A mosquito population replacement model consisting of two differential equations

Releasing Wolbachia-infected mosquitoes to replace wild mosquito vectors has been proved to be a promising way to control mosquito-borne diseases. To guarantee the success of population replacement, the existing theoretical results show that the reproductive advantage from Wolbachia-causing cytoplasmic incompatibility and fecundity cost produce an unstable equilibrium frequency that must be surpassed for the infection frequency to tend to increase. Motivated by lab experiments which manifest that redundant release of infected males can speed up population replacement by suppressing effective matings between uninfected mosquitoes, we develop an ordinary differential equation model to study the dynamics of Wolbachia infection frequency with supplementary releases of infected males. Under the assumption that infected males are released at a ratio $r$ to the total population size during each release period $T$, we find two thresholds $r^{\ast }$ and $T^{\ast }$, and prove that when $0<r<r^{\ast }$, or $r\ge r^{\ast }$ and $T>T^{\ast }$, an unstable $T$-periodic solution exists which serves as a new infection frequency threshold. Increasing the release ratio to $r>r^{\ast }$ and shortening the waiting period to $T\le T^{\ast }$, the unstable $T$-periodic solution disappears and population replacement is always guaranteed.