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 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.
Highlights
Whereas deterministic models of the growth of populations of tsetse fly(Glossina spp). (Diptera: Glossinidae) are adequate for large populations [1, 2], stochastic models are more appropriate when numbers are small, if the population approaches zero through natural processes and/or following attempts to eradicate the fly
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 can be eradicated as long as adult female mortality is raised to levels greater than about 3.5% per day
Summary
Whereas deterministic models of the growth of populations of tsetse fly(Glossina spp). (Diptera: Glossinidae) are adequate for large populations [1, 2], stochastic models are more appropriate when numbers are small, if the population approaches zero through natural processes and/or following attempts to eradicate the fly. At that point the focus changes from attempting to obtain deterministic predictions of future population levels, to predicting the probability that the population will go extinct, and the expected time required in order to achieve this end. Hargrove developed a stochastic model for the life history of tsetse flies (Glossina spp) and thereby provided estimates of the probability of extinction, and expected time to extinction, for these insects [3]. Such estimates were always of interest in situations where there was pressure in favour of area-wide eradication of entire tsetse species [4]. The formulae developed were shown to provide good estimates of the time to extinction in successful operations that had already been carried out
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