A Periodic Delay Differential System for Mosquito Control with Wolbachia Incompatible Insect Technique

Wolbachia incompatible insect technique (IIT) depends on releasing male-only Wolbachia-infected mosquitoes to reduce/eradicate wild type mosquito populations. Mathematical models can be utilized to help investigate the altered mosquito population dynamics under IIT and propose the appropriate releasing strategies. Incorporating seasonal temperature variations and larval competition, a delayed periodic stage structured model is formulated, which is essentially different from existing models since additional density-dependent death rate due to larval competition introduces a density-dependent survival probability of larvae. This novel feature brings new challenges to mathematical analysis and further model extension in studying the dynamics of mosquito borne diseases (MBDs). By assuming continuous proportional releasing strategies, threshold dynamics in terms of the basic offspring number R₀ are established by employing the dynamical system approach. The mosquito population size is shown to either go extinction if R₀<1 or stabilize at a periodic pattern when R₀>1. Then, we numerically explore the seasonal effects on R₀ and the impact of larval competition and released amount of Wolbachia-infected males under two levels of mosquito control objectives, i.e., population eradication and reduction. Overestimates of R₀ are observed if seasonality is ignored. The ratio of released Wolbachia-infected males to wild type males (the overflooding ratio, p_w) and larval competition are shown to play positive roles in reducing both the required time for population eradication and ultimate adult peak abundance. In particular, the initial population size is negatively correlated with the eradication time, which can be shorten to a minimum of 5 years. In summary, IIT alone may be difficult to eradicate/reduce the mosquito population. Other conventional mosquito control approaches such as spraying larvicides should be used in combination to efficiently control mosquitoes.