2.1 Review of the 2-PDE Wolbachia transmission model and its assumptions

The foraging behavior of adult female Aedes aegypti consists of local flights characterized by a random unidirectional movement. A diffusion process can approximate this phenomenon [17]. We consider the reaction-diffusion PDE system proposed in [16] for modeling the spatial dynamics of the vertical transmission of Wolbachia, which depends on the infection status of both male and female mosquitoes [18]. We define and to be the population density (mosquitoes per square meter) of the adult uninfected and Wolbachia-infected mosquitoes at location at time t. Our 2-PDE model for the birth, death, and diffusion of female mosquitoes is

(1a)

(1b)

We assume a constant uniform diffusion coefficient, D, and is the two-dimensional Laplacian operator. One major assumption in our approach is that we only track adult mosquitoes, a consequence of the model reduction process from the 9-ODE to the 2-ODE framework. This leads to the caveat that the model may not predict field trials that disrupt the natural balance among different life stages or the sexual ratio of the mosquitoes [16]. We ameliorate this by considering releases of both male and female mosquitoes in a 1:1 ratio. Throughout the remainder of the paper, any specified quantity of female mosquitoes for the release will also imply the release of approximately an equal number of males. The complete description of the biologically relevant parameters for the model and their corresponding values and dimensions are presented in Table 1.

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Table 1. Model parameters, dimensions, and baseline values.

Parameter values were retrieved from [16]. The adjusted parameters in our reduced 2-PDE model derive from original 9-ODE model parameters through hierarchical reduction detailed in [15]. For example, adjusted reproduction rates and incorporate net effects of aquatic stage dynamics including egg development rates (ψ), larval survival, and emergence rates through relationships such as , where represents aquatic stage death rate. Similarly, adjusted death rates account for overall mortality across all life stages compressed into adult representation. A comprehensive overview of the parameters and their values is available in [11,15].

https://doi.org/10.1371/journal.pntd.0013787.t001

The terms B_u and B_w in the system of Eq 1 represent the per capita birth functions for uninfected and infected mosquitoes, respectively. The parameters and are their death rates, while K_f indicates the carrying capacity of the adult female population. These birth functions are defined as:

(2a)

(2b)

Note that each birth rate function is written as a product of different factors representing the mosquito birth cycle based on infection status. The first one, in Eq 2a and Eq 2b, corresponds to the per capita reproduction rates of uninfected and infected females, respectively. Here, we assume that half of the offspring will develop into the next generation of females (b_f = 1/2). Due to the perfect maternal transmission assumption of Wolbachia in our model, all the offspring produced by the infected females is Wolbachia-infected regardless of the infectious status of the males [12].

It is worth highlighting that the strain wMel of Wolbachia we are working with exhibits a strong cytoplasmic incompatibility (CI) phenomenon [18]. This means that when an infected male mates with an uninfected female, no viable offspring are produced [19], this phenomenon is incorporated in the term B_u of Eq 2a.

Based on the equation reduction process derived in Eq 3.11 in [15] to reduce the number of independent variables, the second factor in Eq 2a, , corresponds to an approximation of the fraction of uninfected males among total males, . Here, M_u and M_w are the populations for the uninfected and infected males, respectively, [15]. Note that if the death rates of the infected and uninfected mosquitoes were the same, , then the ratio of uninfected females and the ratio of uninfected males would be the same. This approximation is biologically intuitive, as it relies on the simplifying yet realistic assumption that an equal number of male and female mosquitoes are born, with the ratio between them only influenced by the differences in their death rates.

All the birth rate terms are regularized by the female carrying capacity, K_f. More precisely, we incorporate the logistic growth model term in both birth functions of Eqs 2. This corresponds to a regularization of the total adult female population and can be interpreted as limiting the availability of breeding sites for adult females [20].

In our framework, we rescale the Wolbachia models state variables relative to the female carrying capacity K_f. This rescaling enhances the model adaptability, enabling results to be applied to regions with varying ecological conditions and population sizes, thereby supporting the broader implementation of Wolbachia-based intervention strategies in diverse settings. With this in mind, if we define and as the Wolbachia uninfected and infected mosquito population sizes, respectively, expressed as fractions of the female carrying capacity, our equations above will be given by:

(3a)

(3b)

It is worth highlighting that the diffusion coefficient D measures the mean absolute deviation of the mosquito flights per day [17]. This additional measurement corresponds to the spatial extension Qu and Hyman proposed in [16] to their previous 2-ODE model [15]. This extension preserved the backward bifurcation dynamics on the 2-ODE system, identifying a critical threshold condition for invasion. More precisely, assuming perfect maternal transmission, the system has a stable Wolbachia-free equilibrium point (WFE), a stable complete infection equilibrium (CIE), and an unstable threshold endemic equilibrium (EE) where both infected and uninfected cohorts coexist. When the infection level is above the threshold, the infection eventually takes off and approaches the CIE; when below this level, the system approaches the WFE, and any minor releases of infected mosquitoes will die out [12].

