Omitting age-dependent mosquito mortality in malaria models underestimates the effectiveness of insecticide-treated nets

Mathematical models of vector-borne infections, including malaria, often assume age-independent mortality rates of vectors, despite evidence that many insects senesce. In this study we present survival data on insecticide-resistant Anopheles gambiae s.l. from experiments in Côte d’Ivoire. We fit a constant mortality function and two age-dependent functions (logistic and Gompertz) to the data from mosquitoes exposed (treated) and not exposed (control) to insecticide-treated nets (ITNs), to establish biologically realistic survival functions. This enables us to explore the effects of insecticide exposure on mosquito mortality rates, and the extent to which insecticide resistance might impact the effectiveness of ITNs. We investigate this by calculating the expected number of infectious bites a mosquito will take in its lifetime, and by extension the vectorial capacity. Our results show that the predicted vectorial capacity is substantially lower in mosquitoes exposed to ITNs, despite the mosquitoes in the experiment being highly insecticide-resistant. The more realistic age-dependent functions provide a better fit to the experimental data compared to a constant mortality function and, hence, influence the predicted impact of ITNs on malaria transmission potential. In models with age-independent mortality, there is a great reduction for the vectorial capacity under exposure compared to no exposure. However, the two age-dependent functions predicted an even larger reduction due to exposure, highlighting the impact of incorporating age in the mortality rates. These results further show that multiple exposures to ITNs had a considerable effect on the vectorial capacity. Overall, the study highlights the importance of including age dependency in mathematical models of vector-borne disease transmission and in fully understanding the impact of interventions.

Malaria is a life-threatening vector-borne parasitic disease, which is endemic in 2 87 countries, mainly in the African Region [1]. The World Health Organization's 3 (WHO) "Global Technical Strategy for Malaria 2016-2030" outlines global targets 4 in the fight against malaria, including a 90% reduction of malaria case incidence 5 by 2030 [2]. Significant progress towards these targets has occurred, where both 6 the number of cases and the number of deaths due to malaria have decreased 7 between 2010-2019, as outlined in a reported published by WHO in 2020 [1]. In 8 2019, there were around 229 million cases of malaria globally, with 94% of them 9 being in the African Region. Additionally, during the same year, 409,000 deaths 10 due to malaria have been estimated worldwide. 11 The vectors responsible for the malaria parasite's transmission, through blood 12 feeding, belong to the Anopheles genus of mosquitoes. The success of the malaria 13 programmes to date is thanks to a range of interventions, most commonly 14 targeted at these vectors. For example, in sub-Saharan Africa around half of the 15 people at risk are sleeping under insecticide-treated nets (ITNs) [1] which are 16 a way to utilise contact pesticides. Between 2000 and 2015, ITNs contributed 17 to the aversion of many cases; 68% of the cases that were prevented due to any 18 intervention are attributed to ITNs, making them a crucial intervention [3]. Bed- 19 nets are currently treated with pyrethroids, and there is evidence of an increase 20 in pyrethroid resistance in malaria vectors, which threatens the elimination 21 efforts [1,4]. Hence, due to mutations and natural selection, mosquitoes develop 22 the ability to resist the harmful effects of insecticides, leading to what is called 23 "insecticide resistance" [4]. We note that there are new dual active ingredient 24 (dual-AI) ITNs against malaria being tested [5][6][7], nevertheless, it is important 25 to evaluate the potential impact that insecticide resistance actually has on the 26 efficacy of current malaria interventions [8]. Alout et al. [9] claim that despite 27 insecticide resistance, vector control is still crucial and can be effective against 28 malaria transmission. This is in agreement with the systematic review and 29 meta-analysis by Strode et al. [10], where the authors concluded that ITNs are 30 more effective than untreated bed-nets, despite insecticide resistance. There has 31 been a mixture of results between different studies [11][12][13][14][15], however, we aim 32 to confirm and add to the existing knowledge regarding long-term impact of 33 insecticide on longevity by using highly resistant field-derived mosquitoes from 34 Côte d'Ivoire in laboratory experiments. 35 By targeting mosquitoes, ITNs target the key point in the transmission cycle, 36 not only reducing the opportunity for blood-feeding on humans, but increasing 37 vector mortality and, therefore, decreasing the number of infectious blood-meals 38 a mosquito will contribute during its life. The use of vector control to reduce or 39 even eliminate infection is well supported by mechanistic transmission models 40 originally developed for malaria by Ross and Macdonald and used extensively ever 41 since [16]. A key metric linked to transmission models is the basic reproduction 42 number, R 0 , which describes the number of secondary cases produced by a single 43 case in an otherwise susceptible population [17]. Garret-Jones [18] took the 44 purely entomological components of R 0 and named them vectorial capacity. It 45 is defined in [19] as "the expected number of infective mosquito bites that would 46 eventually arise from all the mosquitoes that would bite a single fully infectious 47 person on a single day", i.e. the average number of humans that get infected 48 due to one infectious human per day. These metrics have been used to study 49 the dynamics of vector-borne diseases and also quantitatively assess the possible 50 impact of interventions to control them. 51 Contact pesticides were being used at the time Macdonald was researching 52 vector control, and that is when he realised that transmission potential was 53 affected by two important factors relating to mosquito longevity [16]: 54 (a) a mosquito which is infected will only become infectious if it survives the 55 time needed for the pathogen to develop, commonly known as the extrinsic 56 incubation period (EIP), and 57 (b) once the mosquito is infectious it must take a blood-meal in order to 58 transmit the infection on to a host. 59 Hence, Macdonald concluded that the number of infectious bites taken by a 60 mosquito will increase the longer the mosquito survives [16,19]. The longer the 61 EIP is, the less chance a mosquito has to survive it, therefore the younger a 62 mosquito is when it gets infected, the more likely it is to transmit the infection. 63 Thus, the transmission potential relies heavily on the survival of the mosquitoes 64 [16]. Due to Macdonald's analysis, many control programmes aim to reduce the 65 lifespan of mosquitoes [16].
for transmission models and can underestimate the impact of vector control 70 strategies [20][21][22][23][24]. The assumption is often used to simplify, otherwise complex, 71 mathematical models, and not because of its biological relevance [24,25]. On 72 the other hand, it is rare that suitable age-dependent vector mortality data are 73 available to inform more complex models; in addition to average life expectancy 74 with and without ITNs in place, the distribution of life expectancy and how 75 this is impacted by intervention is also required. In the present study, we bring 76 more complex mosquito modelling together with detailed experimental data to 77 demonstrate how we may rethink the way the vectorial capacity is calculated.

