Two-sex mosquito model for the persistence of Wolbachia

Wolbachia is a genus of endosymbiotic bacteria that can infect mosquitoes and reduce their ability to transmit dengue virus. Although the bacterium is transmitted vertically from infected mothers to their offspring, it can be difficult to establish an endemic infection in a wild mosquito population. We developed and analyzed an ordinary differential equation model to investigate the transmission dynamics of releasing Wolbachia-infected mosquitoes to establish an endemic infection in a population of wild uninfected mosquitoes. Our transmission model for the adult and aquatic-stage mosquitoes takes into account Wolbachia-induced fitness change and cytoplasmic incompatibility. We showed that, for a wide range of realistic parameter values, the basic reproduction number is less than one. Hence, the epidemic will die out if only a few Wolbachia-infected mosquitoes are introduced into the wild population. Even though the basic reproduction number is less than one, an endemic Wolbachia infection can be established if a sufficient number of infected mosquitoes are released. This threshold effect is created by a backward bifurcation with three coexisting equilibria: a stable zero-infection equilibrium, an intermediate-infection unstable endemic equilibrium, and a high-infection stable endemic equilibrium. We analyzed the impact of reducing the wild mosquito population before introducing the infected mosquitoes and observed that the most effective approach to establish the infection in the wild is based on reducing mosquitoes in both the adult and aquatic stages.


Introduction
We use a disease transmission model to investigate the conditions for releasing Wolbachia-infected mosquitoes that will establish an endemic infection in a population of wild uninfected mosquitoes.The Wolbachia infected mosquitoes are less able to transmit dengue virus, and the goal of the modeling effort is to better understand how this bacterium can be used to control vector-borne diseases.We observed that the basic reproductive number for the Wolbachia model is less than one for typical model parameters, and so small Wolbachia infestations will die out.However, the model predicts that there is a critical threshold, and if a sufficient number of infected mosquitoes are released, then an endemic Wolbachia-infected population of mosquitoes can be established.We also investigated how this critical threshold can be reduced by first decreasing the population of wild mosquitoes.
Dengue is the world's most significant and widespread arthropod-borne viral disease [1].Each year, 400 million people are infected with dengue virus in more than 100 countries [2], while the other one-third of the world's population is at risk.To date, there are no vaccines or specific therapy available.Traditional control strategies focusing on reducing population of Aedes mosquito vectors have failed to slow the current dengue pandemic, especially in tropical communities [3].The main vector, Aedes aegypti, rebounds in many areas and the secondary vector, Aedes albopictus keeps expanding its geographic distribution, leading to 30 fold increase in cases over the past 50 years [4], which necessitates effective novel alternatives to break dengue transmission cycles [3] targeting Aedes aegypti and Aedes albopictus.
Increasing attention has been paid to controlling the spread of dengue by targeting mosquito longevity by introducing genetically modified mosquitoes or introducing endosymbiotic Wolbachia bacteria to shorten the mosquito lifespan [3,5,6].That is, Wolbachia-infected mosquitoes are released to create a sustained infection in the wild (uninfected) population.If the infection is sustained, then the wild infected mosquitoes will be less effective in transmitting dengue fever.We create and analyze a mathematical model to help understand the underlying dynamics of Wolbachia-infected mosquitoes that are needed to create a sustained endemic Wolbachia infection.Once the Wolbachia disease transmission model is well understood, one of our future goals will be to couple this model with a mosquito-human model for the spread of dengue.

