Wolbachia infection dynamics by reaction-diffusion equations

Dengue fever is caused by the dengue virus and transmitted by Aedes mosquitoes. A promising avenue for eradicating the disease is to infect the wild aedes population with the bacterium Wolbachia driven by cytoplasmic incompatibility (CI). When releasing Wolbachia infected mosquitoes for population replacement, it is essential to not ignore the spatial inhomogeneity of wild mosquito distribution. In this paper, we develop a model of reaction-diffusion system to investigate the infection dynamics in natural areas, under the assumptions supported by recent experiments such as perfect maternal transmission and complete CI. We prove non-existence of inhomogeneous steady-states when one of the diffusion coefficients is sufficiently large, and classify local stability for constant steady states. It is seen that diffusion does not change the criteria for the local stabilities. Our major concern is to determine the minimum infection frequency above which Wolbachia can spread into the whole population of mosquitoes. We find that diffusion drives the minimum frequency slightly higher in general. However, the minimum remains zero when Wolbachia infection brings overwhelming fitness benefit. In the special case when the infection does not alter the longevity of mosquitoes but reduces the birth rate by half, diffusion has no impact on the minimum frequency.