Modelling the $Wolbachia$ Strains for Dengue Fever Virus Control in the Presence of Seasonal Fluctuation

Consider that infection with $Wolbachiacan$ limit a mosquito''s ability to transmit Dengue fever virus through its saliva, a mathematical model describing the transmission of Dengue fever between vector mosquitoes and human, incorporating $Wolbachia$-carrying mosquito population and seasonal fluctuation, is proposed. Firstly, the stability and bifurcation of this model are investigated exactly in the case where seasonality can be neglected. Further, the basic reproductive number $mathcal{R}_0^s$ for this model with seasonal variation is obtained, that is, if $mathcal{R}_0^s$ is less than unity the disease is extinct and $mathcal{R}_0^s$ is greater than unity the disease is uniformly persistent. Finally, numerical simulations verify the theoretical results. Theoretical results suggest that, compared with the mosquito reduction strategies (such as the elimination of mosquito breeding sites, killing of adult mosquitoes by spraying), introducing $Wolbachia$ strains is as effectual to fight against the transmission of Dengue virus.