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To investigate the impact of multiple insecticidal exposures on the mosquitoes 79 and their ability to transmit malaria, we address the following research questions: 80 To answer these, we use data collected in Côte d'Ivoire. Laboratory experiments 86 were conducted on field-derived female Anopheles gambiae s.l. mosquitoes, one 87 of the main malaria vectors in Côte d'Ivoire [26]. The setup of the experiments 88 allowed the comparison of the mosquitoes' survival rates when they were exposed 89 to standard ITNs versus when they were exposed to untreated nets. We fitted 90 various survival functions to these data to estimate biologically realistic mosquito 91 mortality rates and used these to obtain the vectorial capacity estimates of the 92 mosquitoes with and without exposure to ITNs. By including realistic survival 93 in the calculation of the vectorial capacity, we can observe how the assumed 94 effectiveness of anti-vectorial interventions is affected.

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The modelling analysis of the effect on the vectorial capacity with and without 97 the presence of insecticides is conducted using data that were collected in Bouaké, 98 Côte d'Ivoire. Here the focus is on the malaria parasite Plasmodium falciparum, 99 which is one of the six Plasmodium species known to regularly infect humans, 100 and both the most prevalent and deadly parasite in sub-Saharan Africa [27]. We 101 present a detailed outline of the experimental setup, along with a presentation of 102 the data, followed by a breakdown of the computation of the vectorial capacity 103 using these data. The laboratory experiment was conducted on field-derived Anopheles gambiae 114 s.l. at 26 ± 1°C and consisted of two treatments: 115 (a) control (non-exposed): in the presence of an untreated net 116 (b) treated (exposed): in the presence of an ITN 117 where a one-way tunnel with two cages was used; a 'holding' cage for the 118 mosquitoes and a 'host' cage for the volunteer's foot that was covered with a 119 treated or untreated net (see Fig 1). The mosquitoes in the experiment are 120 considered to be extremely resistant to the pyrethroid deltamethrin, which was 121 used to treat the ITNs. More details regarding the experimental setup are 122 presented further on. Depending on the chosen treatment for the experiment, the net is either an unwashed PermaNet 2.0 or an untreated net, measuring 25 × 25cm in both cases. In this setup the mosquitoes have direct access to the foot for blood feeding.

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Mosquito populations 124 Anopheles gambiae s.l. mosquitoes were collected as larvae in natural breeding 125 habitats around Bouaké, in central Côte d'Ivoire, and colonised at the Pierre 126 Richet Institute. These are highly pyrethroid-resistant mosquitoes. The 1014F 127 kdr mutation is almost fixed (≥ 90%), and 1575Y, as well as upregulation 128 of CYP6M2, CYP6P3, and CYP9K1 result in a ≥ 1500-fold resistance to 129 deltamethrin relative to a standard susceptible strain [28]. Larvae were reared 130 at 27 ± 2°C, 60 ± 20% RH and ambient light in metallic bowls of 300 larvae 131 with 1L of deionised water. They were fed daily with TetraMin baby fish food 132 following a standardised 'high food' regime as described in Kulma et al. [29]. 133 Adult mosquitoes were kept in 32.5 × 32.5 × 32.5cm mosquito cages, in the same 134 environmental conditions as the larvae, and maintained on 10% sugar solution. 135 Mosquitoes were four to five days old at their first exposure to insecticide and 136 randomly assigned to a net treatment. Females had constant access to egg laying 137 substrate (a wet cotton pad) and were maintained on a 10% sugar solution cotton 138 that was renewed daily. Sugar was removed to starve mosquitoes for four hours 139 before each experimental run.

