Total abundance.

The carrying capacity of Anopheles species in Africa is often expressed as a function of rainfall. For example, [33] found exponentially weighting the past 4 days of rainfall gave the best fit when modelling the abundance of An. gambiae s.s. and An. arabiensis in Nigeria, an approach subsequently adopted by [37]; whilst [38] used a 7 day moving average of rainfall to model the carrying capacity of the aquatic population of An. gambiae s.s., An. coluzzii and An. arabiensis in Mali.

More complex functional forms invoke additional parameters such as the location (and sometimes length or size) of perennial, intermittent, permanent or human-associated water bodies, as in the models developed by [39] and [40]. The most relevant approach for our purposes, however, is that of [21], who group our two proposed subspecies An. gambiae s.s. and An. coluzzii in a spatially explicit, individual-based model, across an area of West Africa that exhibits significant environmental variation. They predict local larval carrying capacity based on rainfall, as well as level of access to temporary and permanent water courses. We adapt their results for our model of total carrying capacity for the aggregate of An. coluzzii and An. gambiae s.s.

Relative abundance.

An. coluzzii has only relatively recently been described as its own subspecies [29] after its earlier description as a molecular form within An. gambiae s.s. [41, 42]. Despite this taxonomic difficulty, several papers have examined differences in habitat preference between An. gambiae s.s. and An. coluzzii. In particular, [26] note that larval predation and competition has led to selection for temporary freshwater habitats in An. gambiae s.s. and conversely permanent habitats for An. coluzzii. They suggest that this leads to humidity and/or rainfall clines in relative abundance. Other sources provide data suggesting this is the case for rainfall [43–46], and that a particular chromosomal arrangement in An. coluzzii performs well in low-rainfall environments [47]. These conclusions, however, tend to be based on relatively small-scale observations or experiments. Some information on relative abundance at larger scales is available [48–51], but very little modelling has been done to quantify these differences across sub-Saharan Africa. So far only occurrence information has been widely used [52]. In contrast, the relative abundance of An. gambiae s.s. in its former definition (including An. coluzzii) versus An. arabiensis has long since been modelled and estimated across sub-Saharan Africa [53].

A notable exception in this context is [27], who develop a logistic regression model using abundance data of each subspecies across western sub-Saharan Africa to predict relative probability of occurrence of the two subspecies. They use model selection to select a subset of relevant spatial, climatic and land cover variables in their predictions. However, despite acknowledging the likelihood of nonlinear effects of some variables, they use only linear predictors in a logistic regression. We use their work as a starting point to develop a flexible neural network model to incorporate nonlinear relationships, along with additional predictors and newly collated records of relative abundance (VectorBase: [54]) to extend our predictions to include the rest of sub-Saharan Africa.

Adults.

The PDE governing each scalar field representing adult populations for N_age > 1 is given by where subscript s denotes sex (M or F), g genotype, and m mosquito subspecies ((1) An. gambiae s.s. and (2) An. coluzzii). We model three alleles: w for wild–type, c for construct (gene drive system), and r for resistant. This results in a set of six potential genotypes G = {ww, wc, wr, cc, cr, rr}. The vector field V(t, x) represents the wind experienced across the spatial domain at a specified time t and place x. Note that our model makes some modifications to the advection process for biological reasons, specified below in Wind advection. Other parameters are given in Table 1.

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Table 1. Parameter definitions and estimates for final model (N_age = 2).

https://doi.org/10.1371/journal.pcbi.1009526.t001

Here N_age − 1 indicates the adult age bracket. If N_age > 1 then N_age − 2 is the eldest larvae age bracket; the special case of N_age = 1 is considered below (Newborns and N_age − 1 model). The first term on the right-hand side () describes the mortality of adults, while the second term () describes aging from the eldest larvae. The final term represents diffusion () and advection (), with A being the probability of an adult mosquito being advected by wind.

Larvae.

For a ∈ [1, N_age − 2], the populations are governed by (1)

As above, the first term on the right-hand side (−d_JX_a,s,g,m) describes the mortality of this age-bracket of larvae, while the second (b[X_a−1,s,g,m − X_a,s,g,m]) describes aging to/from older and younger age brackets respectively. Note that for N_age ≤ 2 there are no such intermediate larvae.

Newborns and N_age = 1 model.

The PDE governing X_0,s,g,m(t, x) is given by (2) Where N_age = 1, there is no age structure and the full PDE takes the form (3)

Note that for readability, we only use subscripts when referring to mosquito population classes (i.e. X_a,s,g,m) and refer to parameters which differ by population class as functions of the offspring classes. Note that we do not include parental classes (i.e. g_M, g_F, m_M, m_F) in the function names, also for brevity, and that these are only defined explicitly in the function definitions.