Wolbachia Bacteria
The wMel strain of Wolbachia pipientis bacteria is a maternally inherited endosymbiont infecting more than 60% of all insect species.This strain has the ability to alternate host reproduction through parthenogenesis, which results in the development of unfertilized eggs, male killing, feminization, and cytoplasmic incompatibility (CI) [7,8] that prevents the eggs from forming viable offspring.The latter includes strategies for both the suppression and replacement of medically important mosquito populations.Cytoplasmic incompatibility is an incompatibility between the sperms and eggs induced by Wolbachia infection [9] and has received considerable attention as a method to control vector-borne diseases [10].Wolbachia even induces resistance to dengue virus in Aedes aegypti [11] and limits transmission of dengue virus in Aedes albopictus [12].
Uninfected females only mate successfully with uninfected males, while infected females can mate successfully with both uninfected and infected males [13].If a male fertilizes a female harboring the same type of infection, the offspring still can survive [6].When Wolbachia-infected males mate with uninfected females, or females infected with a different Wolbachia strain, then the CI often results in killing the embryos.[9].Therefore, infected females have a reproduction advantage over uninfected females due to protection from CI [14].
To successfully transmit dengue virus, a vector must imbibe virus particles during blood-feeding and survive to the point that the pathogen can be biologically transmitted to the next vertebrate host [15].This time period, called extrinsic incubation period (EIP), varies with ambient temperature, many other climatic factors, and characteristics of the vector-parasite system [16].Vectors that survive long enough to transmit the pathogen are called effective vectors [15].Typically dengue virus has an incubation period as long as two weeks to transmit through Aedes aegypti populations [1].A life-shortening strain of Wolbachia may halve the life span of Aedes aegypti [17].Wolbachia infection may reduce the rate of disease transmission due to the reduction on the lifespan of infected mosquitoes or the interference with mosquito susceptibility to dengue virus.

Existing Mosquito-Wolbachia Models
Ordinary differential equation (ODE) models have been developed to explore key factors that determine the success of applying Wolbachia to dengue control.A single-sex model for Wolbachia infection with both age-structured and unstructured models were presented to study the stability and equilibrium based on the assumption that Wolbachia infection leads to increased mortality or reduced birth rate [18].A model assuming a fixed ratio of females and males addressed how pathogen protection affects Wolbachia invasion [19].Age-structured and unstructured models combining males and females were found to be different in terms of existence and stability of equilibrium solutions [18].A stochastic model for female mosquitoes was developed to investigate the impact of introduction frequency on establishment of Wolbachia [20].
Discrete-time models explored the impact of the type of immigration and the temporal dynamics of the host population on the spread of Wolbachia, assuming equal sex ratio between males and females [21].Discrete generation models for female mosquitoes were built to understand unstable equilibrium produced by reduced lifespan or lengthened development [22].Reaction-diffusion and integro-difference equation model has been used to analyze the impact of insect dispersal and infection spread on invasion of Wolbachia [23].
An ordinary differential equation (ODE) model was developed to evaluate the desirable properties of the Wolbachia strain to be introduced to female mosquitoes, assuming that Wolbachia-infected mosquitoes have reduced lifespan and reduced capability to transmit dengue, and equal fraction of male and female mosquitoes [24].A continuous time non-spatial model and a reaction-diffusion model incorporating lifespan shortening and CI were developed to study factors that determine the spatial spread of Wolbachia through a population of female Aedes aegypti mosquitoes assuming constant population size and perfect maternal transmission of Wolbachia [25].A two-sex deterministic model with deterministic immature life stages and stochastic female adult life stage was developed to understand Wolbachia invasion into uninfected host population [26].
A single strain model, two strain model, and spatial model were developed to study whether multi-stain of Wolbachia can coexist in a spatial context [27].A two-sex ODE model taking into account different death rates, but the same egg laying rates of Wolbachia-infected and Wolbachia uninfected mosquitoes [28] showed the basic reproduction number is always less than one, and the complete infection equilibrium is locally asymptotically stable (LAS) due to positive determinant of the Jacobian matrix for the system.Simulations showed that dengue epidemics will not occur when Wolbachia infection is sufficiently prevalent [28].A two-sex mosquito model assuming complete vertical transmission and equal death rates for male and female mosquitoes was developed and four steady states were found [29].
Most of these models consider either a single-sex model for adult mosquitoes, or assume a fixed ratio between the number of male and female mosquitoes.Also, most of the models assume homogeneous death rates and egg laying rates for Wolbachia-free and Wolbachia-infected mosquitoes.Our model addresses both of these issues.Our emphasis is to understand how Wolbachia infection can be established in a wild population of mosquitoes.

