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On the mathematical model of Rabies by using the fractional Caputo–Fabrizio derivative
Advances in Difference Equations volume 2020, Article number: 382 (2020)
Abstract
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 integerorder derivative and the Caputo and Caputo–Fabrizio fractionalorder 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 warmblooded 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 [2–4]. There are various mathematical models explaining the release of Rabies by ordinary and partial differential equations (see, e.g., [5–7]). The study of disease dynamics is a dominating theme for many biologists and mathematicians (see, e.g., [8–12]). It has been studied by many researchers that fractional extensions of mathematical models of integer order represent 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. [16–28], Ozdemir et al. [29–33], and Talaee et al. ([34]). Recently, many works related to the fractional Caputo–Fabrizio derivative [35] have been published (see, e.g., [8–12, 36–38], and [7]).
In this paper, we first study the integerorder Rabies model with the Caputo–Fabrizio fractionalorder 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 \(b>a\), \(f\in H^{1}(a,b)\), and \(\eta \in (0,1)\). The Caputo–Fabrizio derivative of order η for a function f is defined by
where \(M(\eta )\) is a normalization function such that \(M(0)=M(1)=1\). If \(f \notin H^{1}(a,b)\) and \(0<\eta <1\), then this derivative can be presented for \(f\in L^{1}(\infty ,b)\) as (see [39])
Let \(n\geq 1\) and \(\eta \in (0,1)\). The fractional derivative \(^{CF}D^{\eta +n}\) of order \({\eta +n}\) is defined by \(^{CF}D^{\eta +n} f(t):={ ^{CF}D^{\eta }(D^{n} f(t))}\) [14]. The Laplace transform of the Caputo–Fabrizio derivative is defined by \(L[^{CF}D^{(\eta +n)}f(t)](s)= \frac{s^{n+1}L[f(t)]s^{n}f(0)s^{n1}f'(0)\cdotsf^{(n)}(0)}{s+\eta (1s)}\), where \(0<\eta \leq 1\) and \(M(\eta )=1\) [39]). The Riemann–Liouville fractional integral of order η with \(\operatorname{Re} (\eta ) > 0\) 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 [40–42]. Consider the set \(A=\{F:\exists \lambda , k_{1} , k_{2} \geq 0,F(t)<\lambda \exp ( \frac{t}{k_{j}}),\quad t\in (1)^{j} \times [0,\infty )\}\). The Sumudu transform of a function \(f\in A\) is defined by
for all \(t\geq 0\), and the inverse Sumudu transform of \(F(u)\) is denoted by \(f(t)=ST^{1}[F(u)]\) [41]. The Sumudu transform of the Caputo derivative is given by
where \(m1< \eta \leq m\) [40]. Let F be a function such that its Caputo–Fabrizio fractional derivation exists. The Sumudu transform of F with Caputo–Fabrizio fractional derivative is defined by \(ST(^{CF}_{0}D^{\eta }_{t})(F(t))=\frac{M(\eta )}{1\eta +\eta u}[ST(F(t))F(0)]\) [43].
Let \((X,d)\) be a metric space. A map \(g:X\to X\) is called a Picard operator if there exists \(x^{*}\in X\) such that \(\operatorname{Fix}(g)=\{x^{*}\}\) and the sequence \((g^{n}(x_{0}))_{n\in N}\) converges to \(x^{*}\) for all \(x_{0}\in X\) ([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,
 \(d(N)\)::

the rate function of per capita death, which depends on the population size N,
 a::

the rate of vaccination,
 c::

the rate of Rabies transmission,
 e::

the mortality rate in disease.
The initial conditions are \(S(0)=S_{0}\), \(I(0)=I_{0}\), \(N(0)=N_{0}\). In this model, \(d(N)\) is a continuous increasing function on \(R^{+}\) such that \(d(m)=b\) for some positive constant m.
