Modelling framework to demonstrate elimination of a vector population: tsetse elimination in Chad

33 Every year, over 700 000 people, particularly children under five, die from vector-borne diseases 34 worldwide. Effectively controlling current endemics and preventing new outbreaks requires an 35 integrated approach that can lead to the elimination of both vectors and diseases. In the last two 36 decades, integrating medical interventions and vector control has significantly reduced the 37 incidence of Gambian Human African Trypanosomiasis (gHAT), with the World Health 38 Organization validating eight countries as having eliminated the disease as a public health 39 problem. However, elimination of the tsetse vector has not been confirmed , leaving the possibility 40 of re-emergence. We developed a five-step modelling framework to assess vector elimination by 41 calculating: (i) the probability of vector capture; (ii) the probability of observing a series of zero 42 catches, even without actual elimination; (iii) the probability of natural elimination; (iv) the 43 probability of failing to detect a rebound; and (v) the reinvasion risk. Our case study is g-HAT in 44 Mandoul, Chad and the elimination of G. fuscipes fuscipes. We used vector control from 2014 to 45 2025 with no tsetse detected since 2018. We cannot yet conclude, with more than 90% 46 confidence, that tsetse has been eliminated from Mandoul, nor that any remnant population will 47 be naturally eliminated. However, since vector control was stopped in April 2025, we estimate 48 that with continued sampling over the next two years, and no tsetse detected, elimination could 49 be demonstrated with 99% confidence. Our multi-step modelling framework can be applied to 50 other vectors, providing policymakers with clear guidelines for ongoing and future efforts. 51 52 Significance Statement 53 The World Health Organisation has set the elimination of transmission of several neglected 54 tropical vector-borne diseases, including human African trypanosomiasis (sleeping sickness), as 55 a target for 2030. We show that deliberate elimination of tsetse, the vector, is feasible and can be 56 demonstrated. We draw on our large-scale intervention in Mandoul, Chad where 3000 57 insecticide-treated Tiny Targets were deployed between 2014 and 2025, with no tsetse detected 58 since 2018. While small undetected remnant populations cannot be entirely excluded, they would 59 rapidly rebound in the absence of control, rendering them detectable. If no tsetse are caught over 60 the next two years, it will confirm elimination. This illustrates a pathway for assessing and 61 achieving vector elimination as a cornerstone of disease eradication. 62 63 64

vector-borne disease, elimination, vector control, mathematical modeling, public 65 health policy. 66 67 68 Main Text 69 70

71 Vector-borne diseases (VBDs) remain a major public health challenge worldwide with over 700 72 000 deaths annually, particularly among children under five (1). Sub-Saharan Africa bears the 73 heaviest burden, where malaria, dengue and other neglected tropical diseases (NTDs) affect 74 millions of women and men. To successfully interrupt disease transmission, control approaches 75 must address critical questions: when, where, how and with whom to act. When im plemented 76 effectively, integrated strategies can eliminate diseases to the point where the World Health 77 Organization (WHO) validates them as Eliminated as a Public Health Problem (EPHP) (2). 78 79 Recent progress includes EPHP validation for gambiense Human African Trypanosomiasis (g-80 HAT or sleeping sickness), in Chad in 2024 (3), alongside seven other endemic countries 81 including Guinea (4), and Côte d’Ivoire (5). After EPHP, the WHO goal is elimination of g-HAT 82 transmission by 2030, with the aim of achieving zero cases of the disease by that time (6). 83 Integrated approaches must target not only diagnostics, vaccines and therapeutics for the 84 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 3 parasite and host, but also target arthropod vectors to reduce host-vector contact, and ultimately 85 reduce and even eliminate transmission (5, 7–10). 86 87 Vector control successes include insecticide treated bednets and innovative housing approaches 88 to control mosquitoes (7, 11) and Tiny Targets to control tsetse (Glossina spp), the sole vectors of 89 HAT (12). In various settings and countries, these small insecticide impregnated pieces of fabric, 90 Tiny Targets, have significantly reduced tsetse populations, lowering human-vector contact (5, 91 13, 14) and helping to halt g-HAT transmission (15). Vector ecology also strongly influences VBD 92 epidemiology (16) and vector populations are highly sensitive to environmental and global change 93 factors (17–21) which affect vector elimination prospects. 94 95 Despite progress, a critical question remains: how to know when elimination has been achieved 96 and how to quantify the risk of error. While disease cases or vector captures may reach zero, 97 confirming the complete elimination of either is a far more complex issue. The challenge lies in 98 determining whether the absence of disease or vectors truly reflects elimination, or simply a 99

in our surveillance and detection capabilities. In some cases, the cessation of control 100 efforts has led to a rebound in disease transmission. In some instances, stopping control has led 101 to resurgence: halting control in the 1960s caused an estimated 300,000 cases of g-HAT in the 102 1990s, underscoring the need for a more robust understanding of vector population dynamics (2). 103 104 To address these challenges, we propose a novel mathematical and modelling framework, 105 addressing the three core questions of VBD elimination: (i) Can elimination, in theory, be 106 achieved? (ii) How will it be attempted in practice? (iii) How do we know when it has been 107 achieved? 108 In our study, using tsetse and HAT as an example, we focus on the third question, considering 109 diseases and vectors that can be eliminated with currently available methods (diagnostics, 110 treatments and tiny targets for vector control) and that for tsetse elimination feasibility can be 111 estimated from birth and death rates and the size of the starting population (22–24). 112 113 When a VBD elimination program reaches a stage where zero cases of disease are diagnosed, 114 and/or zero vectors are captured, elimination of the disease and/or the vector may indeed have 115 been achieved. Alternatively, this outcome may simply reflect the lower detection threshold of our 116 sampling methods, below which parasites or vectors remain undetectable. If control efforts are 117 halted while the disease and/or vectors persist – though undetectable with current tools – there is 118 a risk of renewed transmission, rising case numbers, and ultimately a re-emergence of the vector, 119 and/or the disease. We therefore investigate the theory underlying mathematical assessment of 120 the probability that elimination has truly been achieved. We primarily focus is on vector 121 elimination, though clear parallels exist for disease elimination. Specifically, we explore 122 mathematical approaches to estimate the risk incurred when declaring a vector population 123 eliminated. This can be structured as a five-steps approach and modelling framework: 124 I. Before vector control is started, estimate the probability of capturing a vector with the 125 surveillance tools to be used. This may be informed by existing literature when available, 126 or by baseline mark-recapture, or other studies, to calculate the probability of capturing a 127 vector. Seber (25) provides an excellent review of available methods and see Williams et 128 al. (26) for more recent advances in the field. 129 II. Calculate the conditional probability of observing a series of zero catches given that at 130 least one individual vector is still present. If that probability is sufficiently low, we conclude 131 the vector has been eliminated (23); 132 III. If the vector population is very low but not yet eliminated, calculate the probability of 133 natural extinction, purely by chance, without further control efforts (22, 24); 134 IV. Use growth models to estimate the expected vector population at various times after the 135 cessation of control efforts, assuming survival of at least one reproductive female. Failure 136 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 4 to detect a rebound in the vector population after extended periods supports elimination 137 (23). 138 V. Calculate the probability of vector reinvasion to further support the demonstration of 139 vector elimination (27). 