Fractional Order Seirs Model

In this paper, we introduce a fractional order SEIRS model. The local asymptotic stability of equilibrium points is studied. We utilized the Adams-Bashforth predictor corrector method for solving the proposed model. Numerical simulations are presented to show the advantage of introducing a fractional model.


Introduction
Mathematical modeling of the spread of infectious diseases continues to be an area of active research.It has become an important tool in understanding the dynamics of diseases and in decision making processes regarding intervention programs for controlling these diseases in many countries.We replace the real world phenomena of the disease by an abstract model in order to employ the tools of mathematical analysis.This model typically takes a mathematical formulation that involves basic variables and relationships corresponding to the entities and the laws of nature or behavior being observed.It usually takes the form of non-linear ordinary differential equations.The subsequent solution is in the mathematical form and can be reinterpreted back in the original real world setting.Mathematical modeling of complex biological processes is a major challenge for contemporary scientists.The complex systems are characterized by the variability of structures in them, multiscale behavior and nonlinearity in the mathematical description of the mutual relationship between parameters [1].Therefore, the theory of integer-order differential equations is not sufficient to describe their dynamics.In recent years, fractional differential equations become more popular because of its powerful potential applications.A large number of new differential equations models that involve fractional calculus are developed.There have found a number of works, especially in mechanics, biology, chemistry, electrical engineering [2,3,4,5,6].In the literature, a number of methods have been developed for the numerical or analytical solutions for fractional differential equations.We listed some of these methods as follows: Adomian decomposition method [7], the collocation method [8], the fractional differential transform method [9,10], homotopy analysis method [11,12], homotopy perturbation method [13].In biology, it has been deduced that the membranes of cells of biological organism have fractional-order electrical conductance and then are classified in groups of non-integer order models.Fractional derivatives embody essential features of the behavior of the pattern formation in bacterial colonies.Also, it has been shown that modeling the behavior of brainstem vestibule-oculumotor neurons by fractional ordinary differential equations has more advantages than classical integer-order modeling.Fractional derivatives are naturally related to systems with memory which exists in most biological systems [14].For more details about using fractional calculus in modeling complex biomaterials, we refer the reader to [1] and the references therein.Adam Bashforth predictor corrector method has been proposed in [15,16] for solving fractional differential equations.The method can be used for linear as well as nonlinear fractional differential equations.In the present work we use it for the numerical simulation of biological system which is fractional order.We introduce the Caputo derivative of orderα in SEIRS model which was discussed in [17] and we discuss the oscillatory behavior of the endemic level of infected population.To the best of our knowledge, this work represents the first numerical solution for disease model of fractional order by applying the Adams-Bashforth-Moulton method.The rest of the paper is organized as follows.In section 2, some necessary definition and notations related to fractional calculus are presented.The formulation of the model and stability of equilibrium points is discussed in section 3. The general procedure for implementation of the predictor corrector method for the fractional model is discussed in section 4 and the numerical results are presented graphically in section 5. Finally, concluding remarks are given in section 6.

Preliminaries
First, we review some basic definitions of fractional differentiation and fractional integration [3].

Riemann-Liouville fractional integral operator of orderα
The operator ) )

Mathematical Model
The integer order model which is reported in [17] is given by > In this paper, we discuss the model consisting of fractional system of equations which is obtained by just replacing an integer order derivative by a fractional derivative of order 0 1.
α < ≤ Thus, the fractional order system is given by

Asymptotic stability of Equilibrium Points
To calculate the equilibrium points, we put the right hand side of system (2) equal to zero and obtain 0, 1

The Adams-Bashforth-Moulton Method
We utilize Adams-Bashforth-Moulton predictor corrector method [15] for solving system of fractional order differential equations.The implementation of the adams-Bashforth-Moulton method according to the concerned fractional system is as follows: ( ) ( , , , , ) ( ) ( , , , , ), T Implementation of the Adams-Bashforth Moulton method [15] on the proposed model are as follows:

D x t g t x y z u D y t g t x y z u D z t g t x y z u D u t g t x y z u
Predictor values for (5) are n j j n j j j j j h a g t x y z u  Fractional order SEIRS model , ( , , , , ) , ( , , , , ) , ( , , , , ) According to the mathematical analysis of this method in [16] we have order of accuracy``" p where min (2,1 ).p α = +

Numerical results
In this section, we shall discuss the oscillatory behavior of the endemic level of infected population of model (2), on the basis of the numerical results which are obtained by using the parameter values given in Table 1.It is clear from figure 1 that when the recovered individuals lose their infection-acquired immunity and in the absence of disease related death, there are damped oscillations of the infected populations for 1.
α = The endemic equilibrium level is eventually attained and this converges to a steady state that is asymptotically stable.The oscillations decrease for fractional values ofα and we obtain the endemic level at the early stage as compared to the integer order.Similar behavior has been observed when we suppose permanent immunity and disease related death rate of infected individuals.This behavior of infected population is shown in figure 2.
By using the condition (4) and the parameter values given in[17], the corresponding approximate equilibrium values of , , S E I

h
= where T is the upper bound of the interval[0, ].
Corrector values are obtained by using Predictor values as

Figure1
Figure1.Oscillatory behavior of infected population when

Table 1 :
Equilibrium Values for different θ and δ .