Mathematical model of Ehrlichia chaffeensis transmission dynamics in dogs

Ehrlichia chaffeensis is a tick-borne disease transmitted by ticks to dogs. Few studies have mathematical modelled such tick-borne disease in dogs, and none have developed models that incorporate different ticks' developmental stages (discrete variable) as well as the duration of infection (continuous variable). In this study, we develop and analyze a model that considers these two structural variables using integrated semigroups theory. We address the well-posedness of the model and investigate the existence of steady states. The model exhibits a disease-free equilibrium and an endemic equilibrium. We calculate the reproduction number ($T_0$). We establish a necessary and sufficient condition for the bifurcation of an endemic equilibrium. Specifically, we demonstrate that a bifurcation, either backward or forward, can occur at $T_0=1$, leading to the existence, or not, of an endemic equilibrium even when $T_0<1$. Finally, numerical simulations are employed to illustrate these theoretical findings.