In model (1), we are going to replace the integerorder 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 righthand sides of the equations have dimension time^{−1}. When we change the order of the equations to η, the dimension of the lefthand side becomes time\(^{\eta }\). 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 \(A=a^{\eta }\), \(B=b^{\eta }\), \(C=c^{\eta }\), \(D(N)=d^{\eta }(N)\), \(E=e^{\eta }\), and \(Z=z^{\eta }\), 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 \(0<\eta \leq 1\), \(S(0)=S_{0}\), \(I(0)=I_{0}\), 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 righthand side of the equations to zero to determine the equilibrium points of the fractionalorder system (3):
By solving these algebraic equations we obtain the equilibrium points \(E^{0}=(\frac{(B+Z)M}{B+Z+A},0,M)\) and \(E^{*}=(S^{*},I^{*},N^{*})\) such that
Also, by the nextgeneration method [45] we obtain the basic reproduction number \(R_{0}=\frac{CM(B+Z)}{E(A+B+Z)}\). To investigate the stability of equilibrium point, we first consider the fractionalorder linear system
where \(v(t)\in R^{n}\), \(\psi \in R^{n\times n}\), and \(0<\eta <1\).
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 \(E^{0}=(\frac{(B+Z)M}{B+Z+A},0,M)\), we have \(N=M\), so that \(D(N)=D(M)=B\). The Jacobian matrix associated with system (3) in \(E^{0}\) is
and thus the characteristic equation is
The roots of the latter are as follows:
Theorem 3.1
If\(R_{0}<1\), then system (3) is stable at\(E^{0}\).
Proof
Since A, B, and C are positive and \(\eta \in (0,1)\), we easily see that \(\lambda _{2}<0\). By the assumption we have \(R_{0}=\frac{CM(B+Z)}{E(A+B+Z)}<1\). Hence \(CM(B+Z)E(A+B+Z)<0\), and so \(\eta (\frac{E(A+B+Z)+CM(B+Z)}{A+B+Z})<0\) and \(1(1\eta )(\frac{E(A+B+Z)+CM(B+Z)}{A+B+Z})>0\). This implies that \(\lambda _{3}<0\). Thus system (3) is stable at \(E^{0}\). □
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
Let \(b_{1}=ZDAC\I(t)\ \), where I is a bounded function: \(\ I(t)\\leq m_{1}\). So
So \(G_{1}\) satisfies the Lipschitz condition, and if \(0\leq (ZDAC\I(t)\) < 1 \), then \(G_{1}\) is a contraction. □
Similarly, \(G_{2}\) and \(G_{3}\) satisfy the Lipschitz conditions:
where \(b_{2}=BDE+C\S(t)\\) and \(b_{3}=BD\), and S is a bounded function (\(\ S(t)\ \leq m_{2}\)). If \(0\leq (BDE+C\S\) <1\) and \(0\leq BD <1\), 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:
With Lipschitz condition (8), we have
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
Hence
By repeating the method we obtain
At \(t_{1}\), we get
Taking the limit as n tends to ∞, we obtain \(\ \mathrm{B}_{1n}(t)\\rightarrow 0\). Similarly, we can show that \(\ \mathrm{B}_{2n}(t)\\rightarrow 0\) and \(\ \mathrm{B}_{3n}(t)\\rightarrow 0\), 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
So
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 fractionalorder model of Rabies by the Laplace Adomian decomposition method. Applying the Laplace transform to both sides of system (3), we have
and so
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
The first three polynomials are given by
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].
Theorem 5.1
LetΩbe a Banach space, and let\(T:\varOmega \to \varOmega \)be a contraction map with contractive constant\(0< h<1\), ThenThas a unique pointusuch that\(T(u)=u\), where\(u=(S,I,N)\). Let\(u_{0}=u_{0}\in F_{r}(u)\), where\(F_{r}(u)=\{u^{\prime } \in \varOmega :\ u^{\prime }u\< r\}\). Then we have\(u_{n}\in F_{r}(u)\)for allnand\(u_{n}\to u\).