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Human host preparation 141 The volunteers involved in this experiment were not actively infected with malaria. 142 They avoided the use of fragrance, repellent products, tobacco, and alcohol for 143 12 hours before and during testing. For the experiment, feet were washed with 144 unscented soap and rinsed with water the day before a test. The 'host' cages were 145 also washed with soap and rinsed with water every time after a test was conducted 146 -to avoid the accumulation of insecticide particles. Cages were not interchanged 147 between treatments, i.e. a cage used in a control treatment was always used for 148 a control treatment. Note that the data were analysed anonymously.

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Insecticide-treated nets (ITNs) 150 As mentioned, two types of nets were used: an unwashed PermaNet 2.0 (Vester-151 gaard Frandsen SA, DK) and an untreated polyester net (Coghlan's) for the 152 control treatment. The PermaNet 2.0 is a long-lasting insecticidal net made 153 of polyester and coated with 55 mg /m 2 ± 25% deltamethrin. We confirmed net 154 efficacy by exposing sensitive mosquitoes (Kisumu strain) to WHO tubes lined 155 with a piece of ITN; all mosquitoes died within 24 hours when exposed to the 156 ITN, while the untreated nets killed none. For the wild-type mosquitoes, as 157 these exhibit such a high level of resistance, there was negligible knockdown or 158 mortality (< 1% knockdown one hour post exposure and no mortality 24 hours 159 later) from ITN exposure in WHO cone assays [30]. In malaria-endemic settings with high ITN usage, mosquitoes potentially con-162 tact ITNs every time they attempt to feed. To capture this effect, we used a 163 tunnel test in which mosquitoes had to fly a short distance between two cages 164 to locate the host and blood-feed. The tunnel apparatus comprised a standard 165 32.5 × 32.5 × 32.5cm mosquito cage as a 'holding' cage, a 32.5 × 32.5 × 32.5cm 166 mosquito cage as the 'host' cage, and a PLEXIGLAS tube (l = 30cm, d = 14.6cm) 167 forming the tunnel between cages (as shown in Fig 1). The holding cages were 168 initially populated with 120 pupae each. After adult emergence, 50 females and 169 10 males were randomly selected to remain in each holding cage until their death, 170 with the excess removed and discarded. We compared the two net treatments, 171 where the foot of a human host was wrapped in a netting sock so the mosquitoes 172 could land on the foot and feed if they chose. Treatments were replicated five 173 times, giving a total of 500 female mosquitoes. Every four days at around 6pm 174 (dusk) the mosquitoes were exposed to a human foot placed in the 'host' cage. 175 The mosquitoes were allowed to visit the cage for 30 minutes. At the end of 176 30 minutes, the total number of mosquitoes that had taken a full or partial 177 blood-meal was recorded. The tunnel was then dismantled and all mosquitoes 178 returned to their respective holding cages. The surveillance of the mosquitoes 179 started when they were four days old and tests were repeated every four days 180 until all mosquitoes had died. The number of mosquito deaths was recorded daily. 181 The net treatment was randomly allocated to one human host experimenter to 182 ensure there were no biases due to possible differences in attraction between hosts. 183

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During an initial inspection of the data, it was decided that Replicate 1 would 185 be excluded from further analysis. The feeding trend (S1 Fig) of Replicate 1 186 suggests that the mosquitoes were not feeding and could therefore explain the 187 mortality trend (S2 Fig). The differences in Replicate 1 highlight the fact that 188 mosquito behaviour, for example feeding and host searching, likely depends on a 189 lot of parameters that need further exploration in order to improve transmission 190 models. It seems that there is no significant difference between mosquitoes 191 tested in each treatment in Replicate 1, and this is because they almost never 192 visit the cage with the foot, and thus were not exposed to insecticide during 193 their life. If we were to compare Replicate 1 with the other replicates, it seems 194 probable that blood-meals improve longevity, hence why in both treatments of 195 Replicate 1 the mosquitoes die so early. However, the objective here is to better 196 identify differences of the survival rates between the two treatments, the impact 197 on vectorial capacity, and contrasts when taking age dependencies into account. 198 Since Replicate 1 does not follow the behaviour of the mosquitoes in the rest of 199 the replicates, it is removed from further analysis for consistency. 200 In the results that follow, the replicates were all combined together, as one 201 larger, aggregated survival dataset, with data from 200 mosquitoes per treatment 202 being examined in total. Fig 2 shows the data used for the calculations. The 203 relevant data can be found in the supporting information (S1 Table, S2 Table, 204  S3 Table, S4 Table, S5 Table, and S6 Table). 205 Vectorial capacity 206 Garrett-Jones [18] introduced vectorial capacity, denoted as C, in order to 207 estimate the risk of the introduction of malaria. C is commonly defined mathe-208 matically as: where m is the mosquito density relative to humans, α is the biting rate, p is the 210 survival rate, µ is the mortality rate, and n is the duration of the EIP, i.e. the 211 number of days between the day a mosquito gets infected until its bites become 212 infectious, and is able to transmit the infection [22].

