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 then 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 cattle -providing 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. Author summaryWe derive equations for the mean and variance of the probability of, and time to, extinction in population of tsetse flies (Glossina spp), the vectors of trypanosomiasis in sub-Saharan Africa. In so doing we provide the complete proofs for all results, which were not provided in a previously published study. We also generalise the derivation to allow the probability that an offspring is female to lie anywhere in the interval (0, 1). The probability of extinction was most sensitive to changes in adult female mortality. The unusual tsetse life cycle, with very low reproductive rates means that populations PLOS 1/19can be eradicated as long as adult female mortality is raised to levels greater than about 3.5% per day. Simple bait methods of tsetse control, such as insecticide-treated targets and cattle, can therefore provide simple, affordable and effective means of eradicating tsetse populations. The 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, but have an increased chance of surviving in others where they were previous...

The weakest link: uncertainty and sensitivity analysis of extinction probability estimates for tsetse (Glossina spp) populations

“…The aim of this section is to develop a stochastic model for tsetse population growth in the form of a branching process and to use the model to obtain a fixed point equation for extinction probability for tsetse populations ([2], [6], [14], [15]). We develop the branching process focusing only on female tsetse flies [6].…”

“…A recent study [2] employed the theory of branching processes to derive an expession for the extinction probability for closed populations of tsetse. This equation involves numerous parameters representing death, development and fertility rates during the fly’s lifecycle.…”

“…Here we adopt the model developed by Kajunguri et al [2] and Hargrove [6] for the reproductive performance of female tsetse flies inseminated by a fertile male. We then use a framework, developed by Harris [7], to derive a fixed point equation for the extinction probability for a tsetse population.…”

“…We then use a framework, developed by Harris [7], to derive a fixed point equation for the extinction probability for a tsetse population. This approach allows us to obtain the same expression for extinction probability as [2], but it is derived with fewer steps and with less mathematical complexity. We compute local sensitivity indices of extinction probability with respect to all input parameters by allowing the value of daily mortality rate of female pupae ( χ ) to vary between 0.001 per-day and 0.025 per-day.…”

“…In the next section, we present the branching process model developed in [2] and [6] and present an approach based on a method used in [7] to derive a fixed point equation for the extinction probability for a tsetse population. In section 3, we present the local sensitivity analysis for the extinction probability at two fixed baseline values of the input parameters and the mathematical derivation of the sensitivity indices of extinction probability w.r.t all input parameters.…”