In Eq 2, the first term (−d_J X_0,s,g,m) describes mortality of newborns, while the second (−b X_0,s,g,m) describes aging into the youngest age-bracket of larvae (or adults for N_age = 2). The final term describes the birth of newborn larvae of a given sex s, genotype g and subspecies m, given all possibilities of the mother’s and father’s genotype and subspecies (g_M, g_F, m_M and m_F; described in more detail later). For brevity, we describe it as the product of functions describing the relevant biological mechanisms: (4) where the baseline fecundity rate is λ, the expected number of larvae per clutch of eggs per female per day, assumed produced by a mating of wildtype mosquitoes. The J and O terms represent subspecies inheritance and genotype inheritance (including sex) respectively, and are described below. In our main results, we keep the relative fecundity function R(g_M, g_F) constant at R = 1, but this can readily be varied to represent reduced fecundity due to inviability of a gene drive construct—see S3 Fig for an example.

The term models the effect of K_m(x), the (spatially varying) larval carrying capacity for subspecies m, on the fecundity of that subspecies using a logistic function. The inclusion of a max term is to keep the model biologically plausible: without it, C(m) > K_m(x) would mean that negative newborns are produced. Here C(m) is the competition that a newborn of subspecies m experiences from all larval populations: (5) where α_m,m′ represents the effect of competition of subspecies m on subspecies m′, and the max term is here used to ensure that where N_age = 1, the larval carrying capacity instead applies to the adult population.

The function J(m) returns the probability that a male adult of genotype g_M and subspecies m_M successfully mates with a female adult of genotype g_F and subspecies m_F to produce newborns of subspecies m: (6) where the number of available males of a subspecies and genotype is The numerator is the relative number of expected matings between a male of subspecies m_M (through S) and genotype g_M (through G) and the given female, while the denominator normalises the probability.

The relative probability of mating based on subspecies S(m_M, m_F) = 1 if m_M = m_F and k otherwise. The relative fitness based on genotype is: (7) where h_n and s_n are adapted from the [16] model (see Table 1).

We compare two different scenarios for subspecies inheritance H(m_M, m_F, m); maternal inheritance and equal inheritance. For maternal inheritance, the proportion of offspring born of a mating between subspecies H(m_M, m_F, m) = 1 if m_F = m and 0 if m_F ≠ m (mother is always the same subspecies as her offspring). For equal inheritance, the proportion of offspring born of a mating between subspecies H(m_M, m_F, m) = 0.5 if m_M ≠ m_F (parents are different subspecies, i.e. cross-species offspring are equally split between subspecies). In both scenarios, H(m_M, m_F, m) = 1 if m_M = m_F = m (both parents and offspring same subspecies) and 0 if m_M ≠ m and m_F ≠ m (both parents different subspecies to offspring). Where there are more than two subspecies, we also need to specify that H = 0 when m_M ≠ m_F, m_F ≠ m and m ≠ m_M (i.e. parents and offspring are all of different subspecies).

The function O(s, g) gives the probability of sex s and genotype g for the offspring. This probability depends on the genotypes of the parents such that (8) The first term i(⋅) describes the probability of an offspring inheriting genotype g given parents of genotypes g_M and g_F. We again adapt the [16] model to our target genotypes, including the effect of the gene drive construct, and using their parameter values (see Table 1). However, instead of assuming full random mixing of alleles as in their model, we directly model genotypes of each parent (see Table 2).

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Table 2. Possible values for each offspring g′ in inheritance table i(g_M, g_F, g′), with probability of occurrence given by the number in parentheses, where allele probabilities w = 1/2 − k_c/2, c = 1/2 + k_c(1 − k_j)(1 − k_n)/2 and r = k_c(k_n + k_j(1 − k_n))/2.

Table is symmetric, so cells marked with * have the same values as their transposes.

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The second term p(⋅) describes the proportion of male offspring (and thus sex bias mechanisms). In the results presented in the main paper, we keep p constant at p = 0.5, though this can be readily varied to model constructs that induce sex bias in viable offspring (see S4 Fig for an example). The sex ratio is kept constant between subspecies [67].

Diffusion.

Diffusion is modelled by using the nearest-neighbour finite-difference approximation to the Laplacian. That is, with timestep size Δt and cell length Δx, the fraction of diffusing mosquitoes removed from one cell is 4DΔt/(Δx)², where Δx is the cell-size. One quarter of this amount is added to each of the 4 neighbours. If the neighbours happen to be inactive (such as on the model boundary) those mosquitoes are assumed to die instantly.

Wind advection.

