Global analysis of a discrete sirs epidemic model with nonlinear incidence rate

Abstract

In this thesis, we prove some behaviors of a discrete SIRS epidemic model with a nonlinear incidence rate and a distributed time-delay. This model is constructed from the discretization of the corresponding continuous model by using a nonstandard finite difference method. The basic properties including the positivity and the boundedness of the solutions are established. We derive the existence of the disease-free equilibrium and the endemic equilibrium of the model. In addition, by applying Lyapunov function techniques, we prove that the disease-free equilibrium is globally attractive. Moreover, we give a sufficient condition for the permanence of the model. In order to illustrate our analytical results, finally, some numerical simulations are also included.