140 As a case study of vector elimination modelling, we examine tsetse-transmitted g-HAT focusing 141 on efforts to eliminate the tsetse species Glossina fuscipes fuscipes from the Mandoul focus in 142 southern Chad (13) – a site of serious g-HAT outbreaks. Both sexes of the tsetse fly are strictly 143 hematophagous and are thus both capable of transmission. The disease is of particular interest 144 because, while massive outbreaks occurred over the last two centuries, g-HAT seems now to be 145 under control in many countries, with EPHP validated (3–5) and even elimination of transmission 146 in some cases. 147 148 Tsetse provide a relatively simple, and tractable system for estimating the probability of vector 149 elimination because of their slow, and predictable reproductive biology. Each adult female 150 produces, only one larva every 9-11 days (28, 29). The larva nearly the same weight as its 151 postpartum mother, contains the energy, microbiome and materials required to pass through all of 152 the final larval and pupal, stages of metamorphosis, into an adult-sized teneral fly (28, 30, 31). 153 Following larviposition, the free-living pupa does not feed at all. Reproduction continues year-154 round, independent of the availability of environmental water (32). The production of a single 155 offspring at each birth event, separated by roughly 10 days, means that the fate of individual 156 offspring may be regarded as independent, which is an important assumption for probabilistic 157 modelling. The low reproductive rate also means that population growth rates become negative if 158 adult female mortality exceeds 4% per day. Sustained mortality at or above this level leads to 159 elimination (33). In principle, tsetse populations should therefore be relatively easy to eliminate, 160 particularly since they have never shown resistance to insecticide and their simple life cycle 161 makes it feasible to calculate the probability that elimination can indeed be achiev ed. 162 163 Our research provides a modelling framework for assessing the likelihood of vector elimination, 164 using G. f. fuscipes and g-HAT in Chad as a case study. This work contributes to ongoing 165 elimination strategies by evaluating whether current evidence is sufficient to demonstrate that 166 specific vectors, such as tsetse, have been successfully eliminated. 167 168 169

170 Step 1: probability of capturing one G. f. fuscipes in a biconical trap in Mandoul 171 Using data from Big Chamaunga Island, Lake Victoria, Kenya, we estimated the daily probability 172 p that a female G. f. fuscipes in Mandoul is killed by a target or captured by an individual trap. 173 174 On Chamaunga, catches of female G. f. fuscipes declined 104-fold in 1 year (Figure 1) after 175 deployment of 30 Tiny Targets along 1.5 km of shoreline habitat (14). Following Hargrove (33), 176 this rate of decline is consistent with targets killing a proportion p = 0.04 (i.e., 4%) of adult 177 females per day. Accordingly, the proportion q = 1-p that is not killed each day by any of the 30 178 targets is 0.96. The probability that a single target fails to kill a single fly in a day is then 0.96^30 = 179 0.99864 and the proportion of the total population killed by one target is 1 - 0.99864 = 0.0014 or 180 0.14% per day. As a single biconical trap is estimated to catch about half as many tsetse as a 181 Tiny Target (Esterhuizen et al., 2011), each trap should then capture (0.14/2) = 0.07% of the total 182 fly population on the island per day. 183 184 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 5 185 Figure 1. Decline in numbers of female G. f. fuscipes trapped on Big Chamaunga Island, following the deployment of 30 186 Tiny Targets on the 1.5 km perimeter of the island. 187 188 We assume these kill and recapture rates apply generally to riverine tsetse when using Tiny 189 targets and such biconical traps, including the Mandoul populations, independent of habitat area. 190 Implicit in this assumption is that targets cover all available habitat, such that each target’s 0.14% 191 contribution is drawn from a local fraction of the total population. By the same logic , zero catches 192 from a given trap reflect only the absence of tsetse in the immediate neighborhood of that 193 particular trap. 194 In Mandoul, 145 G. f. fuscipes were captured in the 2013 baseline survey, and following the 195 initial deployment of 2713 Tiny Targets in March 2014, the follow up surveillance sessions –in 196 April and June 2014 – resulted in only 2 and 3 flies being captured respectively (13) – suggesting 197 that the population had already declined by 98% (Figure 2). The decline was so rapid that it is 198 difficult to estimate the true rate of decline of the population (Figure 3). Nonetheless, the results 199 are consistent with a population growth rate of -0.767 per month, equivalent to a yearly decline of 200 exp(-0.76712)  10-4 per year, and the overall control effort killing c. 4% per day of the adult 201 female population – as for the Big Chamaunga Island. Hence each target in Mandoul killed (1 - 202 0.96)^(1/2713) = 0.0015% flies per day and if the biconical traps used to sample G. f. fuscipes is 203 about half as efficient as the Tiny Targets (12), each trap is expected to catch about 0.00075% 204 per day of all flies in the Mandoul control area. The probability of capturing one G. f. fuscipes in a 205 biconical trap in Mandoul is of 0.0000075. 206 207 N(m) = 12.22 exp(-0.67m) R² = 0.92 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 Tsetse catch per trap per day (N(m)) Months (m) after targets deployed .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 6 208 Figure 2. Catches of G. f. fuscipes from traps deployed, between November 2013 and October 2023, in the Mandoul 209 focus of southern Chad. Numbers are plotted as the log (base 10) of the catch; zero catches cannot thus be plotted. 210 211 212 Figure 3. Observed catches of adult G. f. fuscipes in biconical traps in Mandoul, and the expected numbers if the growth 213 rate was taken to be -0.767 per month. 214 215 Since this level of mortality in Mandoul should guarantee eventual elimination, we expect that the 216 use of high densities of Tiny Targets tsetse should lead to the elimination of the Mandoul G. f. 217 fuscipes population. The following sections assess the probability that elimination has indeed 218 been achieved – or, conversely, the risk that elimination has not been achieved, despite 219 continued failure to detect any flies using the above calculated probability of capture per trap. 220 221 222 Step 2: Probability of zero tsetse catches, as a function of the numbers of tsetse surviving 223 Suppose now that there was only a single tsetse remaining in Mandoul, and that the area is 224 sampled using 44 biconical traps, every day for 30 days. The probability of observing zero 225 catches from every trap on every day is then ((1-0.0000075)44)30  0.991, so a 99.1% chance of 226 0 20 40 60 80 100 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 Catch (% of baseline) Months since targets deployed Decline in trap catches female G. f. fuscipes Mandoul, Chad Observed Predicted r = -0.767/month .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 7 having a false absence/failing to catch that surviving fly. Similar calculations show that even if the 227 44 traps were run for 160 days, there would still be a 90% chance of failing to catch the surviving 228 fly. Moreover, even with a 4-fold increase in trap efficiency, there would still be a 90% chance that 229 80 days of trapping would fail to catch a surviving fly (Figure 4A). 230 If the number of tsetse surviving were 10, or 100, then the probability of observing a series of 231 zero catches when some flies are still present obviously diminishes (Figure 4 B, C). With 10 232 surviving tsetse, 44 traps run for 30 days would still have a 90% chance of returning a series of 233 zero catches/failing to catch any tsetse (Figure 4B): and, with 100 surviving flies, there would still 234 be a 37% chance of failing to capture any. 235 236 237 Figure 4. Probability of zero catch of tsetse, given that 1, 10 or 100 tsetse (A, B and C from left to right) have survived in 238 the Mandoul area, as functions of trap efficiency and numbers of days trapping. The dotted lines show examples where 30 239 traps, each with 0.00075% efficacy, are run for 44 days. 240 241 242 Step 2´: pursuing vector control and vector surveillance until one is 99% confident of tsetse 243 elimination by relying on probability of capture in surveillance tools 244 If one flies remain and vector control is continued and vector monitoring still relies on the same 44 245 biconical traps, then it would take 13,900 days to be 99% confident that one has not missed 246 capturing that single surviving fly. Even with 4 times more efficient traps it would still take 3475 247 days. 248 If 10 flies remain it would still take 1400 days with our current trap, and 348 days with a four times 249 more efficient trap for a 99% confidence in tsetse elimination. 