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

Much work has focused on the basic reproduction ratio [Formula: see text] for a variety of compartmental population models, but the theory of [Formula: see text] remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of [Formula: see text] for a class of such impulsive models. We first introduce [Formula: see text] and 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 [Formula: see text]. Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.