Analyses of the SIR Epidemic Model Including Treatment and Immigration

Abstract

This paper aims to examine the dynamics of a variation of a nonlinear SIR epidemic model. We analyze the complex dynamic nature of the discrete-time SIR epidemic model by discretizing a continuous SIR epidemic model subject to treatment and immigration effects with the Euler method. First of all, we show the existence of equilibrium points in the model by reducing the three-dimensional system to the two-dimensional system. Next, we show the stability conditions of the obtained positive equilibrium point and the visibility of flip bifurcation. A feedback control strategy is applied to control the chaos occurring in the system after a certain period of time. We also perform numerical simulations to support analytical results. We do all these analyses for models with and without immigration and show the effect of immigration on dynamics.