Wind advection is expected to occur over very short time periods (overnight, as mosquitoes are not believed to travel during daylight; see [32]) and potentially very large distances (hundreds of kilometres, passively carried by the wind with negligible resistance; see [32]). As such, we explicitly trace the trajectories of mosquitoes from each cell during each timestep (1 day), as advected by the wind vector field V(t, x). We use the Cross-Calibrated Multi-Platform (CCMP) Ocean Surface Wind Vector Analyses dataset [68] to define this vector field; the data is available at the required daily timesteps over the time period required. We interpolate their vector field, given at 0.25 degree intervals (approximately 28 km at the equator), to fit our grid. In their own modelling, [32] set up two different scenarios, where mosquitoes mosquitoes travel either 2 hours or 9 hours a night. We explore both scenarios here, and also a third scenario with no wind advection.

Total abundance.

We apply the method of [21] to estimate larval carrying capacity for both species combined. Specifically, we use Eq 1 from their paper: (9) with parameters as estimated in their paper using Markov Chain Monte Carlo simulation with population data (see Table 1). The model incorporates rainfall (r(t, x) in mm per week), as well as the availability of nearby rivers and lakes; these may be either permanent or intermittent (W_p(x) and W_n(x) respectively; details in their paper). It also has scope for modelling permanent larval sites (α₀(x)) but we set this to be zero for our model. Whereas [21] uses settlements as their sites for mosquito populations, we are calculating populations across a square grid, so we assume that each cell contains a settlement as modelled in their paper if and only if there is a human population present in the cell. An implicit assumption in this approach is that areas without human settlements yield zero carrying capacity. This assumption is explored for a sparsely populated region as follows. To calculate K(t, x) at each cell at each required timestep using Eq 9, we estimate human presence or absence by using data produced by [69] and publicly available from the Humanitarian Data Exchange (HDX; https://data.humdata.org/), except in Sudan, South Sudan and Somalia where these data are not available. There we assume human presence in all cells, which as an extreme case would facilitate the spatial spread of populations by local dispersal or advection. The other extreme, assuming human absence, is explored in S7 Fig: it was found to have minimal effect outside of these countries, which take up a relatively small area near the edges of the species range.

We use inland water data from the Digital Chart of the World (DCW) as in their paper, and obtain monthly rainfall data from NASA’s Land Data Assimilation System (https://ldas.gsfc.nasa.gov/fldas). We also set the mosquito population and carrying capacity to zero for each subspecies in locations outside their range as estimated by the Malaria Atlas Project (see [3], updated by the Malaria Atlas Project, accessed 25 May 2021 from https://malariaatlas.org/ and available for use under a Creative Commons Attribution 3.0 Unported License, https://creativecommons.org/licenses/by/3.0/legalcode).

Relative abundance.

Once we have a measure of the combined larval carrying capacity K(t, x), we need to separate this for two subspecies K₁(t, x) and K₂(t, x), which requires knowledge of the relative abundance of the two subspecies at each site. We use available field data collated by [27] on the number of captures of both subspecies at various locations across Africa to flexibly predict a spatially-varying but temporally-constant relative abundance. We call this relative abundance K_r(x) = K₁(t, x)/K(t, x), or the proportion of An. gambiae s.s. at a site.

We first attempt to replicate the results of the [27] logistic regression model by independently sourcing the predictors that they used in their final model: these are Latitude, Distance to Coast, Annual Mean Temperature, Mean Temperature of Wettest Quarter, Mean Elevation, Annual Normalized Vegetation Difference Index (NVDI) and Annual Variation in NVDI. We then apply a logistic regression model to the (slightly different) predictors, as they did. As well as providing an independent verification of the results in their paper, sourcing the predictors ourselves allows us to use more data sources across sub-Saharan Africa, allowing us to extrapolate and test different modelling approaches.

Using our independently sourced version of the predictors, we then apply a dense feed-forward neural network to flexibly model the relative abundance function K_r(x) (see S2 Appendix for details), using the likelihood from a binomial statistical model as our loss function (equivalent to cross-entropy in the Results). We perform leave-one-out cross-validation (jackknifing) on each of these models, comparing the replicated logistic regression model and the original [27] model with the neural network model, to test whether there is an increase in performance by incorporating nonlinear effects on the same set of variables.

