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 then 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. Author summary We derive equations for the mean and variance of the probability of, and time to, extinction in population of tsetse flies (Glossina spp), the vectors of trypanosomiasis in sub-Saharan Africa. In so doing we provide the complete proofs for all results, which were not provided in a previously published study. We also generalise the derivation to allow the probability that an offspring is female to lie anywhere in the interval (0, 1). 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. Simple bait methods of tsetse control, such as insecticide-treated targets and cattle, can therefore provide simple, affordable and effective means of eradicating tsetse populations. The 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, but have an increased chance of surviving in others where they were previously unsustainable due to low temperatures.


Introduction
1 Whereas deterministic models of the growth of populations of tsetse fly(Glossina spp). 2 (Diptera: Glossinidae) are adequate for large populations [1,2], stochastic models are 3 more appropriate when numbers are small, particularly if the population approaches 4 zero through natural processes and/or following attempts to eradicate the fly. At that 5 point the focus changes from attempting to attain deterministic predictions of future 6 population levels, to predicting the probability that the population will go extinct, and  The above considerations prompted us to revisit the original derivations, from which 32 several things became apparent: (i) It was assumed in the original derivation that equal 33 proportions of male and female offspring were produced by female tsetse. The equations 34 presented were correct for this particular case -but require modification for the more 35 general case where the probability (β) that an offspring is female lies anywhere in the 36 interval (0,1). (ii) At a number of points in the development it is claimed that results 37 can be shown by induction, but the proofs are not provided. (iii) An heuristic 38 explanation for one of the equation is misleading because it refers to a number > 1 as a 39 probability. (iv) Finally, the development is restrictive in that it only treats the case 40 where birth and death rates are constant over time. In the current paper we correct the 41 first three problems and suggest ways of overcoming the fourth.

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In this paper, we provide full details of the derivation of the formulae used and also 44 provide a general form of the governing equation in [3], which can accommodate all 45 possible values of β.

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A female tsetse fly generally mates only once; it is thus crucial to include in our model 48 the probability that a female tsetse fly is inseminated by a fertile male. We will also 49 assume that the probability that a deposited pupa is male or female can be anywhere in 50 the open interval (0, 1). Note that, at both endpoints, extinction occurs with probability 51 1.0, because the population will consist only of one gender of fly.

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Parameters and Interpretations 53 λ daily survival probability for adult female tsetse ψ daily mortality rate for adult females = -ln(λ) ϕ daily survival probability for female pupa χ daily mortality rate for female pupae = -ln(ϕ) ν time from adult female emergence to first ovulation(days) probability female is inseminated by a fertile male τ inter-larval period(days) P pupal duration(days) p n,k probability female tsetse fly dies between pregnancy n and (n + 1) and produces k surviving female offspring β probability deposited pupa is female

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The probability p 1,1 that a female survives one pregnancy and produces one 55 surviving female offspring is calculated as follows: First, we know that a female tsetse 56 fly is inseminated by a fertile male with a probability , then survives with probability 57 λ (ν+τ ) up to the time she produces her first pupa, which itself has a probability β of 58 being female. This pupa survives the pupal period with a probability ϕ P , and the 59 mother finally dies with a probability (1 − λ τ ) during the next pregnancy. Thus, 60 combining all these factors, we obtain the probability that a female tsetse fly produces 61 one surviving daughter after surviving one pregnancy as In general the probability that a female tsetse fly produces k surviving daughters after 63 surviving n pregnancies is given by for n > 0, 1 ≤ k ≤ n, and where n k are the binomial coefficients.
It is clear that We notice that M n refers to the mother's survival and q n,k refers to the pupae survival. 70 So we can base our proof by concentrating on the pupal survival since the product of 71 the two gives the result of interest.

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It was actually observed that equation (2) can be proved without resorting to are the same for all pupae, and these outcomes for different pupae are independent, the 77 probability that there are k adult females from n pupae is given by a binomial 78 distribution as Thus, from equation (3), we obtain the expression for p n,k as 80 p n,k = M n .q n,k Note that this reduces to the governing equation in [3] when β = 0.5.

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Remarks: 82 1. The heuristic explanation for equation (2) in [3] is misleading because it terms a 83 number greater than 1 a probability. Nonetheless, the formula is correct for the 84 case he considered, and is also correct more generally with the adjustment of that 85 term, as the proof shows.  Summing equation (2) over n leads to the probability (p k ) that a female tsetse fly 89 produces k surviving female offspring before she dies. Thus PLOS

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Thus, in general The probability that a female tsetse fly produces at least one surviving daughter 92 before she dies can be obtained by summing equation (7) over k > 0, to obtain .
(See Supporting information for detailed proofs of equations (7) and (8)) 94 Thus, the probability that a female tsetse fly does not produce any surviving female 95 offspring before she dies is given by .
Assuming that we start with one female tsetse fly in the initial generation, which 97 produces k surviving offspring, we can write the moment generating function for the 98 next generation as Substituting for p 0 and p k and putting the terms not involving k outside the summation 100 sign we get where A = 1 − λ τ , B = βλ τ ϕ P and C = 1 − λ ν

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The extinction probability can be found by solving the quadratic equation φ(θ) = θ, 103 and it will be the smallest non-negative root [7,8]. Thus the extinction probability is: This is the probability that a female tsetse population, resulting from an initial 105 population of one fly, goes to extinction. If the initial population consists of N flies, 106 then, assuming the independence of the probability of extinction of each female line, the 107 probability of extinction is θ N .