The reason behind using two spatial dimensions in our model relies entirely on the need to guide field releases of Wolbachia infected female mosquitoes. These releases are typically done in a local region or point. The resulting infection wave propagates outward as a radial expansion. With this in mind, we simplify the model by assuming that the effect of environmental barriers such as roads and buildings, wind, and other environmental heterogeneity is minimal and that the spread can be approximated by assuming cylindrical symmetry. While this is a significant simplification that may not capture all urban complexity, it provides a tractable framework for understanding fundamental spatial threshold dynamics. Our diffusion coefficient range (day) implicitly captures some environmental variation, with lower values representing areas with more movement barriers and higher values representing more open environments. Given a coordinate point in the 2D Cartesian space (x,y), we consider the change of coordinates, , to account for this radial expansion effect. This leads to the following cylindrically symmetric system of equations:

(4a)

(4b)

The derivation of the system and the corresponding numerical method for approximating the solution can be found in Sect A in S1 Text and Sect B in S1 Text, respectively.

Our modeling approach has several important limitations. By focusing exclusively on adult mosquitoes, we exclude the aquatic stages, which limits the model’s direct applicability to interventions targeting larvae or pupae. This simplification is appropriate for our emphasis on thermal fogging, which targets adults specifically. To account for the missing aquatic compartments, we follow the hierarchical reduction described in [15], which adjusts adult-stage parameters to reflect life-cycle dynamics.

In particular, the adjusted reproduction rates of uninfected and Wolbachia-infected adults, and , capture the net effects of aquatic-stage dynamics on adult recruitment. Likewise, the adjusted mortality rates, and , aggregate mortality across all life stages into equivalent adult rates. These reduced parameters (Table 1) implicitly incorporate fitness costs across the mosquito life cycle, including Wolbachia-associated effects on larval competition and development times observed in recent studies [21–23]. This framework ensures that differences between infected and uninfected mosquitoes are explicitly reflected in both reproduction and mortality parameters, with the precise mathematical relationships provided in Table 1.

Our model does not incorporate meteorological variables such as temperature and humidity, which significantly affect mosquito population dynamics and Wolbachia density. Temperature stress can lead to Wolbachia curing, incomplete maternal inheritance, and reduced cytoplasmic incompatibility [24–26]. While this represents a significant limitation, our focus on threshold dynamics under different spatial patterns provides foundational insights that can inform more complex models incorporating environmental variability in future research.

We assume uniform diffusion coefficients and minimal environmental heterogeneity. In reality, urban environments impose significant barriers to mosquito movement, including buildings, roads, and varying micro-climates. For simplicity, our model adopts a cylindrically symmetric framework, which provides a tractable yet idealized representation suitable for initial policy guidance. Future models should incorporate anisotropic diffusion or network-based approaches to more accurately capture the effects of complex urban landscapes.

In the following section, we use the mathematical framework previously defined to determine the critical threshold conditions, i.e., the minimal release size of infection, in two spatial dimensions to establish a wave invasion of Wolbachia near the release center [16].

2.2 Characterization of a threshold of infection

Wolbachia deployment trials require consistent monitoring to ensure infected mosquito populations establish and maintain long-term sustainability. Key monitoring strategies include deploying mosquito traps and regular molecular testing to assess Wolbachia presence. These trials typically rely on specific infection frequency thresholds that must be achieved and sustained over several weeks before ceasing releases [27]. For instance, deployments in Rio de Janeiro used BG-Sentinel traps to monitor infection rates weekly during the initial release phase and every 1 to 3 months after establishment [28]. Field trials typically monitor for sustained infection levels above 50-60% before declaring successful establishment, as this level provides a buffer above the theoretical threshold to account for environmental variation and ensure long-term persistence.

In what follows, we propose a computational framework to identify a threshold, defined as the minimum number of Wolbachia-infected mosquitoes that must be introduced into a two-dimensional spatial domain to sustain infection frequencies at or above 60% for at least five years. This 60% threshold is based on field observations indicating that Wolbachia frequencies above this level provide robust protection against arboviral transmission and are less susceptible to stochastic fade-out [18,27]. Field trials have shown that sustaining infection rates above this threshold significantly reduces dengue transmission potential and ensures long-term population replacement [4].

Conceptually, this threshold represents the critical tipping point at which the advantages of Wolbachia infection (via cytoplasmic incompatibility) outweigh the associated fitness costs, enabling sustainable establishment rather than fade-out. Targeting this threshold simplifies operational efforts by reducing the need for continuous weekly monitoring of infection rates and lowering costs linked to labor-intensive tasks such as trap deployment, morphological identification, and molecular testing [8,28].

The analytical results presented in [16] and [29] suggest that the ODE threshold corresponds to a PDE threshold when extended to the spatially homogeneous setting. However, this poses a practical drawback for guiding field releases since it requires a positive infection on an infinite domain.

