Thresholds for disease persistence in models for tick-borne infections including non-viraemic transmission, extended feeding and tick aggregation

Lyme disease and Tick-Borne Encephalitis (TBE) are two emergent tick-borne diseases transmitted by the widely distributed European tick Ixodes ricinus. The life cycle of the vector and the number of hosts involved requires the development of complex models which consider different routes of pathogen transmission including those occurring between ticks that co-feed on the same host. Hence, we consider here a general model for tick-borne infections. We assumed ticks feed on two types of host species, one competent for viraemic transmission of infection, the second incompetent but included a third transmission route through nonviraemic transmission between ticks co-feeding on the same host. Since a blood meal lasts for several days these routes could lead to interesting nonlinearities in transmission rates, which may have important effects. We derive an explicit formula for the threshold for disease persistence in the case of viraemic transmission, also for the case of viraemic and non-viraemic transmission. From this formula, the effect of parameters on the persistence of infection can be determined. When only viraemic transmission occurs, we confirm that, while the density of the competent host has always a positive effect on infection persistence, the density of the incompetent host may have either a positive effect, by amplifying tick population, or a negative (‘‘dilution’’) effect, by wasting tick bites on an incompetent host. With non-viraemic transmission, the ‘‘dilution’’ effect becomes less relevant. On the other hand, if the nonlinearity due to extended feeding is included, the dilution effect always occurs, but often at unrealistically high host densities. Finally, we incorporated the effects of tick aggregation on the hosts and correlation of tick stages and found that both had an important effect on infection persistence, if non-viraemic transmission occurred.