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We develop and analyse an ordinary differential equation model to investigate the transmission dynamics of releasing Wolbachiainfected mosquitoes to establish an endemic infection in a population of wild uninfected mosquitoes. Wolbachia is a genus of endosymbiotic bacteria that can infect mosquitoes and reduce their ability to transmit some viral mosquito-transmitted diseases, including dengue fever, chikungunya, and Zika. 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. Our transmission model for the adult and aquatic-stage mosquitoes takes into account Wolbachia-induced fitness change and cytoplasmic incompatibility. We show that, for a wide range of realistic parameter values, the basic reproduction number, R 0 , is less than one. Hence, the epidemic will die out if only a few Wolbachiainfected 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 zeroinfection equilibrium, an intermediate-infection unstable endemic equilibrium, and a high-infection stable endemic equilibrium. We analyse 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. ARTICLE HISTORY

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We develop and analyze an ordinary differential equation model to assess the potential effectiveness of infecting mosquitoes with the Wolbachia bacteria to control the ongoing mosquito-borne epidemics, such as dengue fever, chikungunya, and Zika. Wolbachia is a natural parasitic microbe that stops the proliferation of the harmful viruses inside the mosquito and reduces disease transmission. It is difficult to sustain an infection of the maternal transmitted Wolbachia in a wild mosquito population because of the reduced fitness of the Wolbachia-infected mosquitoes and cytoplasmic incompatibility limiting maternal transmission. The infection will only persist if the fraction of the infected mosquitoes exceeds a minimum threshold. Our two-sex mosquito model captures the complex transmission-cycle by accounting for heterosexual transmission, multiple pregnant states for female mosquitoes, and the aquatic-life stage. We identify important dimensionless numbers and analyze the critical threshold condition for obtaining a sustained Wolbachia infection in the natural population. This threshold effect is characterized by a backward bifurcation with three coexisting equilibria of the system of differential equations: a stable disease-free equilibrium, an unstable intermediate-infection endemic equilibrium and a stable high-infection endemic equilibrium. We perform sensitivity analysis on epidemiological and environmental parameters to determine their relative importance to Wolbachia transmission and prevalence. We also compare the effectiveness of different integrated mitigation strategies and observe that the most efficient approach to establish the Wolbachia infection is to first reduce the natural mosquitoes and then release both infected males and pregnant females. The initial reduction of natural population could be accomplished by either residual spraying or ovitraps.

Once Aedes aegypti and Aedes albopictus mosquitoes that spread Chikungunya virus, dengue virus, and Zika virus are infected with Wolbachia, they have reduced egg laying rates, reduced transmission abilities, and shorter lifespans. Since most infected mosquitoes are only infectious in the last few days of their lives, shortening a mosquito’s lifespan by a day or two can greatly reduce their abilities to spread mosquito-borne viral diseases, such as Chikungunya, dengue fever, and Zika. We developed a mathematical model to compare the effectiveness of the wMel and wAlbB strains of Wolbachia for controlling the spread of these viruses. The differences among the diseases, mosquitoes, and Wolbachia strains are captured by the model parameters for the mosquito-human transmission cycle. Moreover, the model accounts for the behavior changes of infectious population created by differences in the malaise caused by these viruses. We derived the effective and basic reproduction numbers for the model that are used to estimate the number of secondary infections from the infectious populations. In the same density of Wolbachia-free Aedes aegypti or Aedes albopictus mosquitoes, we observed that wMel and wAlbB strains of Wolbachia can reduce the transmission rates of these diseases effectively.