250 Finally, if 100 flies remain, it would take 140 days with our current trap and 35 days with a four 251 times more efficient trap to be 99% confident of tsetse elimination. 252 253 254 Step 3: Probability of natural elimination of a tsetse remnant population in Mandoul 255 The previous sections underline the risk of inappropriately interpreting even a long series of zero 256 trap catches as proof that the G. f. fuscipes population in Mandoul has been eliminated. If, 257 however, we can at least be confident that the surviving population is small (10 or less) then what 258 are the chances that such a population will be eliminated by chance? The results in Figure 5 259 suggest that, in general, tsetse populations are remarkably resilient to being eliminated by 260 chance. 261 262 For the Mandoul population, if indeed adult female mortality could be maintained at 4% per day, 263 then any population would be eliminated. If, however, the imposed mortality fell to, say 2% per 264 day then the probability that a remnant population of 10 inseminated female tsetse disappears by 265 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 0 20 40 60 80 100 120 140 160 Probability zero catch given 1 fly present Days trapping A. Probability of zero catch | 1 fly surviving vs trap days and trap efficiency P = 0.00075% P = 0.00150% P = 0.00300% 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 20 40 60 80 100 120 140 160 Probability zero catch given 10 flies present Days trapping B. Probability of zero catch | 10 flies surviving vs trap days and trap efficiency 0.001 0.010 0.100 1.000 0 20 40 60 80 100 120 140 160 Probability zero catch given 100 flies present Days trapping C. Probability of zero catch | 100 flies surviving vs trap days and trap efficiency .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 8 chance is <10%. And even a remnant of 16 such flies is only eliminated by chance with 266 probability <10% if female adult mortality is about 3% per day. In general, therefore, we would be 267 unwise to rely on a remnant population being eliminated by chance in Mandoul. 268 269 270 Figure 5. Annual growth rates of a tsetse population as a function of adult mortality and fecundity. Redrawn from 271 “Hargrove J.W. (1988) Tsetse: the limits to population growth. Medical and Veterinary Entomology, 2, 203-217.”. 272 273 274 Step 4: Allowing for the detection of a post-vector-control rebound 275 The expected growth rate (i.e. rebound) of the hypothetical remaining tsetse population, following 276 the removal of the Tiny Targets, can be inferred from Figure 5. 277 At a low starting population number, implied by the inability to catch any tsetse with the 278 surveillance system in place, we may safely assume that density-dependent mortalities among 279 pupae and adults will be at a minimum. If we suppose that female mortality is at a conservative 280 2% per day, and that there are negligible losses among pupae, then Figure 5 suggests that the 281 population could increase by up to 100-fold in the first year. Such rates of increase have been 282 approached for field populations of G. pallidipes in Kenya (34) and G. m. morsitans and G. 283 pallidipes in Zimbabwe (35). We might then expect that the G. f. fuscipes population in Mandoul 284 could grow to a minimum of 1000 flies in two years. Even if the population increased by only 10-285 fold per year, the population should reach the 1000 level in three years. 286 287 If the Mandoul population were to rebound to a level of 1000 flies, 44 traps run for 3 weeks (21 288 days) would catch at least 1 fly with 99.9% certainty (Figure 6). That is to say, the probability of 289 catching zero tsetse would be 0.001, or 0.1%. A zero catch over the whole 3-week period would 290 then imply there was negligible risk in concluding that elimination had been achieved. 291 The above scenario – based on the assumption of sampling a population that was assumed to 292 reach the level of 1000 flies – might take two to three years to provide a decision that elimination 293 has been achieved. 294 295 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 0 1 2 3 4 5 6 7 8 log10(N1/N0) Adult female mortality (%/day) Tsetse population growth rate vs mortality and interlarval period 10-day pregnancy 15-day pregnancy .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 9 296 Figure 6. The probability of observing a series of zero catches of G. f. fuscipes in Mandoul after 297 the removal of all Tiny Targets – given that the population has increased to a total of 1000 flies 298 before the sampling effort is initiated – plotted as a function of trap efficiency and numbers of 299 days of trapping. 300 301 The results in Figure 7 suggest that a demonstration of elimination could be achieved in much 302 less than 3 years. If the sampling procedure continues to involve the use of 44 traps, the initial 303 probability of catching a fly will be small – and one would have <90% confidence that elimination 304 had been achieved elimination even after more than a year (550 days) of consecutive zero trap 305 catches (Figure 7). Thereafter, however, confidence levels grow very rapidly – reaching 90% and 306 95% by days 600 and 650, respectively, and 99% after 2 years. 307 308 Doubling the sampling effort, by using 88 traps every day, makes only a modest difference to the 309 outcome; 1.5 – 2 years of zero catches would still be required to be to be 99% confident that 310 elimination had been achieved. Increasing the trapping effort further would probably be 311 counterproductive. With 132 traps run daily, one could be 99% confident of elimination after 18 312 months of zero catches (Figure 7) but only if the traps were acting independently of each other. 313 This assumption becomes increasingly unlikely as trap density increases (36). 314 315 0.001 0.010 0.100 1.000 0 2 4 6 8 10 12 14 16 18 20 22 Probability zero catch given 1000 flies present Days trapping Probability of zero catch | 1000 flies surviving vs trap days and trap efficiency P = 0.00075% P = 0.00150% P = 0.00300% .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 10 316 Figure 7. Estimates of confidence that series of zero trap catches support a conclusion that the Mandoul population of G. 317 f. fuscipes has been eliminated. Calculated as a function of the number of traps used and the number of days sampling. 318 319 320 Step 5: Vector elimination and reinvasion risk 321 322 The closest population of tsetse, G. f. fuscipes, beyond the borders of Mandoul is in the 323 neighborhood of Timbéri, 50 km distant. We estimate the probability that a tsetse fly could survive 324 long enough to move between the two populations. We use the results of Hargrove & Lange (27) 325 in modelling tsetse dispersal as diffusion in the plane. The fly’s position (x, y, t) in time and space 326 is then defined by a normally distributed random variable with density function and if the 327 population are widely separated, relative to the rate of diffusion, the equations can be much 328 simplified. 329 For our study, Mandoul and Timbéri are separated by order 50 km, and the relative rate of 330 diffusion is of the order of (0.04km)2/day, so that, for t > 0. That being the case, we can make the required simplifications and the probability is then 0 332 (see equation 5 in material and methods). There is no chance that a fly can move between 333 Timbéri and Mandoul in under 100 days, even if it survived for that period. For t > 100 days, the 334 probability of completing the journey increases, but the probability that the fly survives this period 335 decreases very rapidly. The danger of reinvasion due to diffusive movement is thus vanishingly 336 low in any case. 337 338 339

340 341 No G. f. fuscipes have been captured in Mandoul since 2018, following a vector control program 342 that began in 2014 and ended in 2025. It is not yet possible to conclude, with more than 90% 343 certainty that the population has been eliminated, owing to the low probability of catching G. f. 344 fuscipes with current trapping methods. Nonetheless, satellite images and community reports 345 (source: Mahamat M.H., Aldjibert M., and Yoni W., 2025, pers. comm.) strongly suggest that 346 tsetse no longer pose a threat: crops are cultivated in areas previously infested with tsetse and 347 new human settlements are emerging along the Mandoul river. These developments further 348 85% 86% 87% 88% 89% 90% 91% 92% 93% 94% 95% 96% 97% 98% 99% 100% 350 450 550 650 750 850 950 1050 Confidence that tsetse population eliminated Days trapping Confidence series of zero catches means elimination achieved vs number of traps and days deployed 44 traps 88 traps 132 traps .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 11 support the decision by the Chad tsetse control team to remove all Tiny Targets in early 2025. If 349 current levels of surveillance continue for two more years without detecting a single G. f. fuscipes 350 – then the authorities will be able to declare elimination that we will have demonstrated with 99% 351 confidence. This represents one of the few modern examples of upcoming successful vector 352 elimination in the world. 353 354 If a small number of tsetse persist, their population should increase once the mortality imposed by 355 the Tiny Targets is lifted, making detection more likely. This represents a risk-reward situation: 356 while risks appear low given over six years without captures, Tiny targets could be redeployed to 357 quickly suppress any remnant population. The rewards, by contrast, are substantial – cost 358 savings from halting control and the prospect of eventually ending all monitoring. Without 359 stopping control, it would take an impractical 40 years of continuous trapping with current 360