Results
We proposed a compartmental two-sex model to investigate the underlying mechanisms that may contribute to invasion and sustainable establishment of Wolbachia in mosquito populations.We assigned female and male mosquitoes to different classes to understand corresponding roles that they are playing in the spread of Wolbachia in mosquito populations.
We showed that, for a wide range of realistic parameter values, the basic reproduction number, R 0 , for this model is less than one.Hence, the epidemic will die out if only a few Wolbachia-infected mosquitoes are introduced into the wild population.Even though the basic reproduction number is less than one, an endemic Wolbachia infection can be established if a sufficient number of infected mosquitoes are released.This threshold effect can be explained as a backward bifurcation with three coexisting equilibria: a stable zero-infection equilibrium, an intermediate-infection unstable endemic equilibrium, and a high-infection stable endemic equilibrium.
If the number of infected individuals is below the unstable endemic equilibrium, then the infection decays to the zero-infection equilibrium.Conversely, if the number of infected mosquitoes is greater than unstable endemic equilibrium, then the solution tends to the stable high-infection equilibrium.We identified the relationships between dimensionless combinations of model parameters and the initial conditions for Wolbachia to be attracted to the high-infection state.As expected, the number of infected female mosquitoes needed to be released to establish the infection in a wild Wolbachia-free population decreases as R 0 increases to one.We analyzed the impact of reducing the wild mosquito population before introducing the infected mosquitoes.We found that the most effective approach of reducing the number of infected mosquitoes needed to establish a wild Wolbachia-infected population requires reducing wild mosquito populations in both the adult and aquatic stages before the release.This could be accomplished by recursive spraying, or a combination of spraying and larvae control.
Our main findings are: (1) Three equilibria, disease free equilibrium (DFE), endemic equilibrium (EE), and complete infection equilibrium (CIE) coexist when R 0 < 1. Disease free equilibrium is a steady state when all individuals are Wolbachia-free.Endemic equilibrium is a steady state when some individuals are Wolbachia-free, the rest are infected with Wolbachia.Complete infection equilibrium is a steady state when all individuals are infected with Wolbachia.
(2) The backward bifurcation analysis of our Wolbachia transmission model predicts that if R 0 < 1, then there is a critical threshold for the number of infected mosquitoes released in the wild before the infection can be established.If we release too few Wolbachia infected mosquitoes, then Wolbachia infection will die out.(3) Killing both aquatic state (eggs and larvae) and adult mosquitoes before releasing the infected mosquitoes greatly increases the chance that the infection will be established.Our model quantifies what fraction of wild mosquitoes must be killed before releasing Wolbachia infected mosquitoes.
After introducing the mathematical model, we summarize the key results from the analysis and numerical simulations.We conclude with a discussion of the relevance, importance, and future directions for this work.

