Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality

In this paper, non-autonomous SIRS epidemic models with bilinear incidence and disease-induced mortality are studied. Under the quite weak assumptions, the sufficient and necessary conditions on the permanence and strong persistence of the disease and the sufficient condition on the extinction of the disease are established. Some new threshold values of the integral form R0∗, R1∗ and R2∗ are obtained. We prove that the disease is permanent if and only if R0∗>0, and if R1∗≤0 or R2∗<0, then the disease is extinct. As applications of the main results, we discuss the periodic and almost periodic models. The corresponding basic reproductive numbers R0R0 are obtained. We show that if R0>1R0>1, then the disease is permanent and if R0≤1R0≤1, then the disease is extinct.