# Backward Bifurcation Emerging from a Mathematical Model of African Animal Trypanosomiasis Disease in White Rhino Populations

## DOI:

https://doi.org/10.5614/j.math.fund.sci.2022.54.1.9## Keywords:

African Animal Trypanosomiasis, Basic reproduction number, Backward bifurcation, Optimal control, Cost-effectiveness analysis## Abstract

This paper introduces a mathematical model for African animal trypanosomiasis (AAT) in white rhino and tsetse fly populations. The model accommodates two types of interventions, namely infection detection and ground spraying. The dynamical system properties were thoroughly investigated to show the existence of equilibrium points, backward bifurcation, and how they are related to the basic reproduction number. We found that there is a chance that AAT may die out from the population if the basic reproduction number is smaller than one. However, the possible existence of backward bifurcation implies a condition where we may have a stable endemic equilibrium, even when the basic reproduction number is smaller than one. Hence, the basic reproduction number is no longer sufficient to guarantee the disappearance of AAT from the population. Our sensitivity analysis on the basic reproduction number showed that the interventions of infection detection and ground spraying have good potential to eradicate AAT from the population. To analyze the most effective intervention as time-dependent variable, we reconstructed our model as an optimal control problem. Numerical simulations on various scenarios were conducted for the optimal control problem. Cost-effectiveness analysis using the Average Cost-Effectiveness Ratio (ACER) and the Incremental Cost-Effectiveness Ratio (ICER) methods was performed. From the cost-effectiveness analysis, we found that ground spraying is the most cost-effective intervention to combat the spread of AAT in white rhino populations.

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