Analysis of a model of gambiense sleeping sickness in humans and cattle

Human African Trypanosomiasis (HAT) and Nagana in cattle, commonly called sleeping sickness, is caused by trypanosome protozoa transmitted by bites of infected tsetse flies. We present a deterministic model for the transmission of HAT caused by Trypanosoma brucei gambiense between human hosts, cattle hosts and tsetse flies. The model takes into account the growth of the tsetse fly, from its larval stage to the adult stage. Disease in the tsetse fly population is modeled by three compartments, and both the human and cattle populations are modeled by four compartments incorporating the two stages of HAT. We provide a rigorous derivation of the basic reproduction number $ \mathcal{R}_{0} $ . For $ \mathcal{R}_{0}1 $ , HAT persists. Elasticity indices for $ \mathcal{R}_{0} $ with respect to different parameters are calculated with baseline parameter values appropriate for HAT in West Africa; indicating parameters that are important for control strategies to bring $ \mathcal{R}_{0} $ below 1. Numerical simulations with $ \mathcal{R}_0 > 1 $ show values for the infected populations at the endemic equilibrium, and indicate that with certain parameter values, HAT could not persist in the human population in the absence of cattle.