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It is important to note that this form of the vectorial capacity is predicated on 214 some key assumptions. The first is assuming perfect transmission, and it can be 215 overcome by including the product cb, denoting the vector competence [25,31,32]. 216 However, for the purposes of this study, we will also assume perfect transmission. 217 A second assumption is that the bite rate is fixed and constant with age. It is often 218 calculated as the reciprocal of the average length of the gonotrophic cycle [33]. 219 Hence, assuming a gonotrophic cycle of length four, we have α = 0.25 days −1 , i.e. 220 the mosquitoes feed once every four days, which is in line with the experimental 221 setup, where the mosquitoes were allowed to feed every four days. In our 222 calculations, we will use α = 0.25 and keep m as an unknown constant. However, 223 we further investigate the remaining parameters. Another notable assumption is the use of a fixed EIP in Eq (1), with the expression 226 e −µn representing the probability that a mosquito survives the n-day EIP [34]. 227 However, this can be modified to account for non-fixed EIP. A popular choice 228 is to assume the EIP follows an exponential distribution [25]. If the average 229 EIP duration is the same as with the fixed EIP (n), then the vectorial capacity 230 becomes: where σ = 1 n is the incubation rate.

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As mentioned, the EIP represents the time period where the malaria parasites 233 ingested by the mosquito are developing inside it in order to be able to transmit 234 the infection. The Centers for Disease Control and Prevention states that n 235 is at least 9 days, but is dependent on temperature and the different kind of 236 species of the parasites [35]. In a recently published paper by Stopard et al. [36], 237 a mechanistic model fitted to data from [37][38][39][40] gave an estimate for the median 238 (50 th percentile -EIP 50 ), at 27°C, to be 10.2 days. In addition to the median, 239 the authors provide values for the 10 th and 90 th percentile (EIP 10 and EIP 90 , 240 respectively), so, using these we can calculate a mean value for the EIP for a 241 given distribution at this temperature.

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Taking into account the information on the shape of the EIP, it would be 243 more realistic to consider a different distribution which lies somewhere between 244 the exponentially distributed or fixed EIPs. A mathematically neat choice which 245 meets this criteria and is flexible with the addition of a single extra parameter is 246 the Erlang distribution. The Erlang distribution is a special case of the gamma 247 distribution, which has been used for the incubation period in many vector-borne 248 disease models [41][42][43][44][45].

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The probability density function (PDF) of the Erlang distribution [46] is given by: where k is the shape parameter and λ is the rate parameter. The 250 mean is given by k λ , which we set equal to 1 σ . Therefore, with λ = kσ we have: 251 We match EIP 10 and EIP 90 from [36] to our distribution to infer values for k and 252 σ. We then take ⌊k⌋, since the Erlang distribution requires k to be an integer, 253 obtaining k = 31. Using this value for k and the value for EIP 50  The mortality rate is often assumed to be constant, which drastically simplifies mathematical calculations [24]. Nevertheless, it can be seen from our data that this assumption is not the most realistic. Styer et al. argue in [20] that ignoring mosquito senescence results in inaccurate predictions with respect to vector control effectiveness. For a more realistic approach, we consider an agedependent mortality rate. Henceforth, we consider the following functions for µ: Logistic [47]: Gompertz [23]: µ G (a) = g 1 e ag 2 where a ≥ 0 is the age of the mosquitoes, and the (positive) parameters µ const , 257 µ 1 , µ 2 , µ 3 , g 1 , and g 2 are estimated by fitting the data. These functions are 258 often used in demography and population models, since they describe a mortality 259 that increases with age [22,48]. The logistic function has an initial exponential 260 increase and then slows down to reach a plateau, whereas the Gompertz continues 261 to increase exponentially. In the field of survival analysis they are often referred 262 to as 'hazard functions'. For clarity we note that the Gompertz function (Eq (6)) 263 is the hazard function derived from the Gompertz distribution. However, this 264 is not the case for the logistic function (Eq (5)), here this represents a sigmoid 265 curve often referred to as the logistic function.

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Using the data, we need to obtain estimates for the unknown parameters in the 268 mortality functions so that we can calculate the vectorial capacity. In order 269 to investigate the differences in the vectorial capacity between the control and 270 treated cases, and between the various mortality functions, we need to reconsider 271 the way the vectorial capacity is calculated.

