<abstract><p>Releasing <italic>Wolbachia</italic>-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 <italic>Wolbachia</italic>-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 <italic>Wolbachia</italic> 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^* $ and $ T^* $, and prove that when $ 0 &lt; r &lt; r^* $, or $ r\ge r^* $ and $ T &gt; T^* $, an unstable $ T $-periodic solution exists which serves as a new infection frequency threshold. Increasing the release ratio to $ r &gt; r^* $ and shortening the waiting period to $ T\leq T^* $, the unstable $ T $-periodic solution disappears and population replacement is always guaranteed.</p></abstract>

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Releasing Wolbachia-infected mosquitoes to suppress or replace wild vector mosquitoes has been carried out in 24 countries worldwide, showing great promise in controlling mosquitoes and mosquito-borne diseases. To face the instability of Wolbachia infection in different environments during the area-wide application, we should consider the overlapping of two Wolbachia strains. In this case, bidirectional cytoplasmic incompatibility occurs, which results in mating partners infected with exclusive Wolbachia strains producing inviable offspring. To determine the better Wolbachia candidate for release, we develop an ordinary differential equation model to study the global dynamics for competition between two Wolbachia strains. Our theoretical results on the sharp estimate of stable curves completely determine the fate of the two Wolbachia strains, which help choose appropriate strains for release.

To control the spread of mosquito-borne diseases, one goal of the World Mosquito Program’s Wolbachia release method is to replace wild vector mosquitoes with Wolbachia-infected ones, whose capability of transmitting diseases has been greatly reduced owing to the Wolbachia infection. In this paper, we propose a discrete switching model which characterizes a release strategy including an impulsive and periodic release, where Wolbachia-infected males are released with the release ratio α1 during the first N generations, and the release ratio is α2 from the (N+1)-th generation to the T-th generation. Sufficient conditions on the release ratios α1 and α2 are obtained to guarantee the existence and uniqueness of nontrivial periodic solutions to the discrete switching model. We aim to provide new methods to count the exact numbers of periodic solutions to discrete switching models.