On the global stability of the discrete-time epidemic models: A new approach.

Abstract

We developed a unified analytical framework for the global dynamics of discrete-time susceptible infectious susceptible (SIS) epidemic models with nonlinear recruitment. Emphasis was placed on demographic feedback through Beverton-Holt and Ricker-type recruitment, which regulates host population size and thereby shapes transmission and long-term persistence (Persistence allows population densities to approach zero asymptotically, wheras uniform persistence requires them to remain bounded away from zero). Under minimal assumptions, we reduced non-autonomous systems to appropriately defined autonomous limiting systems and used this reduction to obtain a complete global threshold characterization: When the basic reproduction number $ R_{0} > 1 $, the endemic equilibrium existed and was globally asymptotically stable; when $ R_{0}\le 1 $, solutions converged to the disease-free state. The approach extended to periodically forced SIS models, which showed that the threshold and stability conclusions persisted in the periodic non-autonomous setting. The results unified and strengthened prior work and clarify how recruitment dynamics govern persistence in discrete-time epidemic systems.