Seasonal population dynamics of ticks, and its influence on infection transmission: a semi-discrete approach.
Ecological interactions underlying the epidemic of Lyme disease involve a spirochete, a tick (with larval, nymph and adult stages), and two (or more) vertebrate hosts. Juvenile ticks ordinarily feed on mice; adult ticks feed on deer. Mice acquire the spirochete from infected nymphs and then pass the infection to larvae of the next tick generation. Lyme disease may result when a human is inadvertently bitten by an infectious nymph. Our model of the Lyme phenomenon counts the total number of ticks in each stage, the numbers of infected ticks by stage, and the number of infected mice. We fix the total population sizes of deer and mice, assume the ticks self-regulate, and solve the homogeneous-mixing case for equilibrium abundances. A local stability analysis identifies a condition where extinction of the spirochete is stable. Reversing this condition implies that the spirochete can invade the system of ticks and vertebrate hosts. When the spirochete can invade, a positive equilibrium number of infected organisms is locally stable. Spirochete invasion is promoted by a sufficient density of mice suffering low mortality, high susceptibility to infection in both mice and ticks, a high attack rate of ticks on mice, a high density of larval ticks, and low mortality among tick nymphs. Low mouse mortality allows the frequency of infection among nymphs to approach an individual tick's susceptibility when feeding on an infected mouse.