On the mathematical model of Rabies by using the fractional Caputo–Fabrizio derivative

Using the fractional Caputo–Fabrizio derivative, we investigate a new version of the mathematical model of Rabies disease. Using fixed point results, we prove the existence of a unique solution. We calculate the equilibrium points and check the stability of solutions. We solve the equation by combining the Laplace transform and Adomian decomposition method. In numerical results, we investigate the effect of coefficients on the number of infected groups. We also examine the effect of derivation orders on the behavior of functions and make a comparison between the results of the integer-order derivative and the Caputo and Caputo–Fabrizio fractional-order derivatives.


Introduction
The Rabies viral disease, which is characterized by the initial symptoms of fever and tingling at the site of exposure, causes inflammation of the brain in humans and other mammals. Another symptom of this disease is the inability to move parts of the body, harsh movements, uncontrolled excitement, fear of water, confusion, and loss of consciousness. After catching the disease, it usually takes one to three months for the initial symptoms to appear; at the same time, this period can take less than a week or more than a year, It depends on the distance the virus takes during the peripheral nerves to reach the central nervous system; however, this disease can often cause death [1]. Although it is possible to prevent Rabies, it still exists in groups of warm-blooded animals in more than 150 countries, and more than 55,000 people die of the disease each year. Rabies is still one of the most important issues in the field of human health and wildlife management [2][3][4]. There are various mathematical models explaining the release of Rabies by ordinary and partial differential equations (see, e.g., [5][6][7]). The study of disease dynamics is a dominating theme for many biologists and mathematicians (see, e.g., [8][9][10][11][12]). It has been studied by many researchers that fractional extensions of mathematical models of integer order rep-resent the natural fact in a very systematic way such as in the approaches of Akbari et al. [13], Aydogan et al. [14,15], Baleanu et al. [16][17][18][19][20][21][22][23][24][25][26][27][28], Ozdemir et al. [29][30][31][32][33], and Talaee et al. ( [34]). Recently, many works related to the fractional Caputo-Fabrizio derivative [35] have been published (see, e.g., [8][9][10][11][12][36][37][38], and [7]).
In this paper, we first study the integer-order Rabies model with the Caputo-Fabrizio fractional-order derivative and investigate the behavior of the results obtained from the model. We will also investigate the effect of the coefficients in the model on the number of infected animals. Now we recall some fundamental notions. The Riemann-Liouville derivative of order η for a function f is given by The Caputo fractional derivative of order η for a function f via integrable differentiations is defined by Our second notion is a fractional derivative without singular kernel, introduced by Caputo and Fabrizio [35] in 2015. 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 M(η) is a normalization function such that M(0) = M(1) = 1. If f / ∈ H 1 (a, b) and 0 < η < 1, then this derivative can be presented for f ∈ L 1 (-∞, b) as (see [39]) Let n ≥ 1 and η ∈ (0, 1). The fractional derivative CF D η+n of order η + n is defined by [14]. The Laplace transform of the Caputo-Fabrizio deriva- s+η (1-s) , where 0 < η ≤ 1 and M(η) = 1 [39]). The Riemann-Liouville fractional integral of order η with Re(η) > 0 is defined by (see [14]) The Caputo-Fabrizio fractional integral is defined by (see [39]) The Sumudu transform is derived from the classical Fourier integral [40][41][42]. Consider the set A = {F : ∃λ, k 1 , for all t ≥ 0, and the inverse Sumudu transform of F(u) is denoted by [41]. The Sumudu transform of the Caputo derivative is given by where m -1 < η ≤ m [40]. Let F be a function such that its Caputo-Fabrizio fractional derivation exists. The Sumudu transform of F with Caputo-Fabrizio fractional derivative [43]. Let (X, d) be a metric space. A map g : X → X is called a Picard operator if there exists x * ∈ X such that Fix(g) = {x * } and the sequence (g n (x 0 )) n∈N converges to x * for all x 0 ∈ X ( [44]).

Mathematical model of Rabies with a fractional derivative involving a nonsingular kernel
In 2014, Demirci [5] has investigated a model explaining the spread of Rabies with an integer order. In this section, we consider this model with a new approach of fractionalorder derivative. In this SIR model the total population of animals N at time t is divided into three groups, susceptible animals S(t), infected animals I(t), and recovered animals R(t). Rabies is a fatal disease: the animals annually die from this disease, and the only way to control the mortality caused by this disease is vaccination. The desired model is as follows: where the parameters have the following meanings: b: the rate of per capita birth, z: waning immunity, In model (1), we are going to replace the integer-order derivative with the fractionalorder derivative introduced by Caputo and Fabrizio [35] as follows: with initial conditions S(0) = S 0 , I(0) = I 0 , and N(0) = N 0 . In system (2) the right-hand sides of the equations have dimension time -1 . When we change the order of the equations to η, the dimension of the left-hand side becomes time -η . In order to make the dimensions match, we should change the dimensions of the parameters, and the system we obtain eventually is For simplicity, we consider A = a η , B = b η , C = c η , D(N) = d η (N), E = e η , and Z = z η , and we use D instead of D(N). We also assume that there exists a positive constant M such that D(M) = B. Then where 0 < η ≤ 1, S(0) = S 0 , I(0) = I 0 , and N(0) = N 0 . In the next section, we show that system (3) has a unique solution.

