Abstract
A general disease transmission model is developed for a tick-borne disease for cattle. Both the tick and cow are possible hosts for the disease parasite. The two species are therefore classified as infected or uninfected. The tick individual is further classified as sedentary (off-host) or feeding (on-host). First conditions for the persistence of the tick population are investigated. A threshold quantity for the disease is derived which is dependent on both the tick and host population parameters and the two transmission rates: from ticks to cattle and vice versa. From the analysis persistence and non-persistence conditions for both the tick population and the disease are investigated. The effect of the presence of a second host species is introduced as an additional feature in the analysis.
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