Improved estimates for extinction probabilities and times to extinction for populations of tsetse (Glossina spp)

Abstract

A published study used a stochastic branching process to derive equations for the mean and variance of the probability of, and time to, extinction in population of tsetse flies (Glossina spp) as a function of adult and pupal mortality, and the probabilities that a female is inseminated by a fertile male. The original derivation was partially heuristic and provided no proofs for inductive results. We provide these proofs, together with a more compact way of reaching the same results. We also show that, while the published equations hold good for the case where tsetse produce male and female offspring in equal proportion, a different solution is required for the more general case where the probability (β) that an offspring is female lies anywhere in the interval (0, 1). We confirm previous results obtained for the special case where β = 0.5 and show that extinction probability is at a minimum for β > 0.5 by an amount that increases with increasing adult female mortality. Sensitivity analysis showed that the extinction probability was affected most by changes in adult female mortality, followed by the rate of production of pupae. Because females only produce a single offspring approximately every 10 days, imposing a death rate of greater than about 3.5% per day will ensure the eradication of any tsetse population. These mortality levels can be achieved for some species using insecticide-treated targets or cattleproviding thereby a simple, effective and cost-effective method of controlling and eradicating tsetse, and also human and animal trypanosomiasis. Our results are of further interest in the modern situation where increases in temperature are seeing the real possibility that tsetse will go extinct in some areas, without the need for intervention, but have an increased chance of surviving in other areas where they were previously unsustainable due to low temperatures.

Extinction probabilities as a function of temperature for populations of tsetse (Glossina spp.)

“…We follow the general approach described earlier [10,12] but the extinction probability is now obtained with fewer mathematical steps, and in a more compact form. We use the solution to obtain numerical results for extinction probabilities, times to extinction and growth rates for tsetse populations living at various fixed temperatures.…”

“…We achieve this by using existing functions in the literature relating tsetse fly lifecycle parameters to temperature. We modify published versions of the model [10,12] by separating the adult life stage of tsetse fly into immature and mature classes, allowing us to assess differential impacts of temperature in the two stages.…”

“…This assumption is not generally true [11], although the results obtained in [10] are consistent with published results on tsetse biology. The second publication [12] provides a sound mathematical foundation for the results obtained in [10], and assumes a situation where male to female sex ratio can vary anywhere in the open interval (0, 1) and shows how extinction probability depends on the male-female sex ratio in tsetse populations.…”

“…There was no explicit modelling of the effect of temperature on vital rates, nor thus on its effect on extinction probabilities. In this paper, we develop a version of the stochastic branching process model presented in [10] and [12] in which all of the biological process in the tsetse lifecycle are explicitly dependent on temperature.…”

“…The model development and assumptions are similar to those in [12], differing only in the way that mortality and fertility rates are used in the models. In the earlier works, these rates were simply set at constant values.…”