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

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 °C - 32°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°C - 31°C). Small decreases or increases in constant temperature in the neighbourhoods of 16°C and 31°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. Author summary Tsetse 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 < 16°C > 32°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.


10
A bite from a tsetse fly (Glossina spp.) infected with a parasite of the genus 11 Trypanosoma may cause Human African Trypanosomiasis (HAT), commonly called 12 sleeping sickness in humans, or Animal African Trypanosomiasis (AAT), commonly 13 called nagana in livestock. These tropical diseases have ravaged the African continent 14 for centuries. They pose serious public health and socio-economic problems, especially 15 to rural farmers, who rely on their livestock for daily subsistence, draught power and 16 general economic gain. Sleeping sickness is difficult to diagnose and the treatments are 17 often difficult to administer [1]. Vector control plays an important role in the fight 18 against trypanosomiasis [2], and understanding the vector population dynamics is thus 19 crucial. 20 As with all insects, the body temperature of tsetse is largely determined by ambient 21 temperature and all of the flies' physiological processes are determined by the 22 temperatures that the flies experience. The flies use various behavioural devices to 23 mitigate the effects of extreme ambient temperatures, such that the temperatures they 24 actually experience are less extreme than indicated by temperatures measured in, for 25 instance, a Stevenson screen [3,4]. Nonetheless, excessively high, or low, temperatures 26 may be lethal for them [5,6]. This is a serious concern for tsetse because, unlike other 27 insects, the genus Glossina is characterised by a very low birth rate. Consequently, 28 small increases in mortality rates can result in negative growth rates that, if sustained, 29 may drive tsetse populations to extinction. 30 As an example of this effect, a study published in 2018 concluded that, over the 31 previous 40 years, there had been a significant increase in temperatures in the Zambezi 32 Valley of Zimbabwe [7]. Specifically, peak temperatures at Rekomitjie Research Station 33 in Mashonaland West Province, Zimbabwe, increased by c. 0.9°C from 1975 to 2017. 34 The hottest time of the year (November) experienced a higher increase in temperature 35 of c. 2°C during the same period. These increases in temperature may potentially 36 explain the reduction in tsetse populations in some parts of Zimbabwe [8]. If these 37 findings are true also for other parts of Africa, then it is important to estimate the 38 impact of the high temperatures on the probability of, at least, local extinction of tsetse 39 in some parts of the continent.

40
Two published works have estimated extinction probabilities and time to extinction 41 for tsetse populations by assuming fixed environmental states throughout the life 42 history of the flies. The first [9], derived a branching processes model for tsetse 43 populations with the assumption that male and female offspring are produced with 44 equal probability. This assumption is not generally true [10], although the results 45 obtained in [9] are consistent with the body of the literature on tsetse biology. The 46 second publication [11] provides a sound mathematical foundation for the results 47 obtained in [9], and assumes a situation where male to female sex ratio can vary 48 anywhere in the open interval (0, 1) and shows how extinction probability depends on 49 the male-female sex ratio in tsetse populations.

50
In both papers, in order to gauge the importance of various determinants of tsetse 51 population growth, extinction probabilities were calculated for numerous arbitrary 52 combinations of mortality and fertility rates. There was no explicit modelling of the 53 effect of temperature on vital rates, nor thus on its effect on extinction probabilities. In 54 this paper, we develop a version of the stochastic branching process model presented 55 in [9] and [11] in which all of the biological process in the tsetse lifecycle are explicitly 56 dependent on temperature. It was also assumed in the earlier papers that mortality in adult flies was 64 independent of the age of the fly. Evidence suggests that mortality rates are actually 65 markedly higher in recently emerged adult flies than in mature flies [6, [11][12][13][14], and that 66 this difference is particularly severe at extremes of high temperatures [5,6]. To capture 67 this difference in mortality rates, we assume different survival probabilities for young 68 adult flies for the first ν days after emergence, compared with the survival probability of 69 all older flies.

71
We follow the general approach described earlier [9,11] but the extinction probability is 72 now obtained in a simpler form. We use the solution to obtain numerical results for 73 extinction probabilities, time to extinction and growth rates for tsetse populations living 74 at various fixed temperatures. We achieve this by using existing functions in the 75 literature relating tsetse fly lifecycle parameters to temperature. We modify published 76 versions of the model [9,11] by separating the adult life stage of tsetse fly into immature 77 and mature classes, allowing us to assess differential impacts of temperature in the two 78 stages.

