Threshold conditions for a non-autonomous epidemic model with vaccination

In this article, we wish to investigate the dynamical behaviour of an SIRVS epidemic model with time-dependent coefficients. Under the quite weak assumptions, we give some new threshold conditions which determine whether or not the disease will go to extinction. The permanence and extinction of the infectious disease is studied. When the system degenerates into periodic or almost periodic system, the corresponding sharp threshold results are obtained for permanent endemicity versus extinction in terms of asymptotic time. In order to illustrate our theoretical analysis, some numerical simulations are also included in the end.