to reach 99% confident of elimination, and even longer with the present regime of a few 361 trapping days per year. The near-zero risk of reinvasion from neighboring tsetse populations, 362 combined with the improbability of accidental introduction via livestock or motorized transport, 363 further supports cessation of control. No other tsetse population or species appear to be able to 364 recolonize the area vacated by G f. fuscipes. 365 366 Our study highlights the difficulty of proving complete elimination of an isolated vector population 367 when monitoring tools cannot reliably detect rare individuals. The problem scales with area size, 368 as shown in the large-scale efforts to eliminate G. m. centralis in Botswana. There, contiguous 369 blocks of 16,000 sq km in the Okavango Delta were sprayed in 2001 and 2002, and no tsetse 370 have been caught since the completion of the second year of spraying (37). However, trap and 371 fly-round sampling methods would not have detected tsetse at population densities of 1 per sq 372 km, nor could such sampling methods cover the full habitat. The conclusion of elimination relied 373 on the expectation that any surviving tsetse would have rebounded to detectable levels within two 374 years. 375 376 Historically, such a “wait and see” approach relying on years without tsetse captures or reported 377 cases of disease has underpinned declarations of elimination, with the passage of time without 378 new cases providing sufficient evidence of local eradication. This is the basis for accepted cases 379 such as G. p. palpalis on from Principe (38), G. pallidipes in Zululand (39) and G. m. morsitans in 380 Umfurudzi Game Area of Zimbabwe (32) where elimination was inferred without long-term 381 systematic sampling. The passage of years, and then decades, in which no tsetse and no cases 382 of trypanosomiasis were reported, simply made it clear that the flies had indeed been eliminated. 383 In contrast, very small or isolated areas allow quicker demonstration of elimination, as shown for 384 G. pallidipes and G. m. morsitans on Antelope Island, Zimbabwe (35) and G. austeni on Unguja 385 Island (40). Mathematical approaches, including landscape genetics (41) or species distribution 386 modeling (42) have also been developed to help identify such target areas for disease or vector 387 elimination. 388 389 Regardless of the scale or surveillance strategies, our framework provides a formal, probabilistic 390

to demonstrate vector elimination. This creates direct linkages to existing international 391 processes: WHO’s elimination of public health problem (EPHP) (2) and elimination of 392 transmission (EoT) for human African trypanosomiasis (HAT), FAO’s Progressive Control 393 Pathways for Animal African Trypanosomiasis (AAT) (43), and WOAH’s recognition of AAT-free 394 status. All are aiming for a greater integration of mathematical frameworks to move beyond 395 reliance on the “wait and see” paradigm. Discussions are underway on incorporating our 396 approach into official WHO and FAO procedures. 397 398 Although developed for tsetse, the framework is general and applicable to any vectors or even 399 populations of beneficial species, be it plants or animals, at risk of extinction. Our results are 400 therefore relevant to public health experts, policy makers, and conservation practitioners. 401 402 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 12 In estimating the probability that the Mandoul population of G. f. fuscipes has been eliminated, we 403 did not account for the possibility that long-term use of Tiny Targets may have induced behavioral 404 change in tsetse, such as avoidance of these attractive systems. Similarly, selection pressure 405 could have resulted in surviving flies feeding preferentially on host animals that can be accessed 406 with reduced movement, thereby lowering the likelihood that flies are killed by Tiny Targets. 407 There is, however, no evidence that suggest such behavioral or ecological shifts have occurred, 408 and any potential impact on our results would be minimal. Importantly, the selection pressure 409 imposed by our 44 surveillance traps is far weaker than that imposed by 2713 Tiny Targets, 410 meaning that monitoring is less likely to be affected by the evolution of resistance than the vector 411 control efforts. 412 413 We also adopted a conservative approach in our calculations to minimize the risk of incorrectly 414 demonstrating elimination in Mandoul and to ensure our conclusions are not overly optimistic. For 415 instance, in calculating the probability that a small surviving population of tsetse would not be 416 eliminated by chance, we disregarded the possibility that adult virgin females might die before 417 successfully mating, a factor that would further reduce the chance of persistence. In this respect, 418 our estimates represent a worst-case scenario of the results obtained by the tsetse control team 419 in Mandoul. This conservative approach strengthens our argument that the risks of removing the 420 Tiny Targets are outweighed by the benefits of that policy. 421 422

423 Whether or not tsetse have already been eliminated from the Mandoul area, Tiny Targets have 424 successfully reduced tsetse population densities by several orders of magnitude to undetectable 425 levels. In 2024, WHO validated elimination of gHAT as a public health problem in Chad, marking 426 the operation a success (3). With the removal of Tiny Targets, continued monitoring will provide a 427 definitive answer regarding the elimination of tsetse in the Mandoul region. More broadly, the 428 multi-step theoretical approach developed in this study offers a rigorous method for 429 demonstrating vector elimination. This framework is applicable to other disease vectors and 430 provides guidance for policymakers and health authorities in planning, evaluating and sustaining 431 future elimination efforts. 432 433

434 435 Modelling framework for assessing vector elimination 436 Using a decision tree (Figure 8), published methodology (22–24) and our novel modelling 437 framework, we use up to five steps, in sequence, to estimate the probability that a vector 438 population has been eliminated in a specific area: 439 1) Baseline probability of capture with a surveillance tool: if there is no relevant existing literature 440 conduct mark-recapture, or other, studies to calculate the probability of capturing a vect or. 441 Alternatively, calculate the theoretical probability that a given control method could eliminate a 442 vector population that is isolated (i.e., closed to all in- and out-migration) (32) and infer the 443 probability of capture with the surveillance tool from the vector control tool efficacy. 444 2) Probabilistic modelling of elimination: apply a probability model to the results of trapping efforts 445 to reject the null hypothesis that vectors are still present in the area. 446 3) Natural probability of elimination: estimate the probability that a very small residual population 447 will be eliminated by chance. 448 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 13 4) Probability of detecting a rebound: use a model of tsetse population growth to estimate the 449 time required for a very small remnant population to become detectable by trapping. 450 5) Vector elimination and reinvasion risk: finally, evaluate whether the vector population has been 451 successfully eliminated with a specified level of risk, while also accounting for the potential threat 452 of reinvasion. 453 454 Our modelling framework, and the associated decision tree (Figure 8), involves calculating 455 mathematically whether the observed vector surveillance data support a conclusion that 456 elimination has been achieved, what risk would be associated with that conclusion, and the best 457 course of action to progress to elimination. 