The [27] model uses only allopatric sites (those where only one subspecies was detected) to train their models, as did our replication model and initial neural network model. As we are interested specifically in subspecies overlap, we include sympatric sites (where both subspecies were detected), including three new sites [70, 71] publicly available from VectorBase [54]. We also add further predictors of potential interest to the described neural network model: mean annual precipitation (BIO12) and precipitation of wettest quarter (BIO16), which were also of interest to their model but excluded by their model selection process, along with salinity [25] which was discussed in detail by [27] but not included as a model predictor. We then apply forward selection to select the model variables in the final model (see S2 Appendix for details). To ensure convergence in the larger range of scenarios, we here use a 50:50 training-testing split on the data, but otherwise keep the same network topology and approach. Once this process is complete, we then have our final estimate of K_r(x), which we can then apply to the previously calculated K(t, x) to obtain results for both K₁(t, x) and K₂(t, x).

Wind advection.

We use estimates from [32] to indirectly estimate the probability of adult mosquitoes being advected by wind. They estimate that 6 million An. coluzzii mosquitoes cross a 100 km line perpendicular to the prevailing wind direction every year. Over the course of a 2 or 9 hour flight (the two night-time flight times tested in their Methods) their calculated trajectories give displacements of 3–69 km and 47–270 km respectively (means 38.6 and 154.1; 95% mean CIs 37–41 and 140–168). If we assume that each migrating individual completes just one 2 or 9 hour overnight flight each way and are only counted once, the mosquitoes that will cross the imaginary 100 km line will largely come from a rectangle bounded by this line on one side, and another 100 km line positioned either 38.6 or 154.1 km upwind on the other side. So the area over which mosquitoes will cross this line will be roughly 100 km × 38.6 km = 3, 860 km² or 100 km × 154.1 km = 15, 410 km², within which 6 million An. coluzzii migrating mosquitoes are estimated to cross. This gives us a density of 1.55 × 10⁻³ or 3.89 × 10⁻⁴ migrating mosquitoes per square metre per year: converting to relevant units gives us 106.39 and 26.65 mosquitoes per 5 km × 5 km cell per daily timestep. If we can then estimate the overall density of mosquitoes at their capture sites, we can then estimate the daily migration rate. Using the total abundance model from [21], the average carrying capacity for larvae at coordinates 14°N, 6.7°W (located in between the capture sites in Mali) across the simulation period (2005–2015) is 41, 526. Given the other parameters d_J = 0.05 d⁻¹, d_A = 0.1 d⁻¹ and b = 0.1 d⁻¹, at equilibrium we would expect approximately the same number of adults as larvae at carrying capacity. This gives a daily migration rate of 106.39/41, 526 ≈ 2.6 × 10⁻³ (or 1 in 390) for 2 hour migration; and 26.65/41, 526 ≈ 6.4 × 10⁻⁴ (or 1 in 1558) for 9 hour migration.

Initial conditions.

As we expect similar numbers of larvae and adults, we set the initial conditions of the model for each age a, sex s, genotype g and mosquito subspecies m to In cells where both subspecies exist, there will initially be competitive effects reducing numbers of one or both subspecies. In addition, advection and diffusion will affect the equilibrium. For these reasons, we run the model once from the initial condition described for the 11-year time period from 1 January 2005 to 31 December 2015 as a “burn-in” period, in order for the population to approximately reach a dynamic equilibrium that might be encountered in the wild.

Once the burn-in period is complete, we simultaneously introduce 10,000 male mosquitoes of each subspecies (An. gambiae s.s. and An. coluzzii that are heterozygous with the construct (genotype wc) in 15 separate locations, and run the model for another 11-year period. We choose heterozygous mosquitoes to introduce as these have less fitness cost than homozygous mosquitoes (see Eq 7), increasing the chance of spread. The first five of these are placed on islands off the African mainland at the nearest suitable location (see below) to assess the potential for incursion of the genetic construct onto the mainland. The island sites chosen as illustrative examples are:

1. the Bijagós islands (off Guinea-Bissau),
2. Bioko (off Cameroon),
3. Zanzibar (off Tanzania),
4. Comoros (off Mozambique), and
5. Madagascar.

The other ten sites (numbered 6–15) were chosen to be as evenly spaced as possible across the range of An. gambiae s.s. where at least 10,000 mosquitoes of either subspecies is available year-round (the algorithm is described in S1 Appendix).

Relative abundance.

