  Wolbachia is a natural bacterium that can infect mosquitoes and reduce their
ability to transmit mosquito-borne diseases, such as dengue fever, Zika, and
chikungunya. Field trials and modeling studies have shown that the fraction of
infection among the mosquitoes must exceed a threshold level for the infection
to persist. To capture this threshold, it is critical to consider the spatial
heterogeneity in the distributions of the infected and uninfected mosquitoes,
which is created by the local release of the infected mosquitoes. We develop
and analyze partial differential equation (PDE) models to study the invasion
dynamics of Wolbachia infection among mosquitoes in the field. Our
reaction-diffusion-type models account for both the complex vertical
transmission and the spatial mosquito dispersion. We characterize the threshold
for a successful invasion, which is a bubble-shaped profile, called the
"critical bubble". The critical bubble is optimal in its release size compared
to other spatial profiles in a one-dimensional landscape. The fraction of
infection near the release center is higher than the threshold level for the
corresponding homogeneously mixing ODE models. We show that the proposed
spatial models give rise to the traveling waves of Wolbachia infection when
above the threshold. We quantify how the threshold condition and traveling-wave
velocity depend on the diffusion coefficients and other model parameters.
Numerical studies for different scenarios are presented to inform the design of
release strategies.
