Modelling the use of insecticide-treated cattle to control tsetse and Trypanosoma brucei rhodesiense in a multi-host population.

We present a mathematical model for the transmission of Trypanosoma brucei rhodesiense by tsetse vectors to a multi-host population. To control tsetse and T. b. rhodesiense, a proportion, ψ, of cattle (one of the hosts considered in the model) is taken to be kept on treatment with insecticides. Analytical expressions are obtained for the basic reproduction number, R0n in the absence, and R(0n)(T) in the presence of insecticide-treated cattle (ITC). Stability analysis of the disease-free equilibrium was carried out for the case when there is one vertebrate host untreated with insecticide. By considering three vertebrate hosts (cattle, humans and wildlife) the sensitivity analysis was carried out on the basic reproduction number (R(0n)(T)) in the absence and presence of ITC. The results show that R(03)(T) is more sensitive to changes in the tsetse mortality. The model is then used to study the control of tsetse and T. b. rhodesiense in humans through application insecticides to cattle either over the whole-body or to restricted areas of the body known to be favoured tsetse feeding sites. Numerical results show that while both ITC strategies result in decreases in tsetse density and in the incidence of T. b. rhodesiense in humans, the restricted application technique results in improved cost-effectiveness, providing a cheap, safe, environmentally friendly and farmer based strategy for the control of vectors and T. b. rhodesiense in humans.