458 459 460 Figure 8. Modelling framework and decision tree for declaring vector elimination: Step 1. Baseline work to assess the 461 probability of vector capture (when we mean known from experiments or estimated from the literature); Step 2. Calculate 462 the conditional probability of observing a series of zero catches given that there is still at least one individual vector 463 present. If that probability is sufficiently low, we conclude the vector has been eliminated (one only enters step 2 once 464 they start having zero catches of vectors in the surveillance system); alternatively step 2´, calculate how long vector 465 control, and surveillance, should continue in order that the risk of obtaining a false negative is acceptably small; Step 3. If 466 the vector population is very low, even if not yet eliminated, calculate the probability of the remnant vector population 467 being eliminated naturally, purely by chance, without further control efforts; Step 4. Use growth models to estimate the 468 expected vector population at various times after the cessation of control efforts, assuming the survival of at least one 469 reproductive female. Failure to detect a rebound in the vector population after protracted periods supports a conclusion 470 that the vector has already been eliminated. If no rebound is detected then the control team must decide whether vector 471 control should resume or whether other disease control actions are necessary ( depends of hosts cases surveillance, 472 disease control status and logistical and budgetary constraints); Step 5. Vector elimination can then be declared with a 473 specified risk, and the risk of tsetse reinvasion can also be evaluated. 474 475 Applying the novel vector elimination modelling framework to our case study in Mandoul, Chad 476 477 Study area 478 Previously described by Mahamat et al. (13), the Mandoul region covers about 840 km2 in parts 479 of five cantons in Southern Chad (Figure 9). It is an historical focus of g-HAT due to T. brucei 480 gambiense, transmitted in the area by only one species of tsetse, G. f. fuscipes. The mean 481 elevation is ~400 meters and annual rainfall is between 1000 and 1200 mm: a wet season lasts 482 from June to October and a dry season from November to May. Vegetation, consisting of woody 483 savannah with gallery forest along the rivers, has been degraded in parts through agriculture. The 484 population comprises pastoralist livestock keepers and sedentary mixed crop -livestock farmers – 485 cultivating sorghum, sesame and sweet potatoes. 486 487 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 14 488 Figure 9. Map of Mandoul area (Chad within the continent, on the left, and Mandoul within Chad on the right). 489 490 Vector control (Tiny Target deployment) 491 Recent efforts to eliminate the Mandoul focus of g-HAT are based on the use of Tiny Targets (12) 492 (Figure 10), deployed after a baseline trapping survey had determined the numbers, species and 493 distribution of tsetse, thereby delineating the area to be controlled (Mahamat et al., 2017). The 494 Tiny Targets, provided by Vestergaard (Lausanne, Switzerland), comprised 0.25m × 0.25m blue 495 polyester flanked by 0.25m × 0.25m black polyethylene netting impregnated with deltamethrin at 496 300mg/m2 (12). Targets were deployed along the three main arms of the Mandoul River, where 497 tsetse were detected, and over an area up to 4 km beyond where tsetse were caug ht. Targets 498 were suspended from tree branches at 10-20 cm above the ground, using string, or erected with 499 wooden sticks obtained locally (Figure 10). 500 501 502 Figure 10. A Tiny Target – as deployed in the Mandoul focus (A: overall view on the river; B: zoom on a target) 503 504 505 A total of 2713 targets were deployed in January-February 2014 and replaced annually until 506 2022, whereupon 10% of them were removed in 2023, a further 40% removed in 2024, and the 507 last targets removed in April 2025. 508 Vector surveillance 509 In November 2013, prior to target deployment, the above-cited baseline survey involved the 510 deployment of 108 biconical traps (44) across the whole Mandoul area (Figure 11). The traps 511 were left in situ for 48 hours, and the numbers of G. f. fuscipes captured in each trap were then 512 recorded. Thereafter, sampling traps were deployed only at 44 sentinel sites and were operated 513 for two days, twice each year, from 2014 to 2024. 514 515 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 15 516 Figure 11. Map showing the deployment positions of the 108 traps used in the preliminary sampling exercise in Mandoul 517 518 Tiny targets are used to kill tsetse, whereas biconical traps are only used to catch live tsetse as a 519 monitoring system. Hence, from 2014-2025, tiny targets stayed active all year long (albeit being 520 sometimes replaced by new ones) while traps were only deployed two days at a time, a few times 521 a year, and then removed. 522 523 The five-step approach in Mandoul 524 Step 1) Baseline probability of capture 525 Although excellent baseline work, including a comprehensive baseline survey, was carried out in 526 Mandoul, this was a control operation, not a research study – and no trials were carried out to 527 estimate the efficacy of the traps and targets used in the control exercise. Moreover, neither life 528 history nor population dynamics studies were carried out on the population of G. f. fuscipes in the 529 Mandoul area. Accordingly, as detailed below, we rely on theory and past examples from other 530 study sites to parametrize some of our models to obtain the probability of capturing a tsetse with 531 the biconical traps used in Mandoul. 532 533 We first estimate the theoretical probability that a given vector control method could, in theory, 534 successfully eliminate an isolated population of tsetse (32). We then estimate the probabilities of 535 kill/capture after deployment of targets/traps for a series of days in Mandoul. Finally, using 536 information from Big Chamaunga Island, Kenya, we estimate the probability p that a female G. f. 537 fuscipes, alive in the Mandoul at the start of a given day, is killed by a target or captured by an 538 individual biconical trap. We assume that we are indeed dealing with an isolated population in 539 Mandoul, since the nearest tsetse population sampled beyond the borders of the Mandoul focus 540 is ~50km distant (45). 541 Hargrove (2005) (22) calculated the probability (s) of eliminating an isolated tsetse population, as 542 a function of the variables impacting female birth and death rates. The probability is given by the 543 solution of the equation: 544 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 16 s = (B+M+Bsqrt(B+M+B)2 – 4B(M+B))/2B (1) 545 where, for compactness, we write 546  = 1 - u, B = IT and M = 1 – I 547  Daily survival probability for adult females. 548  Daily survival probability for female pupae. 549 u Period between female adult eclosion and first ovulation (days). 550 I Inter-larval period (days). 551 T Pupal duration (days). 552  Probability deposited pupa is female. 553  Probability adult female is inseminated. 554 The probability of elimination is the smaller of the two roots of equation (1). This gives the 555 probability that the line emanating from a single female fly is eliminated. If generation zero 556 consists of N flies, all subject to the same survival probabilities and reproductive rates, the whole 557 population is eliminated with probability sN. 558 Evaluation of equation (1) shows that, if the mortality of adult female tsetse in an isolated 559 population can be maintained at a level of at least 3.5 – 4.0% per day, that population will be 560 eliminated with probability 1.0 (Figure 12) – even if there are no reproductive losses, and 561 regardless of density dependent effects (22). The minimum mortality required to be sustained 562 among adult females, in order to achieve elimination, naturally decreases as pupal mortality 563 increases. Thus, if pupal mortality is of the order of at least 1% per day, as estima ted for G. 564 pallidipes in Kenya (46), a sustained adult mortality of between 2.5 and 3.0% per day will ensure 565 elimination (Figure 12). 566 567 568 Figure 12. Probability of elimination of an isolated tsetse population as a function of the mortality of female adult females 569 and pupae (legend from 0 to 5%). Calculated for values of u = 7 days, I = 9 days, and T = 27 days. Redrawn from 570 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Probability of elimination Adult female mortality (%/day) 0% 1% 2% 3% 4% 5% .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 17 “Hargrove J.W. (2005) Extinction probabilities and times to extinction for populations of tsetse flies Glossina spp (Diptera: 571 Glossinidae) subjected to various control measures. Bulletin of Entomological Research 95, 13-21”. 