For the scope of our simulations, a suitable threshold condition for a realistic field release of infected mosquitoes requires a release over a concentrated region. This type of release is known as point release and allows the model to simulate mosquito dynamics more realistically. In particular, for simulating natural mosquito dispersal dynamics [30], since mosquitoes tend to concentrate around resources such as water, shade, and human habitats [31]. To characterize the infection threshold, Qu and Hyman in [12] used a point-release process that eventually transitioned into a critical, bubble-shaped infection profile. Here, to approximate the geometry of this critical bubble, we will use an inverse squared exponential distribution as an initial smooth release profile of infected mosquitoes. We also assume that the uninfected mosquito population starts at carrying capacity level, i.e., K_f = 3 mosquitoes per m². The corresponding formulations for the initial conditions where the uninfected wild mosquitoes are at the fraction u_o of the carrying capacity and the Wolbachia-infected mosquitoes are released as Gaussian function,

(5a)

(5b)

Unless stated otherwise, we assume that the wild mosquitoes are initially at the carrying capacity, u_o = 1. This assumption considers an initial value close to but not equal to the disease-free equilibrium of the system of equations, whose derivation can be found in [12]. Here, W corresponds to the total number of Wolbachia infected mosquitoes that will be introduced in our domain, K_f is the carrying capacity, and is the standard deviation of the Gaussian release.

Given this initial shape profile, we can now identify the corresponding threshold condition parameterized by its infection level at the release center and with a standard deviation of meters with respect to this center. Our choice for focuses on representing short-range mosquito dispersal expected in urban environments [32]. A graphical representation of the release shape geometry chosen can be found in Fig 1.

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Fig 1. Release shape geometry.

Graphical representation of the squared exponential release over a 2D domain size of . The left plot highlights the release over a subset of the domain (), while the right plot illustrates the Gaussian release standard deviation of meters with respect to the domain center.

https://doi.org/10.1371/journal.pntd.0013787.g001

We define p_thres to be the threshold condition for Wolbachia infection establishment. To compute it, we consider the following formula that characterizes the level of infection at the peak at time t, in terms of the state variables,

(6)

The previous formula should be interpreted as the fraction of Wolbachia infected mosquitos at the release center r₀ = 0 for a given time t. We employ the bisection method [33] on initial infection values and keep track of their peak of infection p_peak for a sufficiently long time (about five years ≈ 2000 days).

The threshold condition, p_thres, picked in this case, is the minimum initial Wolbachia release quantity, obtained by integrating the initial condition formula in Eq 5b, that leads to the curve that stays at a constant level of infection for a longer time, and that satisfies

(7)

The value that satisfies the previous condition is p_thres = 1,072,640 female and male mosquitoes, which corresponds to approximately 35.7% of the female carrying capacity K_f = 3,000,000 (per km²). This analysis was performed with a time step of the PDE solver of days and a Gaussian standard deviation of 35 meters with respect to the release center. It is worth highlighting that the numerical resolution of the threshold depends on the time step chosen for the PDE solver. While the threshold condition may vary slightly with different time steps, our numerical convergence test (detailed in Sect C in S1 Text) confirms our results do not suffer from numerical artifacts and accurately reflect the model dynamics.

The analysis result is depicted in Figs 2 and 3. Here, we present the behavior of both the fraction of infection curves at the release center (Fig 2) and the PDE solution (Fig 3) when releasing mosquitoes at three different release sizes: at threshold level (p_thres = 1,072,640 female mosquitoes, assuming an equal number of males are released), equivalent to 35.7% of the female carrying capacity), represented by the black curve, above the threshold (blue curve), and below the threshold (red curve).

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Fig 2. Numerical computation of threshold condition - infection curves (PDE solver time step , diffusion coefficient ).

Fraction of infection curves for different initial release sizes: releasing at the threshold, i.e, female mosquitoes (assuming same amount of male mosquitoes are also released), which is approximately 35.7% of the female carrying capacity (black curve), above the threshold (blue curve), and below the threshold (red curve). After approximately 250 days of transition, the infection curve at the threshold stabilizes at 0.6 over an extended period, demonstrating critical balance between infection growth and spatial spread. In contrast, the curves for releases above or below the threshold quickly approach the complete infection or no-infection states, respectively.

https://doi.org/10.1371/journal.pntd.0013787.g002

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Fig 3. Numerical computation of threshold condition - spatial profiles (PDE solver time step , diffusion coefficient ).

Behavior of the PDE solution for the Wolbachia-infected state variable w(r,t) (at t = 180 days) when releasing Wolbachia-infected mosquitoes at threshold value (black curve), above threshold (blue curve), and below threshold (red curve). The solution curve at the threshold evolves into a bubble shape at day 180 and preserves this shape for about four years.

https://doi.org/10.1371/journal.pntd.0013787.g003

In Fig 2, we observe that if we release infected mosquitoes below the threshold (red curve), the fraction of infection will decrease until infection dies out. If we release above that threshold, the curve will reach complete infection quickly. But if we release Wolbachia-infected mosquitoes at threshold level, the fraction of the infection curve reaches a steady level of approximately 0.6 for about four years before eventually establishing completely among the population.

