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

Abstract

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 R0. For R0 < 1, the disease free equilibrium is globally asymptotically stable, thus HAT dies out; whereas (assuming no return to susceptibility) for R0 >1, HAT persists. Elasticity indices for R0 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 R0 below 1. Numerical simulations with R0 > 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.

Analysis and simulation of a mathematical model of tuberculosis transmission in Democratic Republic of the Congo

“…In order to identify ways to control diseases in the population, several studies have been conducted in mathematics [9][10][11][12][13][14]. Mathematical models are developed and applied in ecology, and also are used to understand epidemiological phenomena [15,16].…”

A fractional-order Trypanosoma brucei rhodesiense model with vector saturation and temperature dependent parameters

“…Furthermore, the proposed framework consists of two parts (i) the life cycle of the tsetse flies and (ii) the full model that governs HAT transmission dynamics when the tsetse fly population growth persists. Comprehensive details on the modelling of the early stages of the tsetse flies can be found in [13]. As proposed in [13], let the following system summarise the dynamical growth of the tsetse fly:…”

“…Comprehensive details on the modelling of the early stages of the tsetse flies can be found in [13]. As proposed in [13], let the following system summarise the dynamical growth of the tsetse fly:…”

“…r is a threshold quantity that determines growth of the tsetse fly population. It is defined as the likelihood of the fly to survive the pupal stage multiplied by the surviving population of female flies [13].…”

“…For example, the study of Hagrove et al [10] demonstrated that treating cattle with insecticides could be useful on HAT management. Pandey et al [11], Funk et al [12], Ndondo et al [13] and Rock et al [14] among others explored the role of animals as reservoirs on HAT dynamics. Their studies demonstrated that animals play an integral role of HAT dynamics hence they need to be incorporated in the frameworks that seek to explore the intrinsic dynamics of HAT.…”