Pathogen dynamic in a tick-host system: A discrete-time modeling approach
We study the dynamics of a pathogen within a tick-host system, focusing on metrics that describe pathogen invasion and establishment, namely the basic reproduction number, the disease prevalence, and the time to disease establishment. We model the tick-host-pathogen dynamics using a system of difference equations that are constructed according to the life cycle of a three-host hard-bodied tick population. This model incorporates the developmental stages for a tick, the dependence of the tick lifecycle and disease transmission on host availability, and three sources of pathogen transmission. We first establish the global dynamics of the disease-free system. We then apply the model to two pathogens, Borelliaburgdorferi and Anaplasma phagocytophila, using Ixodes ricinus as the tick species to study properties of the invasion and establishment of these diseases numerically.