Mortality and survival functions 273
In order to fit the data, a survival function, S(a), is considered for each mortality 274 function. Its relationship with the mortality function [49] is expressed as: Using Eq (7), we obtain the survival function for each of the mortality functions: We use maximum likelihood estimation to estimate the various parameters for 276 our survival functions (Eqs (8), (9), and (10)) with the help of the Optim.jl [50] 277 package in Julia [51]. Further on, we obtain a 95% Wald confidence interval for 278 each of the parameters. More on the fitting method can be found in the S1 File. 279 The parameter values obtained are shown in Table 1.

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We use Monte Carlo simulations to get 95% confidence intervals for the 281 functions of the parameters reported in the Results section. We do so by 282 simulating 10,000 random samples from the multivariate normal distribution 283 fitted to the parameter estimates. In the multiple parameter cases, we use the 284 estimated parameters as the mean and the inverse of the Fisher Information 285 Matrix as the covariance matrix. We then take the quantile values at 2.5% and 286 97.5% to obtain marginal confidence intervals of the functions of interest.  comparing each treatment. Note that on the survival plots we add the Kaplan-290 Meier estimator, a non-parametric method frequently used in survival analysis, 291 which we obtain using the Survival.jl package.

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Rethinking the vectorial capacity 293 We outline the calculations required to obtain the expected number of bites a 294 mosquito takes after being infected and show how this is linked to the vectorial 295 capacity. This will give a more realistic value for C. To aid our calculation of 296 the expected number of infectious bites, we pose the following four questions: 297 1. What is the probability of a mosquito surviving the EIP given an infectious 298 blood-meal is taken at age a 0 ? 299 2. How many bites will the mosquito take if it has survived the EIP? We attempt to 305 answer these questions in four steps. In each step we explore the three different 306 mortality functions: case (i) being the age-independent (Eq (4)), case (ii) 307 the logistic (Eq (5)), and case (iii) the Gompertz (Eq (6)), and subsequently 308 compare their results. More detailed calculations can be found in the S2 File. Plots for the proportion of surviving mosquitoes at each day for the control (non-exposed) and treated (exposed) cases using the three different survival functions (age-independent, logistic, Gompertz). In the treated column, all mosquitoes are dead by day 33. The fits are extended to day 64 for a better comparison between the two treatments. The shaded area around the control and treated curves represents the 95% confidence interval due to the error propagated from the parameter estimates in each function using Monte Carlo simulations. The shaded area around the Kaplan-Meier estimator represents the pointwise log-log transformed 95% confidence interval.  Mosquito timeline after taking an infectious blood-meal. We assume that a mosquito takes an infectious blood-meal at age a 0 . In order for it to become infectious, it must survive the EIP. After surviving the EIP, at age a 1 , the mosquito will take infectious blood-meals up until its death, at a 2 . [Note: the mosquito might not survive the EIP, hence, it is possible that a 2 < a 1 .] Step 1: P P P(mosquito survives EIP | infectious blood-meal at a 0 ) 310 Case (i): P(surviving EIP | infectious blood-meal at age a 0 ) Case (iii): P(surviving EIP | infectious blood-meal at age a 0 ) e a 0 g 2 (e g 2 t −1) dt (13) Cases (ii) and (iii) cannot be solved analytically, but we integrate them numer-311 ically using the QuadGK.jl package in Julia. The results for the three cases are 312 plotted and shown in Fig 6. 313 From this first step we notice that the age-independent case is not biologically 314 realistic. Comparing it with the age-dependent cases, we can see that as a 0 315 increases, the probability a mosquito will survive the EIP decreases significantly, 316 which makes sense, given that there is evidence mosquitoes senesce both in the 317 data used here, but also from Styer et al. and Ryan et al. [20,24]. We also 318 notice that the Gompertz function's curve approaches zero faster, which could be 319 somewhat more realistic, whereas the logistic function reaches a plateau above 320 zero. Comparing the two treatments, control and treated, we notice that the 321 probability is lower initially with treatment, but also that the age-dependent 322 functions approach zero much faster, which is in line with the mosquitoes having 323 a lower life expectancy in the treated case (Fig 3).  (13) over different ages. The shaded area represents the 95% confidence interval due to the error propagated from the parameter estimates using Monte Carlo simulations.
Step 2: P P P(z = j | mosquito exits EIP at a 1 ) 325 We are interested in the probability mass function (PMF) of the number of bites, z, supposing the mosquito exits the EIP at age a 1 , and dies at age a 2 .
We also plot the PMFs for a specific age (a 1 = 15) which can be found in S3 Fig. 326   Fig 7. Heatmaps of the probability the number of bites is equal to some j given that the mosquito exits the extrinsic incubation period (EIP) at age a 1 . The heatmaps are obtained from the fitted parameters. The error bars on the mean markers represent the 95% confidence interval due to the propagated error for the fitted parameters using Monte Carlo simulations.
In Fig 7 we can see that, for the age-dependent cases, the older the mosquito 327 is when it exits the EIP, the higher the probability that the number of bites it 328 takes is small. The two age-dependent functions give similar results, but the 329 age-independent function shows again that it is not biologically realistic, since it 330 has a constant average and a constant probability across all ages. In the treated 331 case, their is a higher probability that there are low, or even zero, bites compared 332 to the control case; this happens for all choices of mortality function.