79
Tsetse life history 80 The following provides a brief description of the life cycle; fuller accounts are provided 81 in [19]. Unlike most other insects, tsetse have a very low birth rate: they do not deposit 82 eggs, instead producing a single larva every 7 − 12 days [17,20]. The larva buries itself 83 in the ground and immediately pupates, staying underground as a pupa for between 84 30 − 50 days [21], emerging thereafter as an adult with the linear dimensions of a 85 mature adult, but with poorly developed flight musculature, and low levels of fat. Both 86 sexes of tsetse feed only on blood, and the first 2-3 blood-meals are used to build flight 87 muscle and fat reserves, before the mature female can embark on the production of 88 larvae of her own. All of these processes are temperature dependent. The model development and assumptions are similar to those in [11], differing only in 98 the way that mortality and fertility rates are used in the models. In the earlier works, 99 these rates were simply set at constant values. In the present study we use the fact that 100 the rates are almost all known to be functions of the temperature experienced by the 101 flies. Accordingly, instead of setting these rates at arbitrary values, we instead allow the 102 environmental temperature to take various values, which then dictate the values of the 103 rates of mortality and fertility to be used in the model. In particular, the following rates 104 are all temperature dependent; pupal duration and inter-larval period, and the daily survival probabilities for female pupae, adult females that are immature (defined as not 106 having yet ovulated for the first time), and mature adult females.

Model assumptions 108
In what follows all parameters with subscript T are temperature dependent.    5. The larva burrows rapidly into the soft substrate where it has been deposited and 118 pupates [8]. The pupa survives with probability ϕ T = e −χ T per day; where χ T is 119 the daily mortality rate for female pupae.  7. The immature female fly is inseminated by a fertile male tsetse after ν days, with 123 probability .

124
The probability that a female tsetse produces k surviving female offspring is 125 obtained as: The proof of equation (1) can be easily adapted from the proof of equation (7) in the 127 supplementary material of [11]. The difference here is simply that the probability is a 128 function of temperature. We assume that T is time invariant. Our interest is to 129 estimate extinction probabilities for tsetse at fixed temperatures.

130
Equation (1) can be used to derive a function for the extinction using the procedures 131 described in earlier publications [9,11]. A simpler derivation, using the work of 132 Harris [22], is given here. Suppose p k (T ) follows a geometric series for all T , then: where b T , c T > 0.

134
For equation (1) we then have: It follows (from [22], page 9) that the generating function f T (s), for p k (T ) is a 138 fractional linear function, and can be expressed as: Mean of female tsetse population at generation n 141 Substituting for b T and c T in equation (4) and taking the first derivative w.r.t s, at 142 s = 1. 143 Equation (5) is the reproduction number for a female tsetse population. For a population of tsetse living at temperature T°C, the expected number of female tsetse in the population at generation n is denoted by

Remark 1 144
When µ T > 1, the branching process is said to be supercritical with extinction 145 probability q T < 1. If µ T < 1, the branching process is subcritical, which implies, in 146 practice, that each female tsetse produces less than one surviving female offspring on 147 average. Extinction is then certain: i.e., the probability q T = 1. The process is called 148 critical if µ T = 1, and extinction probability is again certain, q T = 1 (see [23] page 36). 149 In other words, for any tsetse population to avoid inevitable extinction, each female fly 150 must produce more than one surviving female offspring in her lifetime.

151
Extinction probability q T

152
The extinction probability is obtained by solving for the fixed points of equation (4), 153 that is, we find s such that f T (s) = s. We therefore need to solve: Substituting for b T and c T in equation (6) and solving for s, the extinction 155 probability s = q T is the smaller nonnegative root of equation (6).

Remark 2
158 (5)). Therefore, whenever the denominator of equation (7) is less 160 than the numerator, extinction probability q T = 1. Hence, for all biologically 161 meaningful parameter ranges, q T is always in [0, 1]. See Remark 1. Furthermore, when 162 the initial population consists of a single female fly, the extinction probability is given 163 by equation (7). If the initial population is made up of N female flies, the extinction 164 probability is given by (q T ) N [11]. extinct is presented in [9], as: where φ(q n−1 (T )) = obtain E(k) as given in equation (8). Note that, since females produce both male and 173 female offspring, extinction of the female population obviously guarantees the extinction 174 of the whole population.

176
Tsetse mortality rates as a function of temperature 177 The relationship between temperature and the instantaneous daily mortality rate of 178 pupae is modelled as a the sum of two exponentials (Fig1) [15].
The relationship between larviposition rate and temperature was modelled using the 190 results from a mark and release experiment conducted at Rekomitjie on G. m. 191 morsitans and G. pallidipes [17].

