Global stability analysis of SEIR model with holling type II incidence function.

Mohammad A Safi,Salisu M Garba

doi:10.1155/2012/826052

Abstract

A deterministic model for the transmission dynamics of a communicable disease is developed and rigorously analysed. The model, consisting of five mutually exclusive compartments representing the human dynamics, has a globally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number (ℛ 0), is less than unity; in such a case the endemic equilibrium does not exist. On the other hand, when the reproduction number is greater than unity, it is shown, using nonlinear Lyapunov function of Goh-Volterra type, in conjunction with the LaSalle's invariance principle, that the unique endemic equilibrium of the model is globally asymptotically stable under certain conditions. Furthermore, the disease is shown to be uniformly persistent whenever ℛ 0 > 1.

Highlights

IntroductionMathematical models have been widely used to gain insight into the spread and control of emerging and reemerging disease
Mathematical models have been widely used to gain insight into the spread and control of emerging and reemerging disease. The dynamics of these models is usually determined by a threshold quantity known as the basic reproduction number, which is defined as the number of secondary cases generated by an infected individual in a completely susceptible population [1,2,3,4,5]
Other methods used in establishing global properties of some epidemic models include Dulac’s criterion to eliminate the existence of the periodic solution and prove the global stability by the Poincare Bendixson theorem [18] and those reported in Kamgang and Sallet [19] and Qiao et al [20]

Summary

Introduction

Mathematical models have been widely used to gain insight into the spread and control of emerging and reemerging disease. Establishing global properties of a dynamical system using Lyapunov function is generally a nontrivial problem This is owing to the fact that there are no systematic methods for constructing Lyapunov function for infectious disease models with standard incidence rate [17]. More general models than SIR or SIRS models assumed that susceptible individuals, once infected, first go through the latent period (in the E class) before becoming infectious; the resulting models are of SEIR or SEIRS type, depending on whether recovered individuals acquired permanent or temporary immunity [33, 34]. Li and Jin [34] consider global stability of an SEIR epidemic model with infectious force in latent, infected, and immune period It should be stated, that the aforementioned three studies [33, 34, 36] considered mass action (bilinear) incidence to model the infection.

Model Formulation

Global Stability of DFE

Existence and Stability for Endemic Equilibrium Point

Findings

Conclusions