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Step 3: E E E(z | infectious blood-meal at a 0 ) 334 We can now calculate the expected number of infectious bites a mosquito takes, 335 given that it takes an infectious blood-meal at age a 0 : Hence, we need to calculate the PMF of the number of bites, given that an infectious blood-meal is taken at age a 0 . To do so, we use the results from Steps 1 and 2. For example, in Case (i) we multiply the results from Eqs (11) and (14), however care must be taken for when j = 0, where we need to consider that we definitely have zero bites if the mosquito does not survive the EIP. Case (i): Cases (ii) and (iii): (15) or Eq (16)] where we use Eq (15) for case (ii) and Eq (16) for case (iii). The above is true 338 for j ̸ = 0. When j = 0, we must add (1 − Eq (12)) to Eq (19) for case (ii), or 339 (1 − Eq (13)) for case (iii), following the same logic as in case (i). This is again 340 integrated numerically (this time using the Cuba.jl [52,53] package in Julia) and 341 put into Eq (17)   The expected number of bites decreases significantly if we consider an age-344 dependent mortality function, as we can see in Fig 8. There is also an obvious 345 difference between the two treatments, where in the treated case the number of 346 bites are a lot lower to begin with. Once again the logistic function seems to be 347 plateauing above zero. The expected number of bites a mosquito will take in its lifetime given it has taken an infectious blood-meal at age a 0 . The shaded area represents the 95% confidence interval due to the error propagated from the parameter estimates using Monte Carlo simulations.
Step 4: E E E(z) 349 We can now use the previous steps to calculate the expected number of infectious 350 bites a mosquito will take in its lifetime: E(z | infectious blood-meal at a 0 ) × P(infectious blood-meal at a 0 ) da 0 Case (i): which, as expected, is the same as Eq (18) as it does not depend on age. Looking 352 back at the vectorial capacity (Eqs (1) and (2)), we can see that this is rep-353 resented there by α µ e −µn and α µ σ σ+µ respectively. This is because the assumed 354 EIP distribution in each case is different (fixed and exponentially distributed), 355 whereas we have assumed an Erlang distribution. 356 We solve Eq (20) for cases (ii) and (iii) numerically and obtain a single 357 number for each. The results for all cases and both treatments can be found 358 in Fig 9. We observe that the expected number of bites is lower if we consider 359 age-dependent mortality functions, where we have 3.26 (2.96, 3.64) for the logistic 360 function and 3.33 (3.03, 3.76) for the Gompertz, versus 4.41 (3.66, 5.40) for the 361 age-independent. With treatment, the numbers are even lower for all cases, where 362 for the age-dependent functions, the expected number of bites is < 1.

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Using all of the above, we can calculate the relative difference in the vectorial 364 capacity between the control and treated cases for the age-dependent and age-365 independent mortality functions. We do so by assuming that the mosquito 366 density, m, and the bite rate, α, are constant. Hence we can use the results from 367 Eq (20) to make our comparisons (Fig 10).

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Between the control and treated cases for each function there is a significant 369 reduction in the vectorial capacity, where the percentage for the age-independent 370 treated case relative to the control is at 42.6% (40.9, 44.3), or, equivalently, 371 there is a reduction of 57.4% (55.7, 59.1). However, we also notice that the 372 age-dependent functions have a bigger decrease: the difference in the reduction 373 percentages between the age-independent and the logistic cases is ≈ 16.32 and 374 between the age-independent and the Gompertz cases is ≈ 15.31. In all previous 375 calculations, we have seen that the age-independent function is not very realistic, 376 whereas the logistic and Gompertz functions behave as we might expect from our 377 understanding of vector senescence. The Gompertz function might be slightly 378 closer to biological realism, however the difference between the two is minimal. 379 Both functions seem to be good candidates, though we might have a stronger 380 preference for using the Gompertz function in future calculations, since it only 381 requires two fitted parameters, compared to three in the logistic function.

