Existence and Uniqueness of Nontrivial Periodic Solutions to a Discrete Switching Model

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.