Global analysis of an epidemic model with nonmonotone incidence rate

Abstract

In this paper we study an epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero as time evolves or the disease persists.

Fig. 1

Nonmonotone incidence function g(I).

Fig. 2

When b = 1.0, d = 0.2, k = 0.2, α = 4.0, γ = 0.3, μ = 0.15, R₀ = 6/7 < 1, S(t) approaches to its steady state value while I(t) and R(t) approach zero as time goes to infinity, the disease dies out.

Fig. 3

When b = 1.0, d = 0.2, k = 0.2, α = 4.0, γ = 0.3, μ = 0.15, R₀ = 20/7 > 1, all three components, S(t), I(t) and R(t), approach to their steady state values as time goes to infinity, the disease becomes endemic.

Fig. 4

The dependence of I^∗ on the parameter α.