Seasonal population dynamics of ticks, and their influence on infection transmission: a semi-discrete approach

A semi-discrete model for tick population dynamics is presented, whereby tick feeding is assumed to occur only during summers of each year. Conditions for existence, uniqueness, and stability of a positive equilibrium were found; the system was then studied numerically using parameter estimates calibrated for the tick Ixodes ricinus in Trentino, Italy, and the sensitivity to parameters was examined. This model was then extended to consider tick-transmitted infection of one species of hosts, while other hosts are incompetent to the infection. Assuming, for simplicity, that the infection is not affecting the total number of either hosts or ticks, a threshold condition for infection persistence was obtained. The dependence of the equilibrium infection prevalence on parameters was studied numerically; in particular, we considered how infection prevalence depends on host densities. This analysis reveals that a dilution effect occurs both for competent and for incompetent hosts. This means that, besides a lower threshold for host densities for infection to persists, there also exists an upper threshold: if host densities were higher than the upper threshold, the infection would go to extinction. Numerically, it was found that the upper threshold was not much higher than observed densities for realistic parameter values.