GLOBAL STABILITY OF AN EPIDEMIC MODEL WITH TWO INFECTED STAGES AND MASS-ACTION INCIDENCE

Abstract

In this research work, we study the global stability of the SIR model which describes the dynamics of infectious disease with two classes of infected stages and varying total population size. The incidence used in the mathematical modeling was the mass-action incidence. The basic reproduction number R0 is computed. If the basic reproduction number is less than one, then the disease-free equilibrium point is locally and globally asymptotically stable. Existence and uniqueness of the endemic equilibrium is established when the basic reproduction number is greater than one and locally stable. We prove that global stability of the disease free equilibrium point using Lyapunov function. Numerical simulations have been carried out applying mat lab. Our result show that if the basic reproduction number R0 is below one the disease free equilibrium point is locally and globally stable in the feasible region, so that the disease dies out. If the basic reproduction number R0 is greater than one a unique endemic equilibrium point is locally asymptotically stable and the disease free equilibrium point is unstable in the interior of the feasible region and the disease will persist at the endemic equilibrium point if it is initially present.

Keywords: Equilibrium Stability, SIR, Basic Reproduction Number, Local and Global Stability