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Mean and variance of female tsetse population at generation n 109 We will use the method of moments to find the mean and variance of the expected 110 number of offspring produced. From these variables we can then derive the mean and 111 variance of the female tsetse population at a given generation n.

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By definition, the m th moment of p k is given by When m = 1, we obtain the first moment as And when m = 2, we obtain the second moment as (See Supporting information for the proofs of equation (12) and equation (13)) 116 The mean, or expected number of surviving daughters of female tsetse fly is and the variance is given by and M (n) and V (n) are the mean and variance of the size of each generation (X n ) 119 respectively with the assumption X 0 = 1. Equations (14) and (15) can be shown easily 120 by induction.

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Time for population of the female tsetse flies to become extinct 122 From the general framework developed by Lange [7, 8] for the probability of extinction 123 of a branching process. We have Where θ n is the probability of extinction at the n th generation and k is the number 125 of offspring. Equation (16) can be rewritten in terms of a moment generating function as 126 Thus, from (17), extinction probabilities can be calculated by starting with 127 θ 0 = 0, θ 1 = φ(θ 0 ), θ 2 = φ(θ 1 ), and continuing iteratively through the generations to 128 obtain 129 θ n = φ(θ n−1 ).

(18)
We also derived the first moments of T , based on the general formula obtained by Feller 130 where (1-θ n ) = P(T > n) and T is the extinction time. The first two moments of T are: 132 and Thus, using equations (10) and (18) and taking θ 0 = 0, we can calculate the values of θ n 134 by iteration. The first two, for example, are: In a situation where there are N surviving females, with N > 1, equations (20) and 137 (21) can be generalised. The probability of extinction at or before generation n is θ n . If 138 we have N surviving females, then the probability that they all become extinct at 139 generation n is (θ n ) N . Thus, and To estimate the mean and variance of the time to extinction for a population of N 142 female tsetse flies, all that needs to be done is to estimate θ n for a population consisting 143 of a single fly, raise each of the values to power N , and obtain the appropriate sums. increased with increasing pupal mortality rate (χ) (Fig S1).

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If the pupal mortality is high enough, then the probability of extinction is high even if 157 the adult mortality is low. For example if χ = 0.03 per day, then there is a greater than 158 40% chance that extinction will happen, even if the adult mortality rate is only 0.01 per 159 day. Even when there was zero pupal mortality, however, extinction was certain when

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When the pioneer population was greater than 1, the relationship with was no 177 longer linear (Fig S4) and, even when the starting population was only 16 inseminated 178 females, the extinction probability was still effectively zero when the probability of Extinction is of course certain if a population consists only of one sex, but the 184 probability of extinction goes to 1.0 more rapidly as the probability (β), that a 185 deposited pupa is female, goes to 0 (all male population) than as it goes to 1 (all female 186 population, Fig 1). For adult female mortality rates very close to zero, the extinction 187 probability goes to 1.0 as β goes to zero: but, for higher adult death rates the limit is  Fig S7 shows that, in the event that eradication is attempted through the release of 204 sterile males, in order to reduce the probability that females are inseminated by fertile 205 males, the eradication process will be much hastened if the mortality of the wild female 206 population is also increased. ]. These species are not strongly attracted by host odour, and the kill rate per 255 target is thus very much lower than for G. pallidipes: the riverine species can, however, 256 be captured on much smaller targets [typically 25 × 25cm] than those required for use 257 with savannah species [up to 2 × 1m]. It is thus economically feasible to deploy much 258 larger numbers of these so-called "tiny targets" and use them to effect significant 259 control of riverine tsetse species. In a trial in northern Uganda where tiny targets were 260 deployed at 20 targets per linear km (giving an average density of 5.7 per sq km), it was 261 possible to reduce the fly population by> 90%. It was noted that this reduction was 262 more than sufficient to break the transmission cycle for Human African 263 Trypanosomiasis in the area, 264 In the same study, experiments on islands in Lake Victoria, Kenya, suggested that 265 tiny targets used at the above density were killing 6% of the female population per day. 266 The suggestion is that a further increase in target density might result in the 267 eradication of populations, without the need to use any ancillary methods to control 268 tsetse or trypanosomiasis.    Supporting information 367 SI 1: Proof of equation (7) 368 When k = 0, we obtain .