Analysis of an epidemiological model with sub-optimal immune reaction and nonlinear incidence rate

In this paper, an epidemiological model with the sub-optimal reaction and the nonlinear incidence is investigated. We analyze the dynamical behavior of the model by constructing the Liapunov and Dulac functions, and combing with the local stability of the corresponding linear system. The proposed model connect two classical models, SIS and SIRS by introducing a parameter σ which describes the sub-optimal immune mechanism of disease. It reflects not only the dynamics of SIS and SIRS, but also the dynamic behaviors of a kind of epidemical models between these two models. Furthermore, by constructing the Liapunov and Dulac functions, we demonstrate that disease-free equilibrium point is globally asymptotical stable when the reproductive number R
₀
<; 1. The endemic equilibrium point is globally asymptotical stable when R
₀
> 1. According to the results of the analysis, we are pleasantly surprised to find that between the classical SIS and SIRS models there are very similar dynamic behavior. However it is obviously discriminate that the date when the endemic equilibrium becomes and the number of individuals will be infected between SIS and SIRS models.