The Finite Element Implementation of a K.P.P. Equation for the Simulation of Tsetse Control Measures in the Vicinity of a Game Reserve

TL;DR

This study models tsetse fly populations near a game reserve using a modified Fisher's equation, demonstrating how different control measures like barriers and mortality rates affect eradication and containment.

Contribution

It introduces a finite element implementation of a K.P.P. equation that accounts for historical population effects and temperature-dependent puparial duration in tsetse control modeling.

Findings

A 2% daily mortality outside the reserve has no lasting impact on populations.

A 5 km barrier with 4% daily mortality can isolate a G. austeni population.

More mobile species are more vulnerable to eradication than sedentary ones.

Abstract

An equation, strongly reminiscent of Fisher's equation, is used to model the response of tsetse populations to proposed control measures in the vicinity of a game reserve. The model assumes movement is by diffusion and that growth is logistic. This logistic growth is dependent on an historical population, in contrast to Fisher's equation which bases it on the present population. The model therefore takes into account the fact that new additions to the adult fly population are, in actual fact, the descendents of a population which existed one puparial duration ago, furthermore, that this puparial duration is temperature dependent. Artificially imposed mortality is modelled as a proportion at a constant rate. Fisher's equation is also solved as a formality. The temporary imposition of a 2 % $\mathrm{day}^{-1}$ mortality everywhere outside the reserve for a period of 2 years will have no lasting effect on the influence of the reserve on either the Glossina austeni or the G. brevipalpis populations, although it certainly will eradicate tsetse from poor habitat, outside the reserve. A 5 $\mathrm{km}$-wide barrier with a minimum mortality of 4 % $\mathrm{day}^{-1}$, throughout, will succeed in isolating a worst-case, G. austeni population and its associated trypanosomiasis from the surrounding areas. A more optimistic estimate of its mobility suggests a mortality of 2 % $\mathrm{day}^{-1}$ will suffice. For a given target-related mortality, more mobile species are found to be more vulnerable to eradication than more sedentary species, while the opposite is true for containment.