Discrete Type SIR Epidemic Model with Nonlinear Incidence Rate in Presence of Immunity

Abstract

Mathematical modeling is very important to describe the dynamic behavior of biological and biomedical systems. The SIR model is the most common mathematical model of epidemics. An epidemic occurs if the number of people infected with a disease is increasing in a population. A numerical discretization for an SIR epidemic model is discussed, where the incidence rate is assumed to be Beddington-DeAngelis type. In particular, we reconsider a SIR epidemic model with Non Linear incidence and treatment rate derived by (Dubey et al. 2015) [1]. We applied Euler method to discretize this model. This discretization leads to a numerical scheme which can be considered as a discrete system. Then we analyzed the dynamics of the obtained discrete system. We developed the model with the focus on the concentration of the basic reproduction number and related stability analysis for the disease-free and endemic equilibrium points. Finally, We have performed numerical simulations to illustrate the disease behavior