572 573 Probabilities of kill/capture after deployment of targets/traps for a series of days in Mandoul 574 For the Chad case study, assume that there is a probability p that a female G. f. fuscipes alive in 575 the Mandoul at the start of a given day is killed by a target during that day. Then the probability 576 that this fly is not killed by a target during that day is q = 1 – p. In what follows we make frequent 577 use of the assumption that the probability that a fly is not killed by any target on a given day is 578 independent of the probability that it was not killed by any target on the previous day. By 579 extension, this assumption means that the fly escapes being killed by a target on n consecutive 580 days with probability Q = qn – and the probability that the fly has been killed by day n is P = 1 – Q 581 = 1 - qn. An analogous argument applies to the probabilities that a fly is captured, or evades 582 capture, by any given trap. 583 584 Estimate probability p that a female G. f. fuscipes, alive in the Mandoul at the start of a given day, 585 is killed by a target/captured by an individual trap – using information from Big Chamaunga 586 Island, Kenya 587 The probabilities of trapping G. f. fuscipes, or killing them using Tiny Targets, were estimated 588 using data from Tirados et al. (14), who monitored the decline in trap catches of G. f. fuscipes, 589 following the deployment of Tiny Targets on Big Chamaunga Island, (-0.426° latitude, 34.233° 590 longitude; surface area 0.2km2; circumference 1.5 km), which lies in the Kenyan section of Lake 591 Victoria. From January 2011 - December 2012, they deployed 30 Tiny Targets at 50m intervals in 592 the shoreline habitat of the island, giving a target density of 20 targets/km. They monitored the 593 impact of deploying targets along the island shore via monthly catches of tsetse, from four 594 biconical traps deployed, between 200 and 300m apart, along the lakeshore, and a further single 595 trap placed at the centre of the island. 596 597 Step 2) probability of elimination based on vector capture 598 We apply a probability model to the results of our surveillance efforts to reject the null hypothesis 599 that insects are still present following a series of days in Mandoul with no tsetse being captured. 600 We define: 601 A Area sampled (km2), assumed isolated (closed to immigration and emigration). 602 N Total insects surviving the eradication attempt, assumed randomly distributed in A. 603  Trap efficiency, i.e., the conditional probability that an insect is caught by a given trap, 604 given that there is only one trap present in the 1-km2 square containing the insect and 605 given that the insect is active. 606 S Number of traps present in all of A. 607 t Number of days for which each trap is operated. 608 With these definitions Hargrove (32) showed that the probability of capturing at least one fly is 609 approximately: 610 C(N, S, , t) = 1 – exp(–StN/A)  StN/A (2) 611 the approximation holding for populations close to elimination, when the exponent is small. 612 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 18 613 Hence the probability (C´) of null trapping, i.e., of catching no tsetse at all, is 614 C´(N, S, , t) = exp(–StN/A)  1 – StN/A (3) 615 We aim to minimize the risk (α) of falsely concluding – from a sequence of zero catches – that 616 tsetse have been eliminated when, in fact, there are still tsetse present. Accordingly, we typically 617 set α to some value close to zero; say, α = 0.01 or α = 0.001. Then, if we find that C´  1 – 618 StN/A < α, we conclude that tsetse have been eliminated, with the attendant risk, α, that our 619

is false. 620 621 Step 3) Probability of natural elimination 622 We estimate the probability that a very small residual population will be eliminated by chance. If 623 only a small number of tsetse survive a control operation, there is a non-zero probability that this 624 remnant population will be eliminated by chance – without the need for further control efforts. The 625 probability that this will occur is calculated using Equation (1) – with the assumption that 626 mortalities among adult and immature females can be much lower than when the population is 627 subjected to control measures. Figure 13 shows that even if only one inseminated female 628 survives, and if the background adult female mortality is 2% per day there is still a 40% chance 629 that the female will give rise to a surviving population. If there are 10 surviving inseminated 630 females, the population will be almost certain to survive – even if adult female mortality is 2.5% 631 per day. 632 633 634 Figure 13. Probability that a small residual tsetse population is eliminated by chance, as a function of adult female 635 mortality and the initial number of females in the residual population. Calculated from Equation (1) with the following input 636 parameters: Time to first ovulation, u=7 days; Inter-larval period, I=9 days. Pupal duration, T=30 days; Probability 637 deposited pupa is female, =0.5; Probability female is inseminated, =1.0. Redrawn from “Hargrove J.W. (2005) 638 Extinction probabilities and times to extinction for populations of tsetse flies Glossina spp (Diptera: Glossinidae) subjected 639 to various control measures. Bulletin of Entomological Research 95, 13-21”. 640 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Probability of elimination Adult female mortality (%/day) Elimination probability vs size of pioneer populationn = 1 n = 2 n = 4 n = 8 n = 16 n = 32 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 19 641 Step 4) Detection of potential rebound 642 If there is insufficient evidence – even after a series of zero catches of tsetse – to conclude that 643 tsetse have been eliminated, or that the remaining small tsetse population may b e eliminated 644 naturally, then there are two options. 645 The first, which we call “ step 2´ ” is to keep the vector control/Tiny Targets in place and to keep 646 sampling with traps until the sequence of zero catches is so long that we can have a high degree 647 of confidence that elimination has been achieved. Hence, using Equation 2, one calculates the 648 sampling duration needed before reaching a high confidence (below the risk ratio) that the vector 649 population has been eliminated. If feasible one could calculate a combination of increased 650 sampling efforts (more traps) and prolonged surveillance efforts. This has the advantage that the 651 presence of the Tiny Targets will ensure a low risk of a recurrence of cases of human 652 trypanosomiasis. It has the disadvantage, however, of incurring, for an unknown/long period, the 653 continued costs of employing the control team, and buying and deploying new Tiny Targets. 654 Moreover, these costs are wasted if the tsetse population has, in fact, already been eliminated. 655 The alternative is to remove all Tiny Targets from the control area, here Mandoul, and to continue 656 with the vector sampling effort – but to stop all control measures and aim at detecting a potential 657 rebound in vector population, which is our actual step 4. There are two possible outcomes of step 658 4: (i) The tsetse population has indeed already been eliminated – in which case the ongoing 659 sampling will fail to catch a fly, regardless of how long the sampling continues: there is then no 660 longer any need to carry out any manner of vector or disease control. (ii) The more interesting, 661 and problematic, possibility is that the surviving tsetse are not eliminated by chance after the 662 Tiny Targets have been removed. If this is the case, we expect the tsetse population to grow 663 steadily, particularly given that there is no longer any risk of the flies being killed by Tiny Targets. 664 Then one has to determine whether vector control should be restarted. 665 Failure to detect a rebound in the vector population after the cessation of control efforts w ill 666 support the conclusion that the vector has already been eliminated. A growth model is used to 667 estimate the expected vector population at various times after the cessation of control efforts, 668 assuming the survival of at least one reproductive female. If no tsetse can be captured, despite 669 predictions of a large population from the growth model, then the vector population can be 670 considered as eliminated. 