## Global behavior and permanence of SIRS epidemic model with time delay.(English)Zbl 1154.34390

Summary: An autonomous SIRS epidemic model with time delay is studied. The basic reproductive number $$R_{0}$$ is obtained which determines whether the disease is extinct or not. When the basic reproductive number is greater than 1, it is proved that the disease is permanent in the population, and explicit formula are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Throughout the total paper, we mainly use the technique of Lyapunov functional to establish the global stability of the infection-free equilibrium and the local stability of the endemic equilibrium but need another sufficient condition.

### MSC:

 34K20 Stability theory of functional-differential equations 34D23 Global stability of solutions to ordinary differential equations 92D30 Epidemiology 93C23 Control/observation systems governed by functional-differential equations
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### References:

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