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Bifurcation and stability analysis of an SIS epidemic model with general incidence rate and saturated treatment function. (English) Zbl 1349.34146

Summary: A saturated treatment function in an SIS model with general incidence rate is introduced. The saturated treatment function adopts a continuous and differential function which describes the effect of delayed treatment when the medical condition is limited and the number of infected individuals is getting larger. The general incidence rate includes the bilinear incidence rate and the standard incidence rate. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given. A backward bifurcation is found when the capacity of the treatment is low. This suggests to improve the efficiency and capacity of the treatment. By mathematical analysis, it is shown that the system undergoes the Hopf bifurcation.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
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