When releasing the infected mosquitoes at the threshold value, the solution, shown in Fig 3, converges to a bubble-shaped profile after a transition period of approximately 180 days. This shape balances the competition between the growth of infection from reproduction of Wolbachia infected females in the reaction term versus the spread of infection from mosquito diffusion.

For the scope of the simulations of field release trials, we aim to release an infection level above the threshold condition profile to establish the wave of Wolbachia. This threshold condition that leads to a critical bubble serves as a theoretical reference for wave initiation. However, it may not be an ideal release design if a faster establishment is required. For this reason, to inform a more practical field release in the upcoming simulations, we consider a target infection level of 90% of Wolbachia infected mosquitoes to be achieved within one year after doing an initial release above the critical threshold.

2.3 Design of release scenarios

In our simulations, we consider releasing Wolbachia-infected mosquitoes over an urban area and use a diffusion coefficient range of 10– to represent realistic daily mosquito dispersal patterns. Evidence suggests that Aedes aegypti prefer to remain close to their breeding sites and typically show limited movement, with most individuals traveling only tens of meters per day and rarely exceeding 150 meters over their lifetime under normal conditions [34–36]. Mark–release–recapture studies further show that most mosquitoes are recaptured within 100–200 meters of release sites, while maximum dispersal distances can reach 300–800 meters under favorable environmental conditions [35,37,38]. Our diffusion coefficient reflects the mean squared displacement per day, which can exceed straight-line flight distances due to repeated movements and search behavior. The upper range of values (day) corresponds to more open areas, such as parks or city blocks, whereas lower values (day) represent more constrained environments, such as backyards or gated communities.

For the upcoming release scenarios, we assume that we release the same amount of Wolbachia infected males and females (ratio 1:1, females:males). Although the current model only accounts for female mosquito populations, with the model reduction process derived in [15], we assume that approximately the same number of male mosquitoes are released.

Building upon these assumptions, we evaluate strategies for improving the establishment of a wave of infection. Specifically, we assess the effects of infection release size, repetitive releases, and insecticide-based pre-release strategies. These strategies are motivated by challenges observed during large-scale releases where wMel failed to establish fully. For instance, in Medellín, Colombia, Wolbachia prevalence declined to 33.5% two years after a release program (2017-2022), with significant variation across urban communes [9]. Similarly, in Rio de Janeiro, Brazil, releases conducted between 2017 and 2019 resulted in only 32% prevalence after 29 months, despite large release efforts [10]. While multiple factors contributed to these outcomes, the spatial variation in establishment success across different urban areas suggests that mosquito dispersal patterns and release strategies play important roles.

These examples highlight the challenges of achieving consistent success in urban environments, particularly when resources for sustained releases or monitoring are limited. To address these issues, we explore the potential of combining Wolbachia release programs with complementary intervention strategies, such as repetitive releases and pre-release insecticide spraying, to mitigate the re-invasion of uninfected mosquito populations and improve the long-term establishment of Wolbachia. Our release protocol simulates a point release at the center of our spatial domain using the Gaussian distribution described in Eq 5b with standard deviation . This represents releasing mosquitoes from a central breeding facility that then disperse naturally according to the diffusion process. In practical terms, this corresponds to establishing monitoring traps within 50–100m of the release site for primary surveillance, with additional traps at 200–500m intervals to track spatial spread patterns. This approach is consistent with field trial protocols used in successful deployments [39].

To enhance infection establishment and minimize the number of infected mosquitoes required for release, a common approach involves reducing the uninfected mosquito population through insecticide spraying before the release of Wolbachia-infected mosquitoes. This method, often called pre-release insecticide application, involves dispersing a liquid fog of insecticide in outdoor areas to target adult mosquitoes directly. In this study, we will refer to this process as thermal fogging. This strategy has been shown to enhance the effectiveness of Wolbachia-based disease control [11,40,41].

In addition to insecticide-based pre-release measures, repetitive releases—in which infected mosquitoes are periodically released in smaller batches—are widely used in field trials [42]. We focus on a 2–5 week release period, chosen to balance biological considerations (such as overlap between mosquito generations) and practical constraints (such as program resources and feasibility). While many successful field programs (e.g., North Queensland, Indonesia) have used longer release periods (8–20 weeks) [5], our findings on the relative effectiveness of batch releases versus single large releases are also applicable to such extended schedules. We further acknowledge that longer release programs may be preferable when resources allow.

While previous studies have compared single large releases to multiple smaller releases [43,44], they often overlook spatial heterogeneity in the release area, which can significantly influence the outcomes.

Using a spatially explicit model, we aim to bridge this gap by investigating how spatial variability affects the efficacy of these strategies. Specifically, we explore the impact of insecticide-based pre-release measures and phased releases on the proportion of Wolbachia-infected mosquitoes required to establish infection across regions with different mosquito dispersal patterns.

2.4 Incorporating pre-release strategies and repetitive releases

Thermal fogging, as a pre-release strategy, is implemented to reduce the uninfected mosquito population prior to releasing Wolbachia-infected mosquitoes. Since the model described in Eqs 4a and 4b only includes adult mosquito compartments, we assume that thermal fogging involves application of insecticide fog to target adult mosquitoes directly.

