Department of Mathematics, Kabale University, Kabale, Uganda.
Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA), University of Stellenbosch, Stellenbosch, South Africa.
PLoS Negl Trop Dis. 2019 Apr 9;13(4):e0006973. doi: 10.1371/journal.pntd.0006973. eCollection 2019 Apr.
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.
一项已发表的研究使用随机分支过程推导出了关于采采蝇（Glossina spp）种群灭绝概率和时间的均值和方差的方程，这些方程是作为成虫和蛹死亡率以及雌性被可育雄性授精的概率的函数。原始推导部分是启发式的，没有为归纳结果提供证明。我们提供了这些证明，以及一种更简洁的方法来达到相同的结果。我们还表明，虽然已发表的方程在采采蝇以相等的比例产生雄性和雌性后代的情况下适用，但对于更一般的情况，即后代中雌性的概率（β）位于区间（0,1）中的任何位置，需要不同的解决方案。我们确认了以前在β=0.5的特殊情况下获得的结果，并表明灭绝概率对于β>0.5时最小，并且随着成年雌性死亡率的增加而增加。敏感性分析表明，灭绝概率受成年雌性死亡率变化的影响最大，其次是蛹的产生率。因为雌性每大约 10 天只产生一个后代，所以每天的死亡率大于约 3.5%将确保任何采采蝇种群的灭绝。对于某些物种，使用经过杀虫剂处理的靶标或牛提供可以实现这些死亡率水平，从而提供了一种简单、有效和具有成本效益的控制和根除采采蝇以及人类和动物锥虫病的方法。我们的结果在现代情况下具有进一步的意义，在现代情况下，温度的升高使得采采蝇在某些地区灭绝的可能性真实存在，而无需干预，但在其他地区，由于低温，它们以前无法生存，因此它们有更大的生存机会。
