Stability Analysis of SIRS Model considering Pulse Vaccination and Elimination Disturbance

It is well known that many natural phenomena and human activities do exhibit impulsive effects in the fields of epidemiology. At the same time, compared with a single control strategy, it is obvious that the multiple control strategies are more beneficial to restrain the spread of infectious diseases. In this paper, we consider pulse vaccination and pulse elimination strategies at the same time and establish an SIRS epidemic model with standard incidence. Firstly, according to the stroboscopic mapping method of the discrete dynamical system, the disease-free $T$ periodic solution of the model under the condition of pulse vaccination and pulse elimination is obtained. Secondly, the basic reproductive number $R_0$ is defined, and the local asymptotic stability of the disease-free $T$ periodic solution is proved by Floquet theory for $R_0<1$ . Finally, based on the impulsive differential inequality theory, the global asymptotic stability of the disease-free $T$ periodic solution is given for $R_0<1$ , and the disease dies out eventually. The results show that in order to stop the disease epidemic, it is necessary to choose the appropriate vaccination rate and elimination rate and the appropriate impulsive period.