We acknowledge that our assumption on thermal fogging as a pre-release intervention has limitations. Thermal fogging targets only the current adult mosquito population and may have limited impact on subsequent generations emerging from desiccated eggs or protected aquatic habitats. This approach avoids the potential issue that repeated insecticide applications can promote resistance, which could reduce long-term effectiveness [45,46]. Furthermore, achieving a substantial reduction in wild mosquito populations through thermal fogging may be challenging in practice due to incomplete coverage, mosquito behaviors such as indoor resting, and environmental factors that influence insecticide efficacy [47].

Our model represents a target scenario for intervention effectiveness. Real-world applications should account for reduced efficacy and the potential development of resistance.

Regarding the implementation of phased releases, we compare the effects of releasing Wolbachia-infected mosquitoes in 2–5 weekly batches versus a single large release. This range is based on several considerations. First, the average lifespan of adult Aedes aegypti mosquitoes (2–4 weeks) ensures that at least one full generation of wild mosquitoes is exposed to Wolbachia infection [48]. Second, operational constraints in field programs often limit continuous releases beyond 4–6 weeks due to resource and logistical challenges. Third, community acceptance is a concern, as releasing millions of mosquitoes simultaneously is likely to face public opposition regardless of infection status. While some field programs have conducted releases over longer periods [5], our analysis focuses on the initial establishment phase, where concentrated releases are most critical for achieving invasion thresholds.

To incorporate multiple levels of thermal fogging intensity, we modify the initial condition of the uninfected mosquito population, u(r,t), which is scaled relative to the female carrying capacity, K_f, and thus satisfies . We define α as the proportion of the initial uninfected female population that is removed by the intervention, where . Consequently, the initial condition for the uninfected population is set as . This represents the fraction of the carrying capacity that remains after fogging has reduced the initial uninfected population to a fraction of its original size.

Regarding the implementation of phased releases, we compare the effect of releasing the total amount Wolbachia-infected mosquitoes distributed in periodic weekly batches versus releasing them all at once. Since the average lifespan of an adult Aedes aegypti mosquitoes ranges between 2-4 weeks [48], we consider repetitive releases ranging between 1-5 weeks for our simulations. This ensures that at least one full generation of wild mosquitoes is exposed to Wolbachia infection [27,42]. This overlap might increase the likelihood of Wolbachia spreading within the population through mating.

3.1 Model results for a fixed release size of Wolbachia infected mosquitoes

The time required to achieve 90% Wolbachia infection in a mosquito population depends on mosquito dispersal rates and the intensity of pre-release interventions, such as thermal fogging. Fig 4 explores this relationship under a resource-constrained release of 3,600,000 Wolbachia-infected female mosquitoes (and an equal number of males) into a area, which is comparable to the field trial deployed in the municipality of Belén in Medellín, Colombia [8]. By holding the release size constant, the analysis isolates the effects of varying mosquito dispersal rates and pre-release reductions of uninfected mosquitoes on infection establishment. Additionally, the diffusion coefficient range used to represent mosquito dispersal rates is contextualized with descriptions of the landscape features associated with the lowest dispersal rate (, typical of enclosed backyards, gated communities, or areas with dense housing and limited connectivity) and the highest dispersal rate (, characteristic of open parks, wide residential blocks, or urban areas with fewer physical barriers) considered (x–axis in Fig 4).

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Fig 4. Heatmap showing the time in months (region labels) and days (right y-axis), required to achieve 90% of Wolbachia infection in a mosquito population, based on a fixed release size of 3,600,000 female infected mosquitoes (and about the same amount of males) in a area.

The x-axis represents the diffusion coefficient in m²/day, reflecting mosquito dispersal rates, while the y-axis indicates the proportion of the wild mosquito population removed through thermal fogging with three different ranges of intensities. The heatmap color code corresponds to the time to reach 90% of infection, with darker shades indicating shorter times. The figure highlights three distinct regions: 0–3 months (dark), 3–9 months (intermediate), and more than 9 months (light). The two dashed white contour lines mark the boundaries between these regions. Removing at least 35% (red line) of the uninfected mosquito population ensures 90% infection is reached within nine months, regardless of mosquito dispersal, while lower removal levels lead to longer times, particularly at large diffusion coefficients.

https://doi.org/10.1371/journal.pntd.0013787.g004

The heatmap in Fig 4 illustrates three distinct temporal regions for infection establishment. The dark region indicates establishment between 0–3 months. In this region, low dispersal rates help achieve 90% of infection within three months regardless of the intensity of thermal fogging applied. This can be attributed to the interplay between mosquito movement and the dynamics of infection spread. Specifically, at low dispersal rates, infected mosquitoes released into the environment are less likely to disperse widely, leading to a higher concentration of infected individuals near the release site. This clustering effect increases the likelihood of local mating of the released mosquitoes, accelerating the spread of Wolbachia.