To compare the effectiveness of the modelling approaches on the jackknifed mosquito data from [27] and VectorBase [54, 70, 71], where the model probability of the subspecies being An. coluzzii at site i of N sites is p_i, and the true probability is s_i (either 0 or 1 depending on subspecies present), we use four measures:

• the actual misclassification rate or “error rate”, where the subspecies at site i is predicted to be An. coluzzii if p_i > 0.5, otherwise An. gambiae s.s.;
• the cross-entropy, which is equivalent to the binomial statistical model used to fit the neural network model, calculated as
  −∑_i[s_i log(p_i) + (1 − s_i) log(1 − p_i)]
• the expected number of misclassifications or errors,
  ∑_i abs(p_i − s_i)/N
• and the root mean square (RMS) error,
  

Table 3 shows that our neural network model performs as well (error rate), slightly better (expected error rate and RMS error) or much better (cross-entropy) on all of the measures. Its much better performance on cross-entropy is likely due to the fact that it is trained specifically to minimise cross-entropy, which may not directly correlate with lower error rates—for example, compared to the other measures, cross-entropy will much more harshly penalise a model for assigning a very low probability to an event which then occurs in the testing data.

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Table 3. Comparison of modelling approaches using four different measures of accuracy.

The “Original” model is that of [27], the “Replication” is our attempt to replicate their model with available data for predictors, and “NN” is our neural network model as described in this paper, but using their predictors. The models are compared with the dataset described in their paper [27] and from VectorBase [54, 70, 71]).

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Fig 1 illustrates the forward selection process. The chosen variables are Mean Annual Temperature, Latitude, Elevation and Distance to Coast—after this point, even the best chosen variable added to the model only increases the mean validation loss.

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Fig 1. The forward selection process for the neural network model that estimates the relative abundance of the two subspecies.

For each round of selection, the validation loss (a measure of how well the model predicts to novel data—lower is better) is shown with a different colour for each predictor. The width of the violin plots reflect the frequency of the validation loss across the 500 individual neural network runs, with the mean shown as a large dot. The forward selection process begins (Round 1) by calculating the validation loss for ten models that each include just a single different predictor. The predictor leading to the “best” model with the lowest mean validation loss is then accepted (Round 1: Mean Annual Temperature). The selection process continues (Round 2), calculating validation loss for nine models including Round 1’s accepted predictor, and one new predictor. Similarly, the predictor leading to the best model is then accepted (Round 2: Latitude). This process is repeated until accepting a new predictor no longer improves the model. In our analysis, the selection process stopped after Round 4.

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Note that the NVDI Coefficient of Variation (CV) actually had the lowest mean validation loss in the second round, but Latitude was instead selected for four reasons:

• The two had mean validation losses that were statistically indistinguishable even after 500 model runs, based on the bootstrap 95% confidence intervals,
• Comparing the results of the forward selection process where either the NVDI CV or Latitude is chosen in the second round, the latter gives much better mean validation loss results in subsequent rounds, demonstrating that the forward selection process is sub-optimally choosing the NVDI CV,
• Data for the NVDI CV is much harder to source, and the variable is a less directly biologically relevant predictor than Latitude, and
• The NVDI CV becomes a much less desirable predictor after Latitude is selected, suggesting that it has little to contribute to the model that is unique to it, and not already contributed by Latitude.

Fig 2 shows the final estimated relative abundance as a mean of 100 neural network model runs using the chosen variables, alongside the standard deviation of these runs. The relative abundance follows a similar pattern to that of [27], with An. gambiae s.s. mostly dominant except in some coastal areas and towards the Sahel. As expected, the variance between model runs is mostly low around the data points in western sub-Saharan Africa upon which the model was trained, with much higher variance around central and southern Africa. Interestingly the model runs also yield low variance around Ethiopia in particular, where they predict An. gambiae s.s. to be dominant.

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Fig 2. Summary statistics of relative abundance based on 100 neural network model runs.

Mean relative abundance (a) is given as the proportion of An. gambiae s.s. (as opposed to An. coluzzii) in the local mosquito population, with low proportions given as white and high proportions as green. Circles represent data points on which the model is trained, filled with colour representing the proportion measured at the given site. The corresponding results from [27] are given for the purposes of direct comparison (b). The standard deviation of relative abundance (c) between model runs is shown in greyscale, with white as low and black as high uncertainty of model estimates at a given site. The rectangle denotes the area of study used in [27]. Note that the relative abundance estimates cover areas where neither subspecies are expected to exist (see later figures). Base map from Natural Earth: https://www.naturalearthdata.com/downloads/10m-physical-vectors/10m-coastline/.

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Total abundance.

Fig 3 demonstrates the resulting estimated larval carrying capacity for each subspecies across the first year of modelling given the relative abundance model above and the [21] total abundance model, combined with information on human presence and the known distributions of each subspecies.

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Fig 3.

Estimated larval carrying capacity of An. gambiae s.s. (left) and An. coluzzii (right), for 2005 (the first year of modelling) in January (Southern Hemisphere summer), April (autumn), July (winter) and October (spring) from top to bottom. Base map from Natural Earth: https://www.naturalearthdata.com/downloads/10m-physical-vectors/10m-coastline/.

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