Analyzing the control of dengue by releasing <em>Wolbachia</em>-infected male mosquitoes through a delay differential equation model

To date, an innovative strategy to control dengue is to release Wolbachia -infected male mosquitoes into wild areas to sterilize wild female mosquito vectors by cytoplasmic incompatibility (CI). To investigate the efficacy of Wolbachia in blocking dengue virus transmission, we develop a deterministic mathematical model of human and mosquito populations in which one dengue serotype circulates. The delay differential equation model captures the respective extrinsic and intrinsic incu-bation periods (EIP and IIP) in the mosquito and human, as well as the maturation delay between mating and emergence of adult mosquitoes, which have received relatively little attention. We analyze the existence and stability of disease-free equilibria, and obtain a sufficient and necessary condition on the existence of the disease-endemic equilibrium. We also determine two threshold values of the release ratio $\theta$, denoted by $\theta_1^*$ and $\theta_2^*$ with $\theta_1^*>\theta_2^*$. When $\theta>\theta_1^*$, the mosquito population will be eradicated eventually. When $\theta_2^*<\theta < \theta_1^*$, a complete mosquito eradication becomes impossible, but virus eradication is ensured at the meantime. When $\theta<\theta_2^*$, the disease-endemic equilibrium emerges that allows dengue virus to circulate between humans and mosquitoes. We carry out sensitivity analysis of the threshold values in terms of the model parameters, and simulate several possible control strate-gies with different release ratios, which confirm the public awareness that reducing mosquito bites and killing adult mosquitoes are the most effective strategy to control the epidemic. Our model provides new insights on the effectiveness of Wolbachia in reducing dengue at a population level.