Description of Model Framework
We developed an ODE model incorporating adult females (F), adult males (M), and an aggregated aquatic (A) stage that includes the egg, larvae, and pupae stages.The population dynamics of mosquitoes without taking into account Wolbachia is in the Appendix, Equation 9.The vertical transmission of Wolbachia from infected females to their offspring is a key factor in establishing an endemic infected population.
Mosquitoes are grouped into six compartments: susceptible aquatic stage, A u , infected aquatic stage, A w , susceptible female mosquitoes, F u , infected female mosquitoes, F w , susceptible male mosquitoes, M u , and infected male mosquitoes, M w .The eclosion rates of susceptible female and male mosquitoes hatching from eggs are ψθA u and ψ(1−θ)A u , respectively.Similarly, the birth rates of infected female and male mosquitoes are ψθA w and ψ(1 − θ)A w .Death rates of uninfected male mosquitoes and infected male mosquitoes are µ a A u and µ a A w .Death rates of uninfected female mosquitoes and infected female mosquitoes are µ f u F u and µ f w F w .Death rates of uninfected male mosquitoes and infected male mosquitoes are µ mu M u and µ mw M w .Development rates of uninfected aquatic stage and infected aquatic stage of mosquitoes are ψA u and ψA w .The model parameters are described in Table 1.
The model (Fig. 1) describing population dynamics of aquatic stage, adult male, and adult female mosquitoes is given by: Because the vertical transmission and birth rates, B * * , depend on the sex of the infected or uninfected parents, the model included the four egg laying situations Here m u = Mu Mw+Mu and m w = 1 − m u are the fractions of uninfected and infected male mosquitoes.B uu is the egg laying rate of uninfected females mating with uninfected males.B uw is the egg laying rate of uninfected females mating with infected males.B wu is the egg laying rate of infected females mating with uninfected males.N A is the total number of aquatic stage of mosquitoes, and K a is carrying capacity of aquatic stage of mosquitoes.These equations reflect the observations in Table 2 that • Mating of uninfected males with uninfected females produce uninfected offspring.
• Mating of infected males with uninfected females leads to death of embryos before hatching due to cytoplasmic incompatibility.• Uninfected males and infected females produce a fraction, denoted by v w , of infected offspring by vertical transmission.
Figure 1.The birthing rates (2) capture that when the uninfected males mate with uninfected females, they produce uninfected offspring.
When infected males mate with uninfected females, then CI causes the embryos to die before hatching.Uninfected males mating with infected females produce a fraction, denoted by vw, of infected offspring by vertical transmission.Cross of infected males with infected females produces a fraction of infected offspring.
• Cross of infected males with infected females produces a fraction of infected offspring.

Model Analysis
We compute the basic reproduction number and the equilibria, and analyze the stability of the equilibrium points.
We use the next generation matrix approach to compute the basic reproduction number [30].Only infected compartments are considered for ease of computation: where F = (F i ) is a vector for new infected, and V = (V i ) is a vector for transfer between compartments.Jacobian matrices for transmission, F , and transition, V , [30] are defined as: where x 0 represents the disease free equilibrium, and x j is the number or proportion of infected individuals in compartment j, j = 1, 2  2. The model allows for Wolbachia to be transmitted vertically from infected parents to their offspring.The offspring of male and female uninfected mosquitoes are infected.Some of the offspring of male and female infected mosquitoes are infected, as are the offspring of an uninfected male and an infected female mosquitoes.The offspring of an infected male and uninfected female mosquito are unviable.
The unique disease free equilibrium is where R 0u = bf φuψ (µa+ψ)µfu is the threshold for Wolbachia-free offspring, bf φu µfu is the total number of eggs laid by uninfected female mosquitoes, ψ µa+ψ is the probability that aquatic stage of mosquitoes survive to the point when they develop into adult mosquitoes.R 0u is the number of female eggs that develop into adult mosquitoes.When R 0u > 1, then the mosquito population may grow; otherwise, the population will decrease.
The Jacobian matrix of F evaluated at DFE is: where . The Jacobian matrix of V evaluated at DFE is: The next generation matrix for infected compartments at disease free equilibrium is: The basic reproduction number is the largest abosulute eigenvalue of F V −1 , denoted by ρ(F V −1 ): Wolbachia-infected and Wolbachia-free and females produce φw µfw and φu µfu eggs during their lifetime, respectively.Hence, φw µfw φu µfu −1 is the ratio of the number of eggs produced by Wolbachia-infected females to the number of eggs produced by Wolbachia-free females during their lifetime.R 0 is geometric mean of vertical transmission rate and the ratio of the total number of eggs produced by Wolbachia-infected females to the total number of eggs produced by Wolbachia-free females.
When the variables are ordered as: T for system of equations ( 1) is: where At disease free equilibrium, D = 0, then the eigenvalues of J are eigenvalues of A and B. Matrices A and B are: If R 0u > 1, then all eigenvalues of A are negative.If R 0 < 1, then all eigenvalues of B are negative.Therefore, the system (1) at disease free equilibrium is LAS whenever R 0u > 1 and R 0 < 1.

