Modeling and Analysis of Mosquito population dynamics and Mosquito-borne disease dynamics

This thesis uses differential equation models to investigate some problems related to mosquito population dynamics and mosquito-borne disease dynamics. It consists of three projects. The first project was motivated by the drastic change in work modes during and after the Covid-19 pandemic. We propose a non-autonomous model for a mosquito-borne disease, specifically, dengue disease transmission between two areas: (A) rural areas where a fraction of the human population lives, characterized by a higher presence of mosquitoes; (B) urban areas where the majority of workplaces (if not all) are found, the rest of the human population inhabits, and fewer mosquitoes are observed. We incorporate periodic mosquito bite patterns and periodic switch of working force in the model. We derive the basic reproduction number, $\mathcal{R}_0$ and determine the epidemic threshold. In particular, we demonstrate how variations in the work shift pattern of the work force will affect the disease's dynamics. In the second project, we propose a mathematical model to describe the interactions of wild mosquitoes and genetically modified mosquitoes that carry {\it Serratia AS1} bacteria. The main concern is whether AS1 can be established in the mosquito population, and if so, in what form: replacing or co-existing with the wild mosquitoes. After verifying the well-posedness, we examine two submodels: one neglects the infection by AS1 in the environment, and the other assumes no cross-vertical transmission of AS1 within mosquitoes. We conduct a thorough analysis for each submodel to obtain conditions for AS1 carrying mosquitoes to replace or suppress the wild mosquitoes, or fail to establish. We also performed numerical simulations to illustrate our theoretical findings. The third project is a continuation of the second, in which we investigate the impact of AS1 on the control of malaria disease. Based on the model in Chapter 3, we first develop a full model that divides the mosquitoes into three compartments: wild, AS1-carrying, and malaria-carrying. By analyzing the dynamics of the model and comparing the results with that for the sub-model that excludes \textit{Serratia AS1} (referred to as the malaria-only model), we explore the role that the AS1 bacterium, introduced as a control measure of malaria, can play in inhibiting or eliminating malaria.