Equilibrium points of the model and asymptotic stability
We set the right-hand side of the equations to zero to determine the equilibrium points of the fractional-order system (3): By solving these algebraic equations we obtain the equilibrium points E 0 = ( (B+Z)M B+Z+A , 0, M) and E * = (S * , I * , N * ) such that Also, by the next-generation method [45] we obtain the basic reproduction number To investigate the stability of equilibrium point, we first consider the fractional- where v(t) ∈ R n , ψ ∈ R n×n , and 0 < η < 1.
Definition 3.1 [46] For system (4) with Caputo-Fabrizio fractional derivative, the characteristic equation is given by The Jacobian matrix associated with system (3) is and thus the characteristic equation of system (3) is Since and thus the characteristic equation is The roots of the latter are as follows: .
Proof Since A, B, and C are positive and η ∈ (0, 1), we easily see that λ 2 < 0. By the

Existence and uniqueness
In this section, we prove that the system has a unique solution. For this purpose, we write system (3) as follows: Using the integral operator introduced by Losada and Nieto [39] on both sides of the equations in the system, we obtain First, we show that the kernels G 1 , G 2 , G 3 satisfy the Lipschitz condition and are contractions.

Theorem 4.1 The kernel G 1 satisfies the Lipschitz condition and is a contraction if
Proof For S and S 1 , we have So G 1 satisfies the Lipschitz condition, and if 0 ≤ (-Z -D -A -C I(t) ) < 1, then G 1 is a contraction.
Similarly, G 2 and G 3 satisfy the Lipschitz conditions: then G 2 and G 3 are contractions.
According to system (7), consider the following recursive forms: with initial conditions S 0 (t) = S(0), I 0 (t) = I(0), and N 0 (t) = N(0). We take the norm of the first equation in the system: Similarly, we obtain Then we can write that To prove the existence of a solution, we state the following theorem.

Theorem 4.2 The Rabies Caputo-Fabrizio fractional model (3) has a solution if there
exists t 1 such that Proof From recursive technique, by Eqs. (9) and (10) we conclude that Then the system has a solution, and also it is continuous. Now we show that the above functions give a solution for model (3). Let , By repeating the method we obtain At t 1 , we get Taking the limit as n tends to ∞, we obtain B 1n (t) → 0. Similarly, we can show that B 2n (t) → 0 and B 3n (t) → 0, and the proof is complete.
To show the uniqueness of a solution, suppose that the system has another solution S 1 (t), I 1 (t), and N 1 (t). Then we have Taking the norm, we have From Lipschitz condition (8) it follows that

Theorem 4.3 The solution of the Rabies model is unique if
Proof Suppose that condition (11) holds. Then S(t) -S 1 (t) = 0, and so S(t) = S 1 (t). Similarly, we can show the same equality for I and N .

Solution of equations by Laplace Adomian decomposition method
In this section, we solve the fractional-order model of Rabies by the Laplace Adomian decomposition method. Applying the Laplace transform to both sides of system (3), we have Using the initial conditions and taking the inverse Laplace transform of these equations, we obtain Suppose that the solutions S, I, N can be written in the form of infinite series and that the nonlinear term SI involved in the model can be decomposed by Adomian polynomials as Recall that the Adomian polynomials P n are defined as P n (t) = 1 Γ (n + 1) The first three polynomials are given by ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ P 0 (t) = S 0 (t)I 0 (t), P 1 (t) = S 0 (t)I 1 (t) + S 1 (t)I 0 (t), P 2 (t) = 2S 0 (t)I 2 (t) + 2S 1 (t)I 1 (t) + 2S 2 (t)I 0 (t).
Using (13), (14), and (15) in model (12), we get   By matching the two sides of equations we obtain Taking the Laplace inverse on both sides of the previous equations, we have , Similarly, we can obtain the remaining terms. Finally, we get the solution in the form of infinite series given by

Convergence analysis
The series obtained from the Adomian method converge rapidly and uniformly to the exact solution of the system. For this purpose, we use the classical method described in [38]. The series of solutions obtained by the Adomian decomposition method (16) can be described as follows (see [47]): We state the convergence conditions of {u n } in the following theorem [38,48,49]. Proof We use mathematical induction. For n = 1, we have If the statement is true for n -1, then u n-1u ≤ h n-1 u 0u . So Also, u 0 ∈ F r (u), and consequently u 0u < r, so we get u nu ≤ h n u 0u ≤ h n r < r. This implies that u n ∈ F r (u). Furthermore, since u nu ≤ h n u 0u and lim n→∞ h n = 0, we have lim n→∞ u nu = 0, and then lim n→∞ u n = u, and the proof is complete.