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Mathematical modelling can contribute by recommending improved or optimised 384 intervention strategies for various (vector-borne) diseases, but there are many 385 challenges that modellers have to overcome to provide policy-relevant insights [54, 386 55]. One must find a balance between creating easy-to-use models and ensuring 387 that biological simplifications do not alter the resultant policy recommendations 388 by over or underestimating the impact of different intervention measures. One 389 such challenge for modelling malaria is to accurately quantify the mortality of 390 the mosquitoes. In the present study, we have demonstrated that using age-391 dependent mortality functions is important; the EIP is typically long compared 392 to a mosquito's lifespan and combined with senescence this has a substantial 393 impact on our calculation of the vectorial capacity. Here, the age-independent 394 assumption provides us with a conservative estimate for the impact of ITNs. This 395 could result in less appropriate recommendations for policy decisions, especially 396 if constant mortality is used in cost-effectiveness analyses or to optimise other 397 policy objectives (e.g. selecting strategies to minimise costs, deaths, etc.).

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This study echos results from other studies (e.g. [22,23,25]), which advocate 399 for incorporation of more biological realism in vector-borne disease modelling. 400 Brand et al. [25] consider different distributions for the EIP to show how R 0 401 changes. On the other hand, Bellan and Novoseltsev et al., [22,23] respectively, 402 highlight the importance of age-dependent mortality. In these studies, however, 403 the traditionally fixed EIP is considered, and the calculations for the vectorial 404 capacity are approached from a different perspective. Bellan [22] incorporates a 405  fixed parameter for the impact of insecticides on the longevity of the mosquitoes, 406 due to the lack of real-life data. Similarly, in [23], the authors consider multiple 407 patterns for age-dependent mortality which are generalised for multiple vector 408 species. This highlights the clear need for better availability of real-world data 409 for different vector-parasite systems to improve modelling predictions. We have 410 shown that without accounting for age the effectiveness of the anti-vectorial 411 intervention against highly-pyrethroid-resistant An. gambiae s.l. mosquitoes 412 would be underestimated.

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The two age-dependent functions that are explored in this paper are often 414 used for mosquito survival analysis [20,22,23,48]. Styer et al. investigated a 415 large-scale mortality study using Aedes aegypti mosquitoes and concluded that 416 the logistic mortality functions fit the data better on most occasions except 417 one, where the Gompertz function was a better fit [20]. Clements and Paterson 418 explored the survival patterns in many different mosquito species and found that 419 their patterns are explained well by using a Gompertz function [48]. Although the 420 logistic and Gompertz are rather flexible functions, other functional forms may 421 be more suitable for different vector species -in this case the general framework 422 presented in here could be readily adapted if required.

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It is important to note that we have included the assumption of perfect trans-424 mission when considering the vectorial capacity. In reality, vector competence 425 and host prevalence does affect the transmission potential. Here, we opted 426 for perfect transmission as we could not estimate the vector competence from 427 our data and field prevalence will be variable depending on factors such as the 428 relative vector to host density. We believe that including a realistic value for 429 vector competence and host prevalence in our calculations (as part of the step 4 430 calculation) would likely show that there is a further proportional decrease in the 431 vectorial capacity when comparing age-dependent to age-independent mortality 432 rates, because the distribution describing the timing of the first infectious blood 433 meal (a 0 ) will shift to older ages and this has a more pronounced effect when 434 there is age-dependent mortality (see Figure 8).

435
The data and results show there is a significant increase in mosquito mortality 436 when several blood-meal opportunities are offered by a host protected with an 437 ITN vs. an untreated net. The consequence for mosquitoes is a reduced vectorial 438 capacity induced by the presence of an insecticide. The fact that standard 439 ITNs do not immediately kill young and unfed mosquitoes following a single 440 and forced exposure to insecticide (which is the standard susceptibility test by 441 WHO [30]) is not sufficient information to assess the efficacy of ITNs against 442 the malaria vectors. In fact, when the EIP and mortality rates are taken into 443 consideration, along with multiple exposures (which is more in line with free-444 flying mosquitoes that are regularly host-searching and feeding), the end result 445 is that ITNs still retain some functionality against resistant mosquitoes and 446 work better than untreated nets. These results are in accordance with other 447 publications [10,12,13]. The experiments in this study were in a laboratory 448 setting, so there may be an underestimation of the effect of insecticide in the 449 field, in which case the effect on the vectorial capacity could be greater. We 450 agree with previous authors, [8,13,56,57], that encourage an update in the way 451 ITN efficacy and resistance are measured. For example, mosquito condition (i.e. 452 one or more blood-feeds), age (we especially care about old mosquitoes and it 453 has been shown previously that resistance declines with age [58]), and exposure 454 history (multiple exposures over time since mosquitoes can encounter nets at 455 each gonotrophic cycle) are some of the factors which combined determine the 456 overall number of mosquitoes in a cohort potentially able to transmit malaria. 457 There are many unanswered questions regarding the behaviour of mosquitoes. 458 One of these relates to feeding and biting patterns. In our model, we have 459 assumed a constant biting rate of one bite every four days on average. We could 460 argue that we could use a different value for each treatment. However, this 461 was not included here since additional data is required to make appropriate 462 estimations for a biting rate. Given that the experiment here gave access to the 463 mosquitoes for feeding every four days, it seems more appropriate to keep this 464 assumption. Having daily access to feeding could give more insight to estimating 465 another biting rate. Furthermore, it would be interesting to investigate feeding 466 and biting patterns that depend on age. 467 There have also been studies where it is suggested that there are parasite-468 induced behavioural changes [59][60][61][62]. This could mean that the feeding and biting 469 patterns of the mosquitoes change significantly before, during, and after the EIP. 470 For example, mosquitoes, having survived the EIP, could bite multiple humans 471 to complete one blood-meal, potentially transmitting the infection to more than 472 one person. Shaw et al. found evidence that the EIP can be shortened if an 473 infected mosquito feeds an additional time [63]; this could mean that if mosquitoes 474 feed during the EIP and the EIP shortens, the result is more infectious blood-475 meals and a larger vectorial capacity, hence an increase in malaria transmission. 476 Therefore, it could be useful to explore other feeding patterns and bite rates.