Proof
We use mathematical induction. For \(n=1\), we have
If the statement is true for \(n1\), then \(\ u_{n1}u\\leq h^{n1}\ u_{0}u\\). So
Also, \(u_{0}\in F_{r}(u)\), and consequently \(\ u_{0}u\< r\), so we get \(\ u_{n}u\\leq h^{n}\ u_{0}u\\leq h^{n}r< r\). This implies that \(u_{n}\in F_{r}(u)\). Furthermore, since \(\ u_{n}u\\leq h^{n}\ u_{0}u\\) and \(\lim_{n\rightarrow \infty }h^{n}=0\), we have \(\lim_{n\rightarrow \infty }\ u_{n}u\=0\), and then \(\lim_{n\rightarrow \infty }u_{n}=u\), 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 CFderivative we obtain
Rearranging, we obtain following equalities:
We obtain the following recursive formulas:
Finally, the solution of system (3) is approximates as follows:
Stability analysis of iteration method
Consider a Banach space \((Y, \\cdot \)\), a selfmap F on Y, and a recursive method \(P_{n+1}=\phi (F, P_{n})\). Denote by \(\varOmega (F)\neq \emptyset \) the fixed point set of F, and let \(\lim_{n\rightarrow \infty }P_{n}=p\in \varOmega (F)\). Let \(\{f_{n}\}\subset \varOmega \) and \(e_{n}=\ f_{n+1}\phi (F, f_{n})\\). If \(\lim_{n\rightarrow \infty }e_{n}=0\) implies that \(\lim_{n\rightarrow \infty }f_{n}=p\), then the recursive procedure \(P_{n+1}=\phi (F,P_{n})\) is Fstable. 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 Fstable.
Theorem 6.1
([44])
Let\((Y , \\cdot \)\)be a Banach space, and letFbe a selfmap of Ysatisfying
for all\(x,y\in Y\)where\(H\geq 0\)and\(0\leq h < 1\). Suppose thatFis PicardFstable.
Suppose that the fractional model of Rabies (3) is connected with the subsequent iterative formula (15). Consider the following theorem.
Theorem 6.2
LetFbe a selfmap defined as follows:
This iterative recursion isFstable in\(L^{1}(a,b)\)if the following conditions are satisfied:
Proof
First, we compute the following equality for \((n,m)\in N\times N\) 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) we obtain
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 \(f_{i}\) are functions from \(ST^{1}[\frac{1\eta +\eta u}{M(\eta )} ST[*]]\). In the same way, we get
where
Then the selfmapping F has a fixed point. Let (20), (21), and (22) hold. We assume that
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 \(b=0.083\), \(c=0.1\), \(e=0.5\), \(a=0\), \(z=0\), \(d=0.01+0.004 N\) and the initial conditions \(S_{0}=9\), \(I_{0}=1\), \(N_{0}=10\) (see [5]). In Figs. 1 and 2, we show the solutions of model (4) for different values of \(\eta =1, 0.9, 0.8, 0.7, 0.6, 0.5\). The figures show that different fractional orders have an important effect on the dynamics of the system and indicate that as \(\eta \rightarrow 1\), the approximate solutions tend to the classic integer solution with \(\eta =1\). Also, figures show that every function has the same behavior for different values of η, but the obtained results for various values of η are different.
Figures 3–7 show the effect of coefficients on the number of infected animals \(I(t)\). In Fig. 3, we plotted \(I(t)\) for different values of b, and we see that increasing b increases the number of infected animals and decreasing b reduces the number of infected animals; \(I(t)\) for different values of z is plotted in 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 nonintegerorder model with \(\eta =0.98\) and the integer order \(\eta =1\) 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 fractionalorder Caputo derivative is closer to the result of integer order, and at other points the result of the fractionalorder 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 Fstability 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 integerorder derivative with the Caputo and Caputo–Fabrizio fractionalorder 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 integerorder derivative, and in some cases the results of the Caputo–Fabrizio derivative are closer to those of integerorder derivative. Therefore, with this information, it is not possible to conclude which type of fractional derivatives is better.
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Acknowledgements
Research of the fourth author was supported by Azarbaijan Shahid Madani University. Also, research of the third author was supported by Miandoab Branch, Islamic Azad University. The authors are thankful to dear referees for their valuable comments which basically improved the final version of this work.
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Aydogan, S.M., Baleanu, D., Mohammadi, H. et al. On the mathematical model of Rabies by using the fractional Caputo–Fabrizio derivative. Adv Differ Equ 2020, 382 (2020). https://doi.org/10.1186/s13662020027984
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MSC
 34A08
 35A22
 35A20
Keywords
 Adomian decomposition method
 Fixed point
 Numerical simulation
 Rabies model
 The Caputo–Fabrizio fractional derivative