Permanence and extinction for a nonautonomous SIRS epidemic model with time delay

In this paper, a nonautonomous SIRS epidemic model with time delay is studied. We introduce some new threshold values _R∗$R_{\ast }$ and R^∗$R^{\ast }$ and further obtain the disease will be permanent when _R∗>1$R_{\ast }>1$ and the disease extinct when R^∗<1$R^{\ast }<1$. Using the method of Liapunov functional, some sufficient conditions are derived for the global attractivity of the system. The known results are extended.