BIFURCATION ANALYSIS OF AN SIRS EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION

Yixian Gao; Weipeng Zhang; Dan Liu; Yanju Xiao

doi:10.11948/2017067

2017 Volume 7 Issue 3

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Yixian Gao, Weipeng Zhang, Dan Liu, Yanju Xiao. BIFURCATION ANALYSIS OF AN SIRS EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1070-1094. doi: 10.11948/2017067

Citation:

Yixian Gao, Weipeng Zhang, Dan Liu, Yanju Xiao. BIFURCATION ANALYSIS OF AN SIRS EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1070-1094. doi: 10.11948/2017067

BIFURCATION ANALYSIS OF AN SIRS EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION

1 School of Mathematics and Statistics, Northeast Normal University, No. 5268 Renmin Street, 130024, Changchun, Jilin, China;
2 School of Mathematics and Statistics, Xidian University, No. 266 Xinglong Section of XiFeng Road, 710126, Xi'an, Shaanxi, China

Fund Project:

Abstract

Abstract

An epidemic model with standard incidence rate and saturated treatment function of infectious individuals is proposed to understand the effect of the capacity for treatment of infective individuals on the disease spread. The treatment function in this paper is a continuous and differential function which exhibits the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. It is proved that the existence and stability of the disease-free and endemic equilibria for the model are not only related to the basic reproduction number but also to the capacity for treatment of infective individuals. And a backward bifurcation is found when the capacity is not enough. By computing the first Lyapunov coefficient, we can determine the type of Hopf bifurcation, i.e., subcritical Hopf bifurcation or supercritical Hopf bifurcation. We also show that under some conditions the model undergoes Bogdanov-Takens bifurcation. Finally, numerical simulations are given to support some of the theoretical results.
- Epidemic model /
- saturated treatment /
- stability /
- bifurcation
MSC: 34C23;34D20

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References

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