How ticks keep ticking in the adversity of host immune reactions
Ixodid ticks are acknowledged as one of the most important hematophagous arthropods because of their ability in transmitting a variety of tick-borne diseases. Mathematical models have been developed, based on emerging knowledge about tick ecology, pathogen epidemiology and their interface, to understand tick population dynamics and tick-borne diseases spread patterns. However, no serious effort has been made to model and assess the impact of host immunity triggered by tick feeding on the distribution of the tick population according to tick stages and on tick population extinction and persistence. Here, we construct a novel mathematical model taking into account the effect of host immunity status on tick population dynamics, and analyze the long-term behaviours of the model solutions. Two threshold values, $${\mathcal {R}}_{11}$$R11 and $${\mathcal {R}}_{22}$$R22, are introduced to measure the reproduction ratios for the tick-host interaction in the absence and presence of host immunity. We then show that these two thresholds (sometimes under additional conditions) can be used to predict whether the tick population goes extinct ($${\mathcal {R}}_{11}<1$$R11<1) and the tick population grows without bound ($${\mathcal {R}}_{22}>1$$R22>1). We also prove tick permanence (persistence and boundedness of the tick population) and the existence of a tick persistence equilibrium if $${\mathcal {R}}_{22}<1<{\mathcal {R}}_{11}$$R22<1<R11. As the host species adjust their immunity to tick infestation levels, they form for the tick population an environment with a carrying capacity very much like that in logistic growth. Numerical results show that the host immune reactions decrease the size of the tick population at equilibrium and apparently reduce the tick-borne infection risk.