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

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.

一项已发表的研究采用随机分支过程（stochastic branching process），以成虫死亡率、蛹死亡率以及雌蝇被可育雄蝇授精的概率为自变量，推导得出采采蝇（Glossina spp.）种群灭绝概率与灭绝时间的均值和方差方程。原始推导过程带有部分启发式特征，且未对归纳得到的结果给出证明。本研究补充了上述证明，并提供了一种更简洁的推导路径以得到相同结论。我们还证明，尽管已发表的方程在采采蝇雌雄后代比例相等的情形下成立，但当推广至更一般的场景——即后代为雌性的概率β落在区间(0,1)内任意值时，需要采用不同的求解方式。我们验证了β=0.5这一特殊情形下的已有结论，并证明当β>0.5时灭绝概率达到最小值，且该最小值的降幅随雌成虫死亡率升高而增大。敏感性分析结果显示，灭绝概率受雌成虫死亡率变化的影响程度最大，其次是蛹的产生速率。由于雌蝇约每10天仅能繁育1只后代，若每日死亡率超过约3.5%，即可确保根除任意采采蝇种群。通过使用经杀虫剂处理的诱杀靶标或牛只，可对部分采采蝇物种达到上述死亡率控制目标，由此提供了一种简便、高效且经济的方法，用于防控并根除采采蝇种群，同时也可防控人兽共患的锥虫病（trypanosomiasis）。在全球变暖的当下，部分区域的采采蝇已存在无需人为干预即可灭绝的现实可能，而原本因低温无法维持种群的区域，采采蝇的存活概率却有所提升；本研究结果对这一现实场景具有重要参考价值。