Complete Vertical Transmission
If R 0 < 1 and R 0w > 1 and the vertical transmission is 100%, (v w = 1), then the ratio of the infected to the uninfected aquatic stage mosquitoes, k, is and there is a unique endemic equilibrium: where R 0w = vwbf ψφw (µa+ψ)µfw .If we further assume that µ mu = µ mw , then k = R −2 0 − 1 and the unique complete infection equilibrium (CIE) is: Jacobian matrix of system of equations (1) at complete Wolbachia infection equilibrium for v w = 1 is: Eigenvalues of J cw are composed of three eigenvalues of A and three eigenvalues of B. The eigenvalues of A are all negative.Characteristic polynomial of B is: If R 0w > 1, all eigenvalues of B are negative.Therefore, the complete infection equilibrium is LAS whenever R 0w > 1, as shown in Figure 2 does not exist Table 4. Threshold condition for existence of disease free equilibrium, endemic equilibrium, and complete infection equilibrium and their stability.R 0u is the threshold for Wolbachia-free mosquito population, and R 0w is the threshold for Wolbachia-infected mosquito population.Only when R 0u > 1, Wolbachia-free may grow, and only when R 0w > 1, Wolbachia-infected population may grow.

Incomplete Vertical Transmission
When vertical transmission is incomplete, i.e., 0 < v w < 1, then at endemic equilibrium, We assume µ mu = µ mw , and let . If 0.5 < v w < 1 and 4v w (v w − 1) < R 0 < 1, two endemic equilibria exist as shown in Figure 2(b), 2(c),  and 2(d).When k = k 1 , the endemic equilibrium is LAS.When k = k 2 , the equilibrium is not LAS and backward bifurcation occurs.If v w ≤ 0.5 and R 0 ≤ 1, then endemic equilibrium does not exist, only DFE exists.When   3. Denote the intersection of two endemic equilibrium, that is, the intersection of the black dashed line and x-axis, as R * 0 = √ 4vwvu.When R 0 < 1 and vw < 0.5, no endemic equilibria exist.When R 0 < 1 and vw > 0.5, as the vertical transmission rate increases so does R * 0 and the LAS equilibrium approaches a constant.If we increase the number of infected females, then the endemic equilibrium may become stable endemic or complete infection equilibrium.If we decrease the number of infected females at endemic equilibrium, then the endemic equilibrium may become disease free equilibrium.WIF denotes Wolbachia-infected female mosquitoes.

