Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency

In this paper, we study a discrete model on Wolbachia infection frequency. Assume that a periodic and impulsive release strategy is implemented, where infected males are released during the first N generations with the release ratio alpha, and the release is terminated from (N + 1)-th generation to T-th generation. We find a release ratio threshold denoted by alpha* (N , T), and prove the existence of a T-periodic solution for the model when alpha is an element of (0, alpha* (N, T)). For the special case when N = 1 and T = 2, we prove that the model has a unique T-periodic solution which is unstable when alpha is an element of (0, alpha* (N, T)). While alpha >= alpha* (N , T), no periodic phenomenon occurs and the Wolbachia fixation equilibrium is globally asymptotically stable. Numerical simulations are also provided to illustrate our theoretical results. One main contribution of this work is to offer a new method to determine the exact number of periodic orbits to discrete models.

Automated summary

This paper studies a discrete model on Wolbachia infection frequency and proposes a periodic and impulsive release strategy. For specific parameter values, the existence of periodic solutions and the stability of a unique solution for the model are proven.