477
For our calculations, we have used the Erlang distribution for the EIP. It is 478 important to note that the mortality data were collected at 26 ± 1°C, whereas 479 the values we obtained for the EIP come from data at 27°C in [36]. Data 480 collected at various temperatures capturing the full distribution of the EIP 481 would be extremely useful. The EIP is affected by many factors, as shown 482 by Ohm et al. [64], where it is emphasised that transmission models can be 483 improved if we have a better understanding of the EIP. Some studies focus 484 on a temperature-dependent EIP [65][66][67], however, given that the experiment 485 our data were collected from was at constant temperature conditions, we have 486 not included this here. Nevertheless, it is important to keep in mind that 487 some factors depend on temperature in real life, so control programmes might 488 need adjustments depending on the time of year. Incorporating temperature 489 dependency is something else that can be explored in the future, following in the 490 footsteps of studies like [66] and [68]. 491 Furthermore, to truly capture the whole picture of malaria in Côte d'Ivoire, 492 bringing together data for human malaria cases and other on-going control 493 strategies with this mosquito data would help calculate the human consequences 494 for malaria transmission and control. This could be done by constructing a 495 modified Ross-Macdonald-type host-vector disease model [69], matching it to the 496 data, and concurrently incorporating an age-dependent mortality rate and an 497 Erlang-distributed EIP for the vectors. In this paper, we have used a modelling framework to investigate the impact of 500 insecticide exposure on mosquitoes and their ability to transmit malaria, along 501 with the impact of age-dependent mortality. Firstly, our results suggest that the 502 mortality rates increase due to insecticide exposure even in mosquitoes classified 503 as highly resistant following the WHO definition. Our analysis found that under 504 a control (no insecticide exposure) scenario there would be a higher expected 505 number of infectious bites by mosquitoes than under a treated scenario (with 506 insecticide exposure). The vectorial capacity is substantially reduced when the 507 mosquitoes are exposed to ITNs based on the experiment conducted, despite 508 being resistant to the pyrethroids used on the nets. In addition, if age dependency 509 is included in our model, the expected number of infectious bites is predicted to 510 have a greater relative reduction by using insecticides than if we use constant 511 mortality.

512
Without detailed vector data on survival with and without insecticides, this 513 type of modelling analysis would not be possible. We strongly advocate for 514 collection, not only of average mosquito life expectancies, but also distributions of 515 survival for other vector-parasite systems where quantitative analyses of different 516 interventions against the disease are desirable. We also suggest that modellers 517 pay close attention to whether more could be done to factor in senescence into 518 vector-borne disease strategy evaluations.

519
The above methodology could be easily used to check the insecticide resistance 520 of mosquitoes from experiments using other pyrethroids and/or mosquito species, 521 if similar experimental data on mosquito survival were available. The results could 522 then be used to further examine how age dependency impacts the effectiveness 523 of various interventions against mosquitoes.

524
Supporting information 525 S1  Fig. Proportion of total fed mosquitoes out of the alive ones for 534 each replicate (access to foot is allowed every four days). The results 535 from Replicate 1 are a lot different than the rest of the replicates. On the control 536 plot (left), even at the beginning where many mosquitoes are still alive, the 537 proportion of those that were fed is (close to) zero, and only at the very end a 538 mosquito actually feeds. Similarly, on the right, almost all of the mosquitoes go 539 through their lifetime without feeding. The error bars represent the propagated uncertainty of the estimated parameters. 548 We can see that the highest probabilities are for the smaller values of j in all 549 cases. The probability the number of bites is closer to zero is higher in the 550 treated than in the control cases. The average number of bites is higher for the 551 control treatment for all three cases, as expected. We also notice that for the 552 treated case and the age-dependent functions the probability that the number 553 of bites is equal to anything above seven is essentially zero. However, for the 554 age-independent graph this probability goes to zero for a much higher value of j, 555 which again shows how unrealistic an age-independent assumption is.