Results
We observed that Wolbachia can persist when R * 0 < R 0 , where R * 0 is the turning point of the backward bifurcation.Three equilibria, namely, disease free equilibrium, endemic equilibrium, and complete infection equilibrium coexist when R 0 < 1, R 0w > 1, v w = 1, and R 0u > 1 as shown in Figure 2(a).The disease free equilibrium is LAS whenever R 0 < 1 and R 0u > 1, and complete infection equilibrium exists and is LAS as long as R 0w > 1.The unique endemic equilibrium is not LAS, which can become disease free equilibrium by decreasing the number of infected individuals or become complete infection equilibrium by increasing the number of infected individuals.Figures 2(b), 2(c), and 2(d) showed that two endemic equilibrium points exist when √ 4v w v u < R 0 < 1 and v w > 0.5, and only one of them is LAS proven by numerical simulations.
When v w is larger, R * is closer to one.The condition for the existence of the equilibria and their stability are summarized in Table 4.
u .When R 0 < 1, the smaller R 0 is, the larger number of infected female mosquitoes are needed to be released for Wolbachia to be endemic.The Wolbachia infection is only sustained if the fraction of WIF mosquitoes is above the red dotted line.
Initial condition thresholds for an epidemic to occur vary with the vertical transmission rate as shown in Table 5.When v w = 0.9, the epidemic will spread if at least 40% of the population are initially infected.When v w = 0.95, the epidemic will spread if at least 34% of the population are initially infected.When v w = 1, the epidemic will spread if at least 28% of the population are initially infected.When the vertical transmission rate is high, the threshold for the number of initially infected individuals to start Wolbachia epidemic is low.
The initial condition threshold for an epidemic to occur varies with the ratio of death rates of Wolbachia-infected male mosquitoes to death rates of Wolbachia-free male mosquitoes, µmw µmu , while fixing other parameters.The larger µmw µmu is, the larger the initial number of infected individuals required to start a Wolbachia epidemic.With higher vertical transmission rate, a smaller percentage of Wolbachia carriers can invade, as shown in Table 5.Similarly, if we increase φ w or µ f u , or decrease φ u or µ f w , then the threshold for initial infection is smaller.Increasing ψ µa+ψ will increase the epidemic threshold for initial infection.The thresholds for fraction of initial infections decreasing with the basic reproduction number is shown in Figure 3.
The reproduction number is very sensitive to the vertical transmission rate, egg laying rates of Wolbachiainfected mosquitoes, egg laying rates of Wolbachia-free mosquitoes, and death rates of Wolbachia-infected female mosquitoes and Wolbachia-free female mosquitoes as shown in Equation (5).The reproduction number varies directly with either the vertical transmission rate, the egg laying rates of Wolbachia-infected mosquitoes, or the death rates of Wolbachia-free female mosquitoes.The reproduction number varies inversely with egg laying rates of Wolbachia-free mosquitoes, or the death rates of Wolbachia-infected female mosquitoes, but towards opposite direction.
We compared five strategies before the release of Wolbachia-infected female mosquitoes: • DFE: releasing Wolbachia-infected female mosquitoes at the disease free equilibrium, • KHA: first killing half of the aquatic stage of mosquitoes, • KHM: first killing half of the wild adult mosquitoes, • KHM2: first killing half of the wild adult mosquitoes, and then killing half of the adult mosquitoes again after two weeks, • KHMA: first killing half of the wild mosquitoes and half aquatic stage of mosquitoes.Initial condition v w = 0.9 5. Initial condition thresholds for epidemic to occur with different vertical transmission rates.When the number is Wolbachiacarrying mosquitoes is above the threshold, Wolbachia-infected mosquitoes can petsit, otherwise, they are wiped out by Wolbachia-free mosquitoes.

Scenario
minimum release ratio when R 0 = 0.85 minimum release ratio when R 0 = 0.9 .007times as many infected female mosquitoes as there are wild infect mosquitoes to establish an infection in the wild that starts at the DFE when R 0 = 0.85.The number of infected mosquitoes that need to be released to establish an infection decreases with the strategies: KHA denotes killing half of the aquatic stage mosquitoes, KHM denotes killing half of adult wild mosquitoes, KHM2 denotes killing half adult of wild mosquitoes once, then kill half adult wild mosquitoes again after two weeks, and finally KHMA denotes killing half adult wild mosquitoes and half aquatic stage of mosquitoes.
The ratios of the minimum number of Wolbachia-infected female mosquitoes that can lead to persistence of Wolbachia to the number of female mosquitoes at disease free equilibrium are listed in Table 6 in decreasing order.Notice that killing the adult mosquitoes only once is not an effective strategy because the aquatic stage mosquitoes hatch and quickly replace the wild uninfected population.Killing both the adult and aquatic (larvae) stage mosquitoes before releasing the infected mosquitoes is the most effective strategy.