199
Extinction probability as a function of fixed temperatures After controlling for generation time, the growth rate of the population as a function 242 of calendar time is seen to take a maximum value of about 2.0% per day at 25°C, falling 243 away increasingly rapidly towards zero as temperatures approach the upper and lower 244 limits of 32°C and 16°C, respectively (Fig7).  Our results confirm findings of earlier studies which suggest that temperature is a key 255 driver of tsetse population dynamics [8,28,29]. We show that constant temperatures Zimbabwe, where extreme temperature events have led recently to significant reductions 259 in tsetse populations [8].

260
As temperatures increase, mortality rates increase for adult female tsetse, and for 261 pupae of both sexes [26,30], but larval production rates also increase and pupal 262 durations decline [18]. In this trade-off, extinction probability initially declines as 263 temperature increase above 15°C (Fig2). In fact even for quite small pioneer 264 populations the extinction probability falls rapidly to zero for temperatures between 17 265 and 27°C. Thereafter, however, increases in mortality rates outweigh the increases in 266 birth rates and the extinction probability increases ever more rapidly as temperatures 267 approach 32°C.  Theoretically, for any population to be able to escape extinction, each female adult 275 must produce strictly more than one female offspring, which must themselves survive to 276 reproduce. In epidemiological terms, we then have that the reproduction number,

277
R o > 1. Figure4 shows that for temperatures below 16°C, female tsetse will not be able 278 to produce enough female flies to sustain the population. If the cold temperature 279 conditions are prolonged, Ro will drop below 1, resulting ultimately in extinction. Thus, 280 as temperatures approach either hot or cold limits, the number of generations that a 281 population can survive goes rapidly to zero even for large pioneer populations (Fig4).

282
The reproduction number reached its highest value of 2. 8 at 19°C (Fig4). This may, 283 initially, suggest that the growth rate is highest at that temperature. However, when we 284 calculated the actual growth rate after controlling for the length of generation for 285 different temperature values, we found that the population attains its maximum growth 286 rate at 25°C. This result agrees with published values in the literature [18], where a 287 different method was used to obtain the same results. It also draws attention to the fact 288 that the reproduction number should be used with caution when comparing two 289 populations with differing lengths of generation.

290
Our findings are in good agreement with experimental, field and modelling studies of 291 the impact of temperature on different tsetse species [8,28,[31][32][33], For instance, in [33] 292 an experiment was conducted on three different strains of Glossina palpalis gambiensis, 293 in a bid to determine critical temperature limits for tsetse survival and their resilience 294 to extreme temperatures. For the three strains, a temperature of about 32°C was 295 reported as the upper limit of survival. Our results showed, similarly, that, if the 296 number of female flies in the population are high enough, tsetse population may escape 297 extinction at temperatures slightly above 31°C but will go extinct at higher 298 temperatures. The experiment also reached the conclusion that temperatures of about 299 24°C are optimal for rearing this species, in good agreement with our modelling results. 300 Limitations of the study 301 Extinction probabilities for tsetse populations have not previously been estimated as a 302 function of temperature. Our modelling framework took into consideration the fact that, 303 as poikilotherms, tsetse mortalities, development rates, larval deposition rates, are all 304 temperature dependent. However, the present study did not consider field situations 305 where temperatures vary continuously with time. A modelling framework which will 306 consider this more realistic situation is under construction.

307
In this study, we considered closed tsetse populations, where there was no in-or-out 308 migration. Estimation of extinction probabilities for populations that are open to 309 migration are markedly more complex and were beyond the scope of the present study. 310 We do note, however, that a preliminary study found that if tsetse populations are in 311 patches, which can compensate for each other, then extinction probabilities are lower 312 than in closed population models [34]. Further work in this area is called for. 313 Finally, we caution that the modelling here is restricted to situations where 314 population numbers lie below the level at which density dependent effects play a 315 significant role. When numbers are larger, however, suppose pupal and/or adult 316 mortality increases with density. Then the temperature-dependent mortalities quoted 317 here provide a lower limit of the true mortality at that time, for any given temperature. 318 Similarly, if density dependence results in a decrease in the birth rate, then the birth 319 rates quoted here provide an upper bound to the true birth rate. For such a population, 320 it then follows that the growth rate at any temperature will be lower than calculated 321 here. Moreover, as temperatures increase, or decrease, towards the upper or lower 322 bounds, respectively, for positive population growth, the population will decline rapidly 323

PLOS
10/15 towards levels where the density-dependent effects fall away. At that point the further 324 dynamics of population growth would be subject to the vital rates used in this study.  Equations (10) and (11) plotted for different temperatures.     After controlling for generation time at different temperature levels.