The reduced reliance on thermal fogging in this region may also stem from the limited overlap between infected and uninfected mosquitoes, meaning that even moderate natural population dynamics (e.g., birth and death) combined with infection spread dynamics suffice to establish high infection rates without requiring intense intervention. Previous studies have also observed this phenomenon of rapid Wolbachia establishment at low dispersal rates due to localized interactions, such as, [27,49].

As the diffusion coefficient increases to dispersal rates of or higher, achieving 90% Wolbachia infection within 3–9 months (intermediate region in Fig 4) or beyond 9 months (light region in Fig 4) requires incorporating thermal fogging as a supplementary intervention in the release program. It is worth noting that at dispersal rates between , infection establishment dynamics indicate that a medium intensity of thermal fogging (at least 35% represented by the horizontal red line in Fig 4) is the minimum required to achieve 90% of infection within 9 months post-release.

At higher dispersal rates, the released infected mosquitoes mix more rapidly with the uninfected populations over larger areas within the domain, causing infection dilution near the release center and making it harder to establish. Thermal fogging helps by reducing the number of uninfected mosquitoes, increasing the relative proportion of Wolbachia-infected individuals in the population and speeding up the spread process. This reduction directly affects the threshold necessary for infection establishment, making achieving it more feasible.

This analysis offers insights for designing release strategies under resource constraints on the number of Wolbachia-infected mosquitoes, while accounting for mosquito dispersal rates shaped by the landscape features of the release domain. Specifically, for dispersal rates between , typical of areas with restricted movement such as backyards or gated communities, Wolbachia mosquitoes remain concentrated locally after release, which may benefit infection establishment without the use of pre-release insecticides. In contrast, for dispersal rates between , associated with more open environments like residential blocks, parks, or urban greenbelts, incorporating pre-release interventions—removing at least 35% of the uninfected mosquito population—is recommended to enhance infection establishment.

3.2 Model results for a fixed level of insecticide efficacy (thermal fogging) sprayed on uninfected mosquito population

Based on the minimum level of insecticide efficacy identified in the previous scenario (35% intensity of thermal fogging), we now focus on a scenario restricting insecticide use to this minimum intensity level. This analysis examines how release size and dispersal rates interact to determine the time required to establish 90% Wolbachia infection. By holding the level of insecticide efficacy constant, this analysis isolates the effects of dispersal rates and release size on infection establishment, identifying potential regimes where infection can be established within nine months using fewer mosquitoes than the reference value used in our simulations (3,600,000 females and about the same number of males).

The heatmap in Fig 5 illustrates three distinct temporal regions for infection establishment under a fixed thermal fogging killing efficacy (35% intensity) applied to the uninfected population. Infection establishment can occur within 0–3 months (dark region), between 3–9 months (intermediate region), or take more than 9 months (light region).

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Fig 5. Heatmap showing the time required to achieve 90% Wolbachia infection in a mosquito population under restricted insecticide efficacy (35% intensity of thermal fogging) sprayed on uninfected mosquitoes in a area.

The x-axis represents the diffusion coefficient (), reflecting mosquito dispersal rates. The y-axis indicates the percentage of Wolbachia-infected mosquitoes to release relative to the baseline reference (3,600,000 females and about the same amount of males). The color scale corresponds to the time to reach 90% infection, with darker shades indicating faster establishment. Three regions are highlighted: 0–3 months (dark), 3–9 months (intermediate), and >9 months (light). The two dashed white contour lines mark the boundaries between these regions, corresponding to the 3-month and 9-month thresholds. The horizontal solid red lines indicate example dispersal rate ranges at which it is possible to achieve 90% of infection within nine months with smaller release quantities.

https://doi.org/10.1371/journal.pntd.0013787.g005

The regional differentiation described above allows us to identify two types of infection spread regimes based on dispersal rate ranges (see Fig 5). The first regime corresponds to dispersal rates between /day, where even a small release quantity of Wolbachia-infected mosquitoes is sufficient to establish infection within nine months. At these lower dispersal rates, where mosquito movement is more limited, the uninfected and infected cohorts are more likely to encounter each other and mate. This increased likelihood of interaction, combined with the reduced competition for mates and resources due to the pre-release killing of 35% of the uninfected cohort, enhances the reproductive success of the Wolbachia-infected mosquitoes via cytoplasmic incompatibility. These factors together explain why a smaller number of Wolbachia-infected individuals is needed to establish the infection within the population successfully [50].

The second regime corresponds to diffusion coefficients greater than , where mosquito dispersal is higher, leading to a more extensive spread of both infected and uninfected individuals. In this case, the dashed contour line in Fig 5 represents the minimum number of Wolbachia-infected mosquitoes that must be released (relative to the reference value of (3,600,000 females) to achieve 90% infection within nine months.

At higher diffusion rates, mosquitoes disperse over a larger area more quickly, reducing the frequency of encounters between infected and uninfected individuals. This dilution effect diminishes the impact of cytoplasmic incompatibility in suppressing the uninfected population. As a result, the pre-release intervention of killing 35% of the uninfected mosquitoes is no longer sufficient on its own. To overcome this challenge, the release size of Wolbachia-infected mosquitoes must be increased to ensure sufficient contact between cohorts and to maintain the spread of Wolbachia throughout the population.

