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

Abstract

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 R 0 ∗ , R 1 ∗ and R 2 ∗ are obtained. We prove that the disease is permanent if and only if R 0 ∗ > 0 , and if R 1 ∗ ≤ 0 or R 2 ∗ < 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 R 0 are obtained. We show that if R 0 > 1 , then the disease is permanent and if R 0 ≤ 1 , then the disease is extinct.