Discussion
We developed a model considering two sex of aquatic stage and adult mosquitoes, diversity in the death rates of Wolbachia-infected mosquitoes and Wolbachia-free mosquitoes, and egg laying rates of Wolbachiainfected female mosquitoes and Wolbachia-free female mosquitoes.The general model is not constrained to particular weather condition, specific Wolbachia strains, or specific mosquito species, and it can be easily adapted to Aedes aegypti or Aedes albopictus at any location with parameters calibrated using realistic environmental factors, such as temperature and rainfall etc.We found conditions for the existence of multiple equilibria and backward bifurcation.If vertical transmission is complete, then a unique endemic equilibrium exists and is not LAS when R 0 < 1 and R 0u > 1. Backward bifurcation occurs when endemic equilibrium changes into disease free equilibrium if we decrease the initial number of infected individuals, or it reaches another LAS equilibrium if we increase the number of initially infected individuals.When vertical transmission is incomplete but more than 50%, and R 0 < 1, two endemic equilibria coexist, but only one endemic equilibrium is LAS such that it becomes disease free equilibrium or endemic equilibrium by perturbation.Since Wolbachia-infected mosquitoes are less capable of transmitting dengue virus, complete Wolbachia-infection is the ideal case for dengue control.
When R 0 > 1, Wolbachia can spread with a small number of initially infected mosquitoes, although very slowly.Population replacement may occur.When R 0 < 1, Wolbachia can spread if initial number of infected individuals exceeds a threshold, which depends on vertical transmission rate, ratio of egg laying rates of Wolbachia-infected females to egg laying rates of Wolbachia-free female mosquitoes, and ratio of death rates of infected female mosquitoes to death rates of uninfected female mosquitoes.A smaller number of Wolbachia-infected female mosquitoes is needed to be released for persistence of Wolbachia if a population suppression strategy is implemented before the release.
The reproduction number is the product of the vertical transmission rate, ratio of the egg laying rates of Wolbachia-infected mosquitoes to egg laying rates of Wolbachia-free mosquitoes, and the ratio of death rates of Wolbachia-free mosquitoes to death rates of Wolbachia infected mosquitoes.If the total number of eggs produced by Wolbachia-infected mosquitoes through vertical transmission is more than the total number of eggs produced by Wolbachia-free mosquitoes, then R 0 > 1, such that complete infection or endemic equilibrium is LAS.
The number of Wolbachia-infected mosquitoes required for sustained Wolbachia infection depends on vertical transmission rate, v w , the ratio of the number of eggs laid by Wolbachia-infected mosquitoes to the number of eggs laid by Wolbachia-free mosquitoes, φw φu , the ratio of the death rates of Wolbachia-free females to death rates of Wolbachia-infected females, µfu µfw , and the ratio of the death rates of Wolbachia-free males to death rates of Wolbachia-infected males µmu µmw .These parameters depend on pariculiarWolbachia strain.If the life span of a mosquito is shorter, then the mosquitoes will lay fewer eggs.Once we know the specific Wolbachia strain that a specific mosquito species, such as Aedes aegypti or Aedes albopictus is carrying, we can estimate the number of infected individuals needed to be released for sustainable Wolbachia establishment using this model.
We find that reducing both the aquatic stage and adult uninfected mosquitoes before releasing Wolbachiainfected female mosquitoes is the most effective strategy to reduce the number of Wolbachia-infected female mosquitoes needed for Wolbachia persistence (Table 6).The second most effective strategy was to repeatedly kill the wild uninfected mosquitoes (to reduce both the adult and the aquatic stage mosquitoes) before releasing the infected mosquitoes.
The model and analysis can help in understanding how Wolbachia can invade and persist in mosquito populations.In future research, we will couple our model to a dengue fever transmission model and analyze the impact that a bacteria infected mosquito population has on the spread of dengue.
for his many helpful comments.