671 The growth of an isolated population of tsetse is determined by the balance between the rates of 672 larval production, and development – and by the rates of immature and adult mortality, whether 673 natural or imposed by human intervention. Hargrove (33) estimated growth rates of tsetse 674 populations, as functions of birth and death rates, by calculating dominant eigenvalues of 675 appropriate Leslie matrices. Those results, summarized in Figure 5, are valid for any tsetse 676 population; it is only necessary to stipulate the appropriate levels of the birth and death rates. In 677 setting development rates for tsetse in the Mandoul area, we assume a mean daily temperature 678 of 28.4C, based on its hot dry equatorial location. In the absence of field estimates of the effects 679 of temperature on various development rates in G. f. fuscipes, we use relationships measured for 680 G. m. morsitans and G. pallidipes, which deposit their first larva at the age of about 14 days, and 681 subsequent larvae at 8-day intervals (32, 47, 48). Pupal duration for female G. m. morsitans, at a 682 constant temperature of 28.4C in the laboratory, is 21 days (49, 50). In the field, however, 683 temperatures in typical larviposition sites are about 2C cooler on average than ambient (51, 52). 684 Accordingly, we assume a temperature of 26.4C during pupal development, giving an expected 685 pupal duration of 24 days. We estimate possible growth rates for tsetse populations in the 686 Mandoul area, before and after the use of Tiny Targets, for a wide range of adult mortalities 687 (Figure 13). 688 689 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 20 Having predicted the growth of the tsetse population, one can then use Equations (2) and (3) of 690 step 2 to determine when we will have a 90%, 99% or 99.9% confidence that our zero catches, 691 and failure to detect a rebound, actually means elimination. 692 693 Step 5) Vector elimination and reinvasion risk 694 Finally, we evaluate whether the vector population has been successfully eliminated with a 695 specified level of risk, while also accounting for the potential threat of reinvasion. The above 696 methodologies, for estimating the probability of elimination of a tsetse population, apply to 697 situations where the tsetse population is isolated. If, however, we conclude that tsetse have been 698 eliminated from the Mandoul, we still need to estimate the probability that t he area could be 699 repopulated by tsetse invading from distant populations. The nearest tsetse population, beyond 700 the borders of the Mandoul focus is ~50km distant (45). 701 Dispersal in tsetse is generally modelled as a random walk (53–55) or, equivalently, as a diffusion 702 process (27). This last paper was published only as a hard copy and is not generally available. 703 Accordingly, it is included here as Supplementary file S1 and is used to estimate the probabili ty 704 that a tsetse fly, present at time 0 at a random point in a given area, will be found in some distant 705 neighborhood at time t later, given that it is still alive. In making these calculations for G. f. 706 fuscipes, we use – as a first approximation – Rogers’ (54) estimate that G. f. fuscipes moves an 707 average of 137 m (150 yds) per day and assume that this rate of movement does not vary 708 significantly with age. 709 The closest population of tsetse, G. f. fuscipes, beyond the borders of the Mandoul focus is in the 710 neighbourhood of Timbéri, some 50 km distant. We estimate the probability that a tsetse fly could 711 survive long enough to move between the two populations. We use the results of Hargrove & 712 Lange (27) in modelling tsetse dispersal as diffusion in the plane, starting at the origin when time t 713 = 0, with coefficient of diffusion σ2. The fly’s position (x, y, t) in time and space is then defined by 714 a normally distributed random variable with density function 715 f(x,y,t) = (1/(2πg)) exp(-(x2 + y2)/2g) 716 (4) 717 where g = g(t) = ∫ 𝜎2(𝑠) 𝑑𝑠 𝑡 0 718 As a first approximation we assume σ2 is independent of the fly’s age and position in the plane, 719 so that g = kt, where k is a constant. Consider a fly starting its dispersal at a point chosen 720 uniformly from the interval [a, b]. Then at some time t it will be in the interval [c, d] with probability: 721 1/(b – a) [Φ ((d – z)/√𝑘𝑡) (𝑧 − 𝑑) − Φ ((𝑐 – 𝑧)/√𝑔)(𝑧 − 𝑐) 722 + √𝑘𝑡/2𝜋 (exp(-(c – z)2/2kt) – exp(-(d – z)2/2kt)) ]𝑎 𝑏 723 (5) 724 Hargrove & Lange (1989) note that if the intervals [a, b] and [c, d] are widely separated, relative to 725 the rate of diffusion, such that (c-a)/(kt)0.5 >> 0, then Equation (5) can be much simplified 726 because: 727 ((c-a)/(kt)0.5  1 728 exp(-(c – a)2/2kt)  0 729 730 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 21 731 Acknowledgments 732 Dedicated to the memories of Hugh Barclay (1941 – 2022), who pioneered efforts to support the 733 declaration of elimination of tsetse populations, of Jean-Baptiste Rayaisse (1967-2020) who was 734 a fantastic tsetse entomologist, a hard worker, deeply committed to trypanosomiasis control, and 735 with a great sense of humour, and to Ali Bachar Alkatib, killed by bees during the first target 736 deployment in Mandoul. We acknowledge inputs from Profs Glyn Vale and Steve Torr on other 737 case studies and to guide parametrization. 738 739 Funding 740 The authors gratefully acknowledge the financial support of the European Union’s Horizon 2020 741 research and innovation programme under grant agreement n°101000467, acronym ‘’COMBAT’’ 742 (Controlling and Progressively Minimizing the Burden of Animal Trypanosomosis), and the 743 financial support of the Gates foundation (INV-001785) through the “TRYPA-NO” consortium. 744 JWH acknowledges ongoing support from CERI-SACEMA at Stellenbosch University. PB has 745 acknowledged support from Open Philanthropy (SYMBIOVECTOR), the Bill and Melinda Gates 746 Foundation (INV0225840). 747 International Centre of Insect Physiology and Ecology (ICIPE) also receives funding and support 748 from The Swedish International Development Cooperation Agency (Sida); the Swiss Agency for 749 Development and Cooperation (SDC); the Australian Centre for International Agricultural 750 Research (ACIAR); the Norwegian Agency for Development Cooperation (Norad); the German 751 Federal Ministry for Economic Cooperation and Development (BMZ); and the Government of the 752 Republic of Kenya. The views expressed herein do not necessarily reflect the official opinion of 753 the donors. 754 755 Author Contributions 756 JH conceptualization, data curation, formal analysis, investigation, methodology, software, 757 validation, visualization, Writing – Original Draft Preparation; MHM investigation, validation, 758 resources, Writing – Original Draft Preparation, MA investigation, validation, resources, Writing – 759 Original Draft Preparation; WY investigation, validation, resources, Writing – Original Draft 760 Preparation, DS investigation, validation, Writing – Review & Editing; JD investigation, resources, 761 Writing – Review & Editing; ES investigation, resources, Resources, Writing – Review & Editing; 762 IK Methodology, Resources, visualization, Writing – Review & Editing; AM Resources, Writing – 763 Review & Editing; PB formal analysis, visualization, Writing – Original Draft Preparation; PS 764 conceptualization, Funding Acquisition, Project Administration, methodology, validation, Writing – 765 Original Draft Preparation; AMGB conceptualization, data curation, Funding Acquisition, Project 766 Administration, formal analysis, investigation, methodology, software, validation, visualization, 767 Writing – Original Draft Preparation 768 Competing Interest Statement 769 The authors declare having no competing interests 770 771 Data, Materials, and Software Availability 772 All necessary data are available within the supplementary material or upon request and all 773 equations are available within the main text. 774 775 Supplementary material 776 • Supplementary file S1 (Hargrove & Lange, 1989) paper 777 • Supplementary file S2 processed data tsetse capture baseline Mandoul in 2013 and 778 capture surveys 2014-2024 (available upon request) 779 • Supplementary file S3 processed data on catches of female G. f. fuscipes, on Big 780 Chamaunga Island, Lake Victoria, Kenya (available upon request) 781 782 783 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 22

