Modeling the invasion and establishment of a tick-borne pathogen.

We develop a discrete-time tick-host-pathogen model to describe the spread of a disease in a hard-bodied tick species. This model incorporates the developmental stages for a tick, the dependence of the tick life-cycle 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, Borellia burgdorferi and Anaplasma phagocytophila, using Ixodes ricinus as the tick species to study properties of the invasion and establishment of a disease numerically. In particular, we consider the basic reproduction number, which determines whether a disease can invade the tick-host system, as well as disease prevalence and time to establishment in the case of successful disease invasion. Using Monte Carlo simulations, we calculate the means of each of these disease metrics and their elasticities with respect to various model parameters. We find that increased tick survival may help enable disease invasion, decrease the time to disease establishment, and increase disease prevalence once established. In contrast, though disease invasion is sensitive to tick-to-host transmission and tick searching efficiencies, neither disease prevalence nor time to disease establishment is sensitive to these parameters. These differences emphasize the importance of developing approaches, such as the one highlighted here, that can be used to study disease dynamics beyond just pathogen invasion, including transitional and long-term dynamics.