Dynamics of a two-sex model for the population ecology of dengue mosquitoes in the presence of Wolbachia

The release of Wolbachia-infected mosquitoes into the population of wild mosquitoes is one of the promising biological control method for combating the population abundance of mosquitoes that cause deadly diseases, such as dengue. In this study, a new two-sex mathematical model for the population ecology of dengue mosquitoes and disease is designed and used to assess the population-level impact of the periodic release of Wolbachia-infected mosquitoes. Rigorous analysis of the model, which incorporates many of the lifecycle features of dengue disease and the cytoplasmic incompatibility property of Wolbachia bacterium in mosquitoes, reveal that the disease-free equilibrium of the model is locally-asymptotically stable whenever a certain epidemiological threshold, known as the reproduction number of the model (denoted by R_0W), is less than unity. The model is shown, using centre manifold theory, to undergo the phenomenon of backward bifurcation at R_0W=1. The consequence of this bifurcation is that Wolbachia may not persist, or dengue disease may not be effectively-controlled, when R_0W is less than unity. Such persistence and elimination will depend on the initial sizes of the sub-populations of the model. Two mechanisms were identified for which the backward bifurcation phenomenon can be removed. When backward bifurcation does not occur, the associated non-trivial disease-free equilibrium is shown to be globally-asymptotically stable when the reproduction number of the model is less than unity. Numerical simulations, using data relevant to dengue transmission dynamics in northern Queensland, Australia, shows that releasing Wolbachia-infected mosquitoes every three weeks, for a one-year duration, can lead to the effective control of the population abundance of the local wild mosquitoes, and that such effective control increases with increasing number of Wolbachia-infected mosquitoes released (resulting in the reduction of over 90% of the wild mosquito population from their baseline values). Furthermore, simulations show that releasing only adult male Wolbachia-infected mosquitoes provide more beneficial population-level impact (in terms of reducing the population abundance of the wild mosquitoes), in comparison to releasing adult female Wolbachia-infected mosquitoes. Increasing the frequency of Wolbachia release (e.g., from the default release frequency of every three weeks to weekly) does not significantly affect the effectiveness of the Wolbachia-based control program in curtailing the local abundance of the wild mosquitoes. Finally, it was shown that the cytoplasmic incompatibility property of Wolbachia bacterium does not significantly affect the effectiveness of the Wolbachia-based mosquito control strategy implemented in the community.