Appendix
The system of equations for mosquito population dynamics is: The parameters for this model are described in Table 1.
Theorem 4.1 The zero equilibrium for mosquito population dynamics is LAS when R 0u < 1, while the steady state is LAS when R 0u > 1, where R 0u = φθψ µfu(µa+ψ) .
Proof.Jacobian matrix of system ( 9) is: The Jacobian matrix for no-infection equilibrium is: The characteristic polynomial of J 0 is: If R 0u < 1, then all eigenvalues are negative, the zero equilibrium is LAS.
The Jacobian matrix at steady state is: Characteristic polynomial of J ss is: If R 0u > 1, then all eigenvalues of J ss are negative, and the non-zero steady state is LAS.
Proof.We follow the approach in [36].First we consider the subsystem: According to the definition of tridiagonal feedback [36], system 12 is a monotone tridiagonal feedback system with Poincaré-Bendixson property [37].
Recall Theorem 2 in [38].If the systems of ODEs dx/dt = f (x), x ⊂ D satisfies: (1) The system exists on a compact absorbing set K ⊂ D.
(2) A unique equilibrium point E exists and is LAS.
(4) Each periodic orbit of the system is asymptotically stable.
Then E is globally asymptotically stable in D.
To prove that each periodic orbit Ω = p(t) : 0 ≤ t ≤ w of system ( 12) is asymptotically stable, we follow [39] and Theorem 3 in [36].We need to prove that the linear system dz(t) dt = J F (p(t))z(t) is asymptotically stable, where J [2] F is the second additive compound matrix of the Jacobian matrix J F associated with system (12).For system (12), J [2] F = − ψ + µ a + φ Ka F + µ f u .We build the following linear system with one equation and the right hand side is the compound matrix of the Jacobian matrix J F .dX dt = − ψ + µ a + φ K a F + µ f u X.
The right derivative of V along the solution paths (X) and (A, F ) is: D + (V (t) = −(ψ+µ a + φ Ka F +µ f u )|X|, which implies that V (t) → 0, and X(t) → 0 as t → ∞.Therefore, the linear system 12 is asymptotically stable, and the solution (A, F ) is asymptotically orbitally stable with asymptotic phase.
By the same argument in [36], the system 12 is uniformly persistent in D ⊂ R 2 .The zero equilibrium (0, 0, 0) is isolated and the largest compact invariant outside D is (A * , F * ), which is absorbing and the system 12 is uniformly persistent [40].The conditions for Theorem 2 in [38] are all satisfied.Therefore, (0, 0) is GAS whenever R 0u < 1, and (A * , F * ) exists and is GAS when R 0u > 1.
is an invariant region under the flow induced by (9).
Proof.The proof directly follows the proofs for Lemma 4.2 and 4.3 in [42].
which is the intersection of unstable and stable endemic equilibrium.When R 0 > 1, a unique endemic equilibrium point exists with k = k 1 .It is proven to be LAS by numerical simulations.vw = 0.75.

Figure 2 .
Figure 2. Bifurcation diagrams for Wolbachia vertical transmission.φu and µ f u are varying, other parameter values are the same as those baseline values in Table3.Denote the intersection of two endemic equilibrium, that is, the intersection of the black dashed line and x-axis, as R * 0 = √ 4vwvu.When R 0 < 1 and vw < 0.5, no endemic equilibria exist.When R 0 < 1 and vw > 0.5, as the vertical

Figure 3 .
Figure 3. Thresholds for fraction of infected individuals vary with reproduction number.A u0 +A w0 = A 0 u , F u0 +F w0 = F 0 u , M u0 +M w0 = M 0 u .When R 0 < 1,the smaller R 0 is, the larger number of infected female mosquitoes are needed to be released for Wolbachia to be endemic.The Wolbachia infection is only sustained if the fraction of WIF mosquitoes is above the red dotted line.

Table 1 .
(1)ction of births that are male mosquitoes = 1 − b f .mw:Fraction of the male mosquitoes that are infected = M w /(M w + M u ).m u :Fraction of the male mosquitoes that are uninfected = 1 − m w .vw:Fraction of infected mosquito eggs produced by infected female mosquitoes.vPercapita development rate of mosquito eggs.Time −1 µ a :Per capita death rate of aquatic stage of mosquitoes.Time −1 µ f u : Per capita death rate of uninfected female mosquitoes.Time −1 µ f w : Per capita death rate of infected female mosquitoes.Time −1 µ mu : Per capita death rate of uninfected male mosquitoes.Time −1 µ mw : Per capita death rate of infected male mosquitoes.Time −1 State variables and parameters for the model(1) , • • • , m. u : Fraction of uninfected mosquito eggs produced by infected female mosquitoes = 1 − v w .φ u : Per capita egg laying rate by Wolbachia-free mosquito eggs.Number of eggs/time φ w : Per capita egg laying rate by Wolbachia-infected mosquito eggs.Number of eggs/time ψ:

Table 6 .
Different population suppression strategies applied before release of Wolbachia infected female mosquitoes can reduce the minimum number of Wolbachia-infected mosquitoes that can lead to persistence of Wolbachia.The top row indicates that you have to release 1