It is worth noting that it is still possible to achieve infection with release quantities smaller than the reference value (3,600,000 for female and male mosquitoes) in this regime. These minimum release values, indicated by the dashed contour line (see Fig 5), increase as the dispersal rate rises, reflecting the greater challenge posed by higher mosquito mobility. Notably, infection establishment within nine months becomes unachievable for dispersal rates approaching /day, even when releasing the full reference quantity of Wolbachia-infected mosquitoes. This indicates that at very high dispersal rates, the spread of uninfected mosquitoes becomes too rapid for the Wolbachia-infected cohort to establish the infection within the desired time frame.

The previous analysis provides recommendations for appropriate release sizes of Wolbachia-infected mosquitoes in different dispersal scenarios, considering the resource limitations on the efficacy level of insecticide used for pre-release interventions. Specifically, in areas with limited mosquito dispersal (/day), such as small courtyards or residential streets with high fences, smaller release quantities of Wolbachia-infected mosquitoes would be sufficient for effective infection spread. However, larger release sizes would be necessary to ensure successful infection establishment in areas with higher mosquito dispersal (greater than /day).

3.3 Effect of repetitive releases

Field trials often require periodic releases of batches of infected mosquitoes [42]. With the following simulations, we aim to study the impact of releasing a certain number of infected mosquitoes split into multiple batches released weekly over five consecutive weeks. All the releases had the same number of infected mosquitoes. Each batch of released mosquitoes was the total released size divided by the number of batches, and they were released at regular time intervals. The results of these simulations are summarized in Table 2, and the behavior of the corresponding fraction of infection curves can be visualized in Figs 6 and 7.

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Fig 6. Fraction of Wolbachia infection over time without pre-release intervention.

Performance comparison of releasing infected mosquitoes at the threshold level all at once or distributed in 2–5 weekly batches. Domain size . Diffusion coefficient day. The standard deviation of Gaussian release of meters.

https://doi.org/10.1371/journal.pntd.0013787.g006

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Fig 7. Fraction of Wolbachia infection over time with pre-release intervention.

Performance comparison of releasing infected mosquitoes at the threshold level all at once or distributed in 2–5 weekly batches. The impact of applying a pre-release mitigation strategy (thermal fogging 35%) was evaluated. Domain size . Diffusion coefficient day. The standard deviation of Gaussian release of meters.

https://doi.org/10.1371/journal.pntd.0013787.g007

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Table 2. Simulation results when releasing Wolbachia-infected mosquitoes in multiple batches.

Comparison of the number of days required to achieve 90% infection when releasing Wolbachia-infected mosquitoes at the threshold level (1,072,640 females and males) all at once or distributed in 2–5 weekly batches over an area with a mosquito dispersal rate of day. All five scenarios were evaluated with and without pre-release mitigation (Thermal Fogging 35%). The table shows the time to reach the threshold infection level for single-release and multi-batch mosquito releases. Pre-release mitigation significantly reduces the time to reach 90% infection, particularly for the single-release strategy, where thermal fogging reduces the days to 185 compared to 5 years without mitigation. Multi-batch releases (2–5 batches) generally show a quicker progression towards reaching 90% of infection, with the shortest times observed in the 5-batch release scenario, especially when combined with pre-release mitigation.

https://doi.org/10.1371/journal.pntd.0013787.t002

To evaluate the performance of these scenarios, we measured the number of days required for Wolbachia to infect 90% of the mosquito population. In the first scenario, shown in Fig 6, no pre-release mitigation strategy was applied. We observe that releasing the infected mosquitoes in multiple batches (2−−5 weekly batches) reduces the time to establishment to within a year, compared to releasing them all at once at the threshold level, where infection is not established within this time frame.

On the other hand, in Fig 7, we observe that if 35% of the wild uninfected population is killed before releasing the mosquitoes in 2−−5 batches, the time to establish infection is reduced by approximately 23 days compared to the case with no pre-release intervention. A more significant reduction is seen in the single-release case, where thermal fogging speeds up the establishment by approximately 4 years.

The results are summarized in Table 2. These findings highlight the importance of applying a pre-release mitigation strategy and incorporating a multiple-release approach in field trials to accelerate the establishment of Wolbachia in the wild mosquito population.

It is worth noticing that including the effect of spatial mosquito dispersion in the field release model has important consequences. The multiple-release analysis performed in [44] uses an ODE model that assumes the fraction of infection among mosquitoes is homogeneous in space. This ODE model showed that without pre-release mitigation, releasing all the infection at once may not be as effective as splitting the release into multiple batches, which is consistent with our results. The benefit of leaving a gap between releases is related to the limited environmental resources available to sustain the uninfected population, not treated with any insecticide, as well as the introduction of additional infected mosquitoes, which overpopulate the region and increase the competition for resources when released in a single batch.

