0.125 × 10 -4 , β 2 = 0.4 × 10 -4 , μ = 0.475 × 10 -5 , and we take its modification parameter as θ = 0.99. Also, the initial values are considered as S h (0) = 800, I h (0) = 200, S m = 600, I m = 300. Figure 1 shows susceptible people S h and Fig. 2 shows infected people I h for the integerorder derivative ν = 1 and fractional-order derivative ν = 0.98, 0.96, 0.94, 0.92, .09. As Fig. 1 shows, the behavior of S h in both types of integer-order and fractional-order derivative is the same and decreasing, that is, over time, all healthy people are exposed to the disease, but the obtained numerical values are different, and as the derivative order decreases, the resulting numerical value increases.
In Fig. 2, you can see that the behavior of I h is the same in both derivatives, and the resulting numerical values are different. As the derivative order decreases, the resulting numerical value for I h increases, and this difference in the obtained value is significant  Figure 2 also shows that I h passes the peak in the first 100 days and the number of infected people gradually decreases and tends to the equilibrium point. Figures 3 and 4 show susceptible mosquitoes S m and infected mosquitoes I m , respectively. In these diagrams, you can see that the behavior of the functions is the same in both derivatives and the resulting numerical values are different. These figures also show that over time the population of healthy mosquitoes decreases and they are more exposed to the disease, while the number of infected mosquitoes increases.

Conclusion
In this paper, a mathematical model for the transmission of Zika virus between humans and mosquitoes is presented using the Caputo fractional-order derivative. The region of the feasibility of system (2), the equilibrium points, and the reproduction number have been determined, and the stability of the equilibrium point E 0 has been checked. Using a fixed point theory, the existence of a unique solution for model (2) has been proven. In the numerical section, the answers of system (2) are calculated using the Euler method, and the results are compared for the integer-order model and the fractional-order model in numerical results. The results show that the behavior of the obtained functions in both types of derivatives is the same, but the resulting numerical values are different, especially the difference in values increases over time.