  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.