784 1. World Health Organization, Vector-borne diseases. WHO (2024). 785 2. J. R. Franco, et al., The elimination of human African trypanosomiasis: Achievements in 786 relation to WHO road map targets for 2020. PLoS Negl. Trop. Dis. 16, e0010047–e0010047 787 (2022). 788 3. A. Boisson-Walsh, Chad eliminates gambiense sleeping sickness. Lancet Infect. Dis. 24, 789 e551–e551 (2024). 790 4. R. Nuwer, Why the last cases of sleeping sickness will be the hardest to eliminate. Nature 791 (2025). https://doi.org/10.1038/D41586-025-00013-6. 792 5. D. Kaba, et al., Towards the sustainable elimination of gambiense human African 793 trypanosomiasis in Côte d’Ivoire using an integrated approach. PLoS Negl. Trop. Dis. 17, 794 e0011514–e0011514 (2023). 795 6. J. R. Franco, et al., The elimination of human African trypanosomiasis: Monitoring progress 796 towards the 2021–2030 WHO road map targets. PLoS Negl. Trop. Dis. 18 (2024). 797 7. P. Barreaux, et al., Priorities for Broadening the Malaria Vector Control Tool Kit. Trends 798 Parasitol. 33, 763–774 (2017). 799 8. P. A. Hancock, et al., Mapping trends in insecticide resistance phenotypes in African malaria 800 vectors. PLoS Biol. 18 (2020). 801 9. J. Hemingway, et al., Tools and Strategies for Malaria Control and Elimination: What Do We 802 Need to Achieve a Grand Convergence in Malaria? PLoS Biol. 14 (2016). 803 10. A. F. Read, P. A. Lynch, M. B. Thomas, How to make evolution-proof insecticides for malaria 804 control. PLoS Biol. 7, e1000058–e1000058 (2009). 805 11. A. M. G. Barreaux, et al., The role of human and mosquito behaviour in the efficacy of a 806 house-based intervention. Philos. Trans. R. Soc. B Biol. Sci. 376, 20190815–20190815 807 (2021). 808 12. J. Esterhuizen, et al., Improving the cost-effectiveness of visual devices for the control of 809 riverine tsetse flies, the major vectors of human African trypanosomiasis. PLoS Negl. Trop. 810 Dis. 5 (2011). 811 13. M. H. Mahamat, et al., Adding tsetse control to medical activities contributes to decreasing 812 transmission of sleeping sickness in the Mandoul focus (Chad). PLoS Negl. Trop. Dis. 11, 813 e0005792–e0005792 (2017). 814 14. I. Tirados, et al., Tsetse control and Gambian sleeping sickness; implications for control 815 strategy. PLoS Negl. Trop. Dis. (2015). https://doi.org/10.1371/journal.pntd.0003822. 816 15. K. S. Rock, et al., Update of transmission modelling and projections of gambiense human 817 African trypanosomiasis in the Mandoul focus, Chad. Infect. Dis. Poverty 2022 111 11, 1–13 818 (2022). 819 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 23 16. S. English, et al., “Incorporating Vector Ecology and Life History into Disease Transmission 820 Models: Insights from Tsetse (Glossina spp.)” in Population Biology of Vector-Borne 821 Diseases, (Oxford University Press, 2020), pp. 175–188. 822 17. A. M. G. Barreaux, P. Barreaux, K. Thievent, J. C. Koella, Larval environment influences 823 vector competence of the malaria mosquito Anopheles gambiae. MalariaWorld J. 7 (2016). 824 18. A. M. G. Barreaux, C. M. Stone, P. Barreaux, J. C. Koella, The relationship between size and 825 longevity of the malaria vector Anopheles gambiae (s.s.) depends on the larval environment. 826 Parasit. Vectors 11, 485–485 (2018). 827 19. A. J. Mackay, J. Yan, C. H. Kim, A. M. G. Barreaux, C. M. Stone, Larval diet and temperature 828 alter mosquito immunity and development: using body size and developmental traits to track 829 carry-over effects on longevity. Parasit. Vectors 16, 1–13 (2023). 830 20. J. S. Lord, J. W. Hargrove, S. J. Torr, G. A. Vale, Climate change and African 831 trypanosomiasis vector populations in Zimbabwe’s Zambezi Valley: A mathematical 832 modelling study. PLOS Med. 15, e1002675–e1002675 (2018). 833 21. J. S. Lord, et al., Effects of maternal age and stress on offspring quality in a viviparous fly. 834 Ecol. Lett. ele.13839-ele.13839 (2021). https://doi.org/10.1111/ELE.13839. 835 22. J. W. Hargrove, Extinction probabilities and times to extinction for populations of tsetse flies 836 Glossina spp. (Diptera: Glossinidae) subjected to various control measures. Bull. Entomol. 837 Res. 95, 13–21 (2005). 838 23. H. J. Barclay, J. W. Hargrove, Probability models to facilitate a declaration of pest -free status, 839 with special reference to tsetse (Diptera: Glossinidae). Bull. Entomol. Res. 95, 1–11 (2005). 840 24. E. B. Are, J. W. Hargrove, J. Dushoff, Does Counting Different Life Stages Impact Estimates 841 for Extinction Probabilities for Tsetse (Glossina spp)? Bull. Math. Biol. 83, 1–13 (2021). 842 25. G. A. F. Seber, The estimation of animal abundance : and related parameters, Second 843 edition. (Griffin, 1982). 844 26. B. K. Williams, J. D. Nichols, M. J. Conroy, Analysis and Management of Animal Populations: 845 Modeling, Estimation and Decision Making. (2002). 846 27. J. W. Hargrove, K. Lange, Tsetse dispersal viewed as a diffusion process. Trans. Zimb. Sci. 847 Assoc. 64, 1–8 (1989). 848 28. L. R. Haines, et al., Big Baby, Little Mother: Tsetse Flies Are Exceptions to the Juvenile 849 Small Size Principle. BioEssays 2000049–2000049 (2020). 850 https://doi.org/10.1002/bies.202000049. 851 29. J. W. Hargrove, M. O. Muzari, S. English, How maternal investment varies with 852 environmental factors and the age and physiological state of wild tsetse Glossina pallidipes 853 and Glossina morsitans morsitans. R. Soc. Open Sci. 5, 171739–171739 (2018). 854 30. K. Heller, Tsetse Flies Rely on Symbiotic Wigglesworthia for Immune System Development. 855 PLoS Biol. 9, e1001070–e1001070 (2011). 856 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 24 31. B. L. Weiss, J. Wang, S. Aksoy, Tsetse Immune System Maturation Requires the Presence 857 of Obligate Symbionts in Larvae. PLOS Biol. 9, e1000619–e1000619 (2011). 858 32. J. W. Hargrove, Tsetse eradication: sufficiency, necessity and desirability Telephone. DFID 859 Anim. Health Programme 133-+ix (2003). 860 33. J. W. Hagrove, Tsetse: the limits to population growth. Med. Vet. Entomol. 2, 203–217 861 (1988). 862 34. D. A. Turner, R. Brightwell, An evaluation of a sequential aerial spraying operation against 863 Glossina pallidipes Austen (Diptera: Glossinidae) in the Lambwe Valley of Kenya: aspects of 864 post-spray recovery and evidence of natural population regulation. Bull. Entomol. Res. 76, 865 331–349 (1986). 866 35. G. A. Vale, J. W. Hargrove, G. F. Cockbill, R. J. Phelps, Field trials of baits to control 867 populations of Glossina morsitans morsitans Westwood and G. pallidipes Austen (Diptera: 868 Glossinidae). Bull. Entomol. Res. 76, 179–179 (1986). 869 36. R. D. Dransfield, The Range of Attraction of the Biconical Trap for Glossina pallidipes and 870 Glossina brevipalpis. Int. J. Trop. Insect Sci. 1984 55 5, 363–368 (1984). 871 37. P. M. Kgori, S. Modo, S. J. Torr, The use of aerial spraying to eliminate tsetse fro m the 872 Okavango Delta of Botswana. Acta Trop. 99, 184–199 (2006). 873 38. J. W. W. S., The eradication of sleeping sickness from Principe. Nature 98, 311–312 (1917). 874 39. R. Du Toit, Trypanosomiasis in Zululand and the control of tsetse flies by chemical means. 875 (1954). 876 40. M. J. B. Vreysen, et al., Glossina austeni (Diptera: Glossinidae) Eradicated on the Island of 877 Unguja, Zanzibar, Using the Sterile Insect Technique. J. Econ. Entomol. 93, 123–135 (2000). 878 41. J. Bouyer, et al., Mapping landscape friction to locate isolated tsetse populations that are 879 candidates for elimination. Proc. Natl. Acad. Sci. U. S. A. 112, 14575–14580 (2015). 880 42. A. H. Dicko, et al., Using species distribution models to optimize vector control in the 881 framework of the tsetse eradication campaign in Senegal. Proc. Natl. Acad. Sci. 111, 10149–882 10154 (2014). 883 43. O. Diall, et al., Developing a Progressive Control Pathway for African Animal 884 Trypanosomosis. Trends Parasitol. 33, 499–509 (2017). 885 44. A. Challier, C. Laveissière, Un nouveau piège pour la capture des glossines (Glossina: 886 Diptera, Muscidae) déscription et essais sur la terrain. Cah. ORSTOM 11, 251–262 (1973). 887 45. S. Ravel, et al., Population genetics of Glossina fuscipes fuscipes from southern Chad. Peer 888 Community J. 3, e31–e31 (2023). 889 46. D. J. Rogers, S. E. Randolph, Estimation of rates of predation on tsetse. Med. Vet. Entomol. 890 4, 195–204 (1990). 891 47. J. W. Hargrove, Reproductive rates of tsetse flies in the field in Zimbabwe. Physiol. Entomol. 892 19, 307–318 (1994). 893 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint 25 48. J. W. Hargrove, Towards a general rule for estimating the stage of pregnancy in field ‐caught 894 tsetse flies. Physiol. Entomol. 20, 213–223 (1995). 895 49. R. J. Phelps, P. M. Burrows, Puparial duration in glossina morsitans orientalis under 896 conditions of constant temperature. Entomol. Exp. Appl. 12, 33–43 (1969). 897 50. J. W. Hargrove, G. A. Vale, Models for the rates of pupal development, fat consumption and 898 mortality in tsetse (Glossina spp). Bull. Entomol. Res. 1–13 (2019). 899 https://doi.org/10.1017/S0007485319000233. 900 51. R. J. Phelps, P. M. Burrows, Lethal temperatures for puparia of glossina morsitans orientalis. 901 Entomol. Exp. Appl. 12, 23–32 (1969). 902 52. P. J. Jackson, R. J. Phelps, Temperature regimes in pupation sites of Glossina morsitans 903 orientalis Vanderplank (Diptera). Rhod. Zamb. Malawi J. Agric. Res. 5, 249–260 (1967). 904 53. E. Bursell, “Dispersal and concentration of Glossina” in The African Trypanosomiases, H. W. 905 Mulligan, Ed. (George Allen and Unwin, 1970), pp. 382–394. 906 54. D. Rogers, Study of a Natural Population of Glossina fuscipes fuscipes Newstead and a 907 Model of Fly Movement. J. Anim. Ecol. 46, 309–309 (1977). 908 55. J. W. Hargrove, Tsetse Dispersal Reconsidered. J. Anim. Ecol. 50, 351–351 (1981). 909 910 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 14, 2025. ; https://doi.org/10.1101/2025.09.09.675028doi: bioRxiv preprint