Modeling and control of mosquito-borne diseases with Wolbachia and insecticides

Mosquitoes cause more human suffering than any other organism. It is estimated that over one million people worldwide die from mosquito-borne diseases every year. With the continuous efforts of many researchers, Wolbachia gets more and more attention due to its characteristics of maternal transmission in mosquito population and it may cause cytoplasmic incompatibility (CI) which makes healthy females cannot fertilize normally after mating with infected males. In this paper, mathematical models are established to study Wolbachia transmission in mosquito population, and integrated mosquito control strategies are explored. Firstly, a classical ordinary differential system with general birth and death rate functions is established to describe the maternal transmission and CI effect. It is shown that the replacement strategy that the Wolbachia-uninfected mosquitoes are replaced by the infected ones is determined by the initial infection frequency. And Wolbachia spreads more easily for greater maternal transmission and CI rate. Moreover, all the wild mosquitoes will eventually be infected with Wolbachia if the maternal transmission is complete. Secondly, an impulsive state feedback control model is constructed to describe the integrated mosquito control. Besides Wolbachia, insecticides are sprayed when the quantity of mosquitoes reaches some Economic Threshold. The existence and stability of Wolbachia replacement periodic solution are discussed. Finally, some discussions are done and the future research directions are prospected.