  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.