On the other hand, when there was pre-release mitigation, the ODE model results in [44] suggest that releasing the infected mosquitoes all at once was more effective than splitting them into batches. However, in our simulations, we observe the opposite behavior, i.e., releasing the infection in batches could lead to a more effective strategy than releasing them all at once. Indeed, with our spatial model, where we assume a limited mosquito dispersal, represented by ranges with small diffusion coefficients 10–100 m²/day, releasing Wolbachia infected mosquitoes in batches allows for a more localized establishment of the infection within specific regions of the habitat. This is because mosquitoes interact primarily with nearby individuals, resulting in clusters of Wolbachia infected mosquitoes in the regions where they were released.

Now, suppose we add the assumption that a fraction of uninfected individuals are killed with thermal fogging before the release. This will decrease the infected cohort’s competition for resources in these localized areas, creating a favorable environment for this population to breed and transmit Wolbachia to their offspring.

3.4 Sensitivity analysis

The parameter values of the reduced model in Table 1 correspond to baseline estimates that carry uncertainty with respect to the biological measurements and vary across Wolbachia strains, climatic conditions, mosquito species, etc. To account for this uncertainty level in our parameters, we use sensitivity analysis to quantify the relative significance of the model parameters of interest (POIs) toward the output quantities of interest (QOIs).

Based on the method followed in [51], we define the normalized sensitivity index (SI) of a quantity of interest QOI, q(p), with respect to a parameter of interest POI, p, as

(8)

evaluated at the baseline value . This dimensionless index is interpreted as the impact of percentage change: if the parameter of interest p changes by around the baseline, then the quantity of interest q changes by

Given that our simulations employed the infection threshold (p_thres), computed in Sect 2.2, as a foundational metric for designing effective Wolbachia release strategies, we assess the sensitivity of this threshold to variations in the baseline values of biologically relevant parameters in the reduced 2-PDE model used in our simulations. The results of this analysis are summarized in Table 3.

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Table 3. Local sensitivity analysis of the threshold of infection p_thres, with respect changes in relevant model parameters.

Normalized sensitivity index (SI) of key biological parameters in the reduced 2-PDE model with respect to the threshold of infection (p_thres), considered as the quantity of interest (QOI). The baseline values of the parameters are provided in parentheses. Positive and negative indices indicate parameters that increase or decrease the threshold, respectively. The sensitivity index was calculated by increasing each baseline quantity of the parameters of interest by 1%. The infection threshold is more sensitive to the per capita reproduction rate of infected and uninfected cohorts, followed by the release radius.

https://doi.org/10.1371/journal.pntd.0013787.t003

If the diffusion coefficient, representing mosquito dispersal, increments by 1%, the number of mosquitoes required for achieving a sustainable infection should increase by approximately 5%. Additionally, if the standard deviation of the Gaussian release, denoted by , increases by 1%, the infection threshold decreases by approximately 8%. This suggests that releasing mosquitoes over a wider area, compared to the baseline value, may require fewer Wolbachia-infected mosquitoes to establish an infection.

An increase of 1% in the death rate of uninfected mosquitoes leads to an approximate 7% reduction in the infection threshold. At the same time, the same perturbation applied to the infected cohort results in a 7% increase in the threshold. This latter outcome aligns with the fact that a shorter lifespan means infected mosquitoes die more quickly. To maintain or increase the number of infected mosquitoes in the environment, more infected mosquitoes would need to be released to compensate for the higher death rate, directly impacting the infection threshold.

The infection threshold is highly sensitive to the per capita reproduction rates of both Wolbachia-infected and uninfected mosquito cohorts. A small increase in the reproduction rate of uninfected mosquitoes leads to a higher threshold of infection (by 20%). This suggests that when the uninfected mosquito population grows, it makes it more difficult for Wolbachia to spread, potentially because the balance of infected and uninfected mosquitoes shifts, diluting the proportion of infected mosquitoes in the population and thereby raising the threshold for successful infection establishment. Conversely, the same increase in the reproduction rate of Wolbachia-infected mosquitoes results in a lower infection threshold (by 20%). This indicates that increasing the reproductive capacity of infected mosquitoes facilitates the spread of Wolbachia, necessitating fewer infected mosquitoes for successful establishment and thereby reducing the threshold of infection.

Regarding the percentage of thermal fogging efficacy as a parameter of interest, we observe two distinct behaviors in the threshold sensitivity depending on the mosquito dispersal regime. In regions with low mosquito dispersal (day), the infection threshold is less sensitive to small increments in thermal fogging efficacy, resulting in a modest reduction of only 0.44% in the threshold. This finding aligns with our simulation results, which suggest that in areas with limited mosquito dispersal, the use of insecticide may not be essential for successful Wolbachia establishment due to the potential local concentration of Wolbachia-infected mosquitoes after release. In contrast, for high mosquito dispersal regimes (day), the threshold is more sensitive to thermal fogging efficacy, with a more substantial 4% reduction in the threshold for the same small increment in efficacy. This is consistent with our simulation results, highlighting the need for pre-release intervention strategies to support infection establishment in regions with high mosquito dispersal rates. These findings emphasize the importance of considering the local ecological context, particularly the mosquito dispersal patterns when designing and implementing Wolbachia-based control strategies.