Numerical solution of stochastic SIR model by Bernstein polynomials

In this paper, we present numerical method based on Bernstein polynomials for solving the stochastic SIR model. By use of Bernstein operational matrix and its stochastic operational matrix we convert stochastic SIR model to a nonlinear system that can be solved by Newton method. Finally, a test problem of SIR model is presented to illustrate our mathematical findings.


Introduction
In modeling the spread process of infectious diseases, many classical epidemic models have been proposed and studied, such as SIR, SEIR and SIRS models.The SIR infections disease model is an important biologic model and has been studied by many authors [1][2][3][4][5][6][7][8][9][10].In 1927, W. O. Kermack and A. G. Mckendrick created a model in which they considered a fixed population with only three compartments: susceptible, S(t); infected, I(t); and removed, R(t).The compartments used for this model consist of three classes; S(t) is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease, I(t) denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category, R(t) is the compartment used for those individuals who have been infected and then removed from the disease, either due to immunization or due to death.Those in this category are not able to be infected again or to transmit the infection to others.One of the most basic SIR models is as follows: where the parameter Ʌ, , , ,  are positive constants and S(t), I(t), R(t) denote the number of the individuals susceptible to the disease, of infected members and of members who have been removed from the possibility of infection through full immunity, respectively Λ is assumed that the natural death are assumed to be equal (denoted by constant µ); β is the daily contact rate;  represent the disease transmission coefficient and the http://www.ispacs.com/journals/jiasc/2016/jiasc-00102/ International Scientific Publications and Consulting Services rate of recovery from infection.Here we assume that stochastic perturbations are of a white noise type which by this way, above model will be deduced to the form { () = (Ʌ − ()() − () +  1 () 1 (), () = (()() + ( +  + )() +  2 () 2 (), () = (() − () +  3 () 3 (), (1.2) where   () are independent standard Brownian motions and   2 ≥ 0 represent the intensities of   (),  = 1, 2, 3.
The objective of this study is use of Bernstein polynomials for numerical solution of SIR model that has been presented in eq.(1.2).Bernstein polynomials and their operational matrix have been frequently used in the solution of integral equations, differential equations and approximation theory [11].This paper is organized as follow: In the next section we review Bernstein polynomials.Section 3, introduces stochastic integration operational matrix related to Bernstein polynomials.In section 4, Bernstein polynomial is applied to solve SIR model.In section 5, numerical results are shown.Finally section 6 provides the conclusion.

Product operational matrix
It is always necessary to evaluate the product of φ(x) and φ(x) T , that is called the product matrix of Bernstein polynomials basis.2), by integration we will have the following equations: By substituting above approximation in system (4.3)we get By replacing S T φ(t)φ T (t) with S T I ̂φ(t), we have: where, X ̂= diag(X), S T I ̂P are 2m-vectors.By replacing ≅ with =, we have After solving this nonlinear system (4.6), we find S, I, R and finally S(t), I(t), R(t) are approximated.http://www.ispacs.com/journals/jiasc/2016/jiasc-00102/ International Scientific Publications and Consulting Services

Conclusion
This paper suggested a numerical method for solving SIR model by using Bernstein polynomials and their operational matrices, also we derived and used the stochastic operational matrix of Bernstein polynomial to transform our SIR model to a nonlinear system of equation.The accuracy is comparatively good in comparison with methods that are applied directly to solve nonlinear stochastic differential equation.

Table 1 :
Mean, standard deviation and confidence interval for S(t).

Table 2 :
Mean, standard deviation and confidence interval for I(t).