An Epidemic Model for Tick-Borne Disease with Two Delays

Abstract

We have considered an epidemic model of a tick-borne infection which has nonviraemic transmission in addition to the viremic transmission. The basic reproduction number ${\Re }_{\mathrm{0}}$, which is a threshold quantity for stability of equilibria, is calculated. If ${\Re }_{\mathrm{0}}\le \mathrm{1}$, then the infection-free equilibrium is globally asymptotically stable, and this is the only equilibrium. On the contrary, if ${\Re }_{\mathrm{0}}>\mathrm{1}$, then an infection equilibrium appears which is globally asymptotically stable, when one time delay is absent. By applying a permanence theorem for infinite dimensional systems, we obtain that the disease is always present when ${\Re }_{\mathrm{0}}>\mathrm{1}$.