Basic reproduction ratios for periodic and time-delayed compartmental models with impulses

Much work has focused on the basic reproduction ratio R0 for a variety of compartmental population models, but the theory of R0remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of R0 for a class of such impulsive models. We first introduce R0and show that it is a threshold parameter for the stability of the zero solution of an associated linear system. Then we apply this theory to a time-delayed computer virus model with impulse treatment and obtain a threshold result on its global dynamics in terms of R0 Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.