Abstract
This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.
Highlights
In the mathematical modeling of epidemic transmission, there are several factors that substantially affect the dynamical behavior of the models
We focus on the bifurcation analysis of an SIS epidemic model with bilinear incidence rate and saturated treatment
In many epidemic models, the basic reproduction number, which is the key concept in epidemiology, can be decreased below unity to eradicate the disease
Summary
In the mathematical modeling of epidemic transmission, there are several factors that substantially affect the dynamical behavior of the models. This means that they use the maximal treatment capacity to cure infective individuals so that the disease can be eradicated They found that the model undergoes saddlenode bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation. Eckalbar constructed an SIR epidemic model with a quadratic treatment function; that is, T(I) = max{rI−gI2, 0}, r, g > 0 They found that the system has as many as four equilibria, with possible bistability, backward bifurcations, and limit cycles. It is really important to adequately stress the interesting connection recently established between the choice of saturated treatment functions in epidemic models and the occurrence of backward bifurcation in the related system dynamics. Motivated by the above points, we will consider the following SIS model with bilinear incidence rate and saturated treatment function: dS dt.
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