Stability of solution
In this section, we first introduce a special solution to the Rabies model by using the Sumudu transform and then prove the stability of the method by a fixed point theorem.

Special solution by using Sumudu transform
Here we provide a special solution to the Rabies model. Applying the Sumudu transform to system (3), we get By the definition of the Sumudu transform of CF-derivative we obtain

Stability analysis of iteration method
Consider a Banach space (Y , · ), a self-map F on Y , and a recursive method P n+1 = φ(F, P n ). Denote by Ω(F) = ∅ the fixed point set of F, and let lim n→∞ P n = p ∈ Ω(F). Let {f n } ⊂ Ω and e n = f n+1φ(F, f n ) . If lim n→∞ e n = 0 implies that lim n→∞ f n = p, then the recursive procedure P n+1 = φ(F, P n ) is F-stable. Suppose that our sequence {f n } has an upper bound. If Picard's iteration P n+1 = FP n satisfies all these conditions, then it is F-stable.
Theorem 6.1 ([44]) Let (Y , · ) be a Banach space, and let F be a self-map of Y satisfying Suppose that the fractional model of Rabies (3) is connected with the subsequent iterative formula (15). Consider the following theorem.
Proof First, we compute the following equality for (n, m) ∈ N × N to prove that F has a fixed point: Applying the norm on both sides, we obtain Because of the same role of both solutions, we consider From Eqs. (16) and (17) Since S n , I n , N n are convergent sequences, they are bounded, so there exist L 1 , L 2 , L 3 for all t such that From Eqs. (18) and (19) we obtain where In the same way, we get where (1 + (B -D -E)f 6 (ν) + CL 2 f 7 (ν) + CL 1 f 8 (ν)) < 1, (1 + (B -D)f 9 (ν) -Ef 10 (ν)) < 1.
Then the self-mapping F has a fixed point. Let (20), (21), and (22)  Then all conditions of Theorem 5.1 hold, and this completes the proof.

Numerical results
In this section, we give a numerical simulation of the results of solving the Rabies model (3) by using the Laplace Adomian decomposition method (LADM). This method provides the solution of the system in the form of infinite series. We have chosen the parameters b = 0.083, c = 0.1, e = 0.5, a = 0, z = 0, d = 0.01 + 0.004N and the initial conditions S 0 = 9, I 0 = 1, N 0 = 10 (see [5]). In Figs. 1 and 2 Fig. 4, and we see that increasing z reduces the number of infected animals. Figure 5 shows the effect of Rabies transmission on the number of infected animals, and as w can see, c has a direct effect on I(t), and a slight increase or decrease causes a significant increase or decrease in the number of infected animals. Also, one way to control the disease is vaccination, which affects the number of infected animals as shown in Fig. 6. With the increase in vaccination rate a, the number of infected animals rapidly decreases. The disease mortality rate also has a direct effect on the number of infected animals, and as Fig. 7 shows, increasing or decreasing e causes a similar behavior of I(t).
A comparison between the noninteger-order model with η = 0.98 and the integer order η = 1 is also given in Tables 1-3. The results of all three types of derivatives for S, I, N shown in Tables 1-3 indicate the similar behavior of functions in all three types of derivatives, but they are different in terms of values, and this difference is not uniform. At some points the value derived from the fractional-order Caputo derivative is closer to the result of integer order, and at other points the result of the fractional-order Caputo-Fabrizio derivative is closer to that of integer order.

Conclusion
In this work, we have studied the mathematical model of Rabies by the concept of Caputo-Fabrizio fractional derivative. We solved the fractional differential equations by the Laplace Adomian decomposition method. The equilibrium points are calculated, and the existence and uniqueness of the solutions are studied by fixed point theorem. The stability of the method was investigated with the F-stability approach. Eventually, some numerical results are presented for different values of η to show the effect of the fractional order. The effect of the model coefficients on the number of infected animals I(t) has been investigated using figures, so that we observed that b, c, and e have a direct effect on I(t), whereas a and z have an inverse effect on I(t). Finally, we compared the results of the integer-order derivative with the Caputo and Caputo-Fabrizio fractional-order derivatives. We observed that the behavior of S(t), I(t), N(t) in all three types of derivatives is the same, but the resulting numerical values are different. Also, in some cases the results of the Caputo derivative are closer to those of integer-order derivative, and in some cases the results of the Caputo-Fabrizio derivative are closer to those of integer-order derivative. Therefore, with this information, it is not possible to conclude which type of fractional derivatives is better.