Extinction probabilities, times to extinction, basic reproduction number and growth rates for tsetse (Glossina spp) populations as a function of temperature

Abstract

Increases in temperature over recent decades have led to a significant reduction in the populations of tsetse flies (Glossina spp) in parts of the Zambezi Valley of Zimbabwe. If this is true for other parts of Africa, populations of tsetse may actually be going extinct in some parts of the continent. Extinction probabilities for tsetse populations have not so far been estimated as a function of temperature. We develop a time-homogeneous branching process model for situations where tsetse flies live at different levels of fixed temperatures. We derive a probability distribution pk(T) for the number of female offspring an adult female tsetse is expected to produce in her lifetime, as a function of the fixed temperature at which she is living. We show that pk(T) can be expressed as a geometric series: its generating function is therefore a fractional linear type. We obtain expressions for the extinction probability, expected number of female offspring per female tsetse, and time to extinction. No tsetse population can escape extinction if subjected, for extended periods, to temperatures outside the range 16 {degrees}C - 32{degrees}C. Extinction probability increases more rapidly as temperatures approach and exceed the upper and lower limits. If the number of females is large enough, the population can still survive even at high temperatures (28{degrees}C - 31{degrees}C). Small decreases or increases in constant temperature in the neighbourhoods of 16{degrees}C and 31{degrees}C, respectively, can drive tsetse populations to extinction. Further study is needed to estimate extinction probabilities for tsetse populations in field situations where temperatures vary continuously.\n\nAuthor summaryTsetse flies (Glossina spp) are the vectors of the African sleeping sickness. We derived an expression for the extinction probability, and mean time to extinction, of closed populations of the flies experiencing different levels of fixed temperatures. Temperatures play a key role in tsetse population dynamics: no tsetse populations can escape extinction at constant temperatures 32{degrees}C. The effect of temperature is more severe if tsetse populations are already depleted. Increasingly high temperatures due to climate change may alter the distribution of tsetse populations in Africa. The continent may witness local extinctions of tsetse populations in some places, and appearances in places hitherto too cold for them.