Models to assess the effects of non-identical sex ratio augmentations of Wolbachia-carrying mosquitoes on the control of dengue disease

The introduction of endosymbiont Wolbachia into laboratory-reared mosquito populations, which are then released to mix with natural populations to prevent the mosquito vectors from reproducing and thus break the transmission cycle of dengue disease, is an innovative new technology. Field trials of Wolbachia-carrying mosquitoes have now been implemented in many countries where there have been the outbreaks of dengue disease. A mathematical model is proposed to investigate the effects of non-identical sex ratio releases of Wolbachia-carrying mosquitoes on the control of dengue transmission. Firstly, we analyzed the existence and stability of equilibria for the system and proved the existence of forward and backward bifurcations. Secondly, bifurcation diagrams, the basins of attraction of the equilibria and the effects of mosquito augmentation for the system with imperfect and perfect transmission rates were obtained. Thirdly, three possible results for mosquito augmentation were summarized for different parameter regions. Further we explored an uncertainty and sensitivity analysis of solutions to estimate the effects of different parameter values on the success or failure of population replacement. Based on the above analysis, we considered a series of relevant issues such as (a) whether or not mosquito augmentation can ensure the success of population replacement? (b) If not, what are the parameter regions for the success or possible success of population replacement? (c) How does the initial density of natural mosquitoes and the quantity of mosquito augmentations affect the success of population replacement? (d) Whether all population replacements are effective for reducing the spread of dengue virus in the end? The results of this study will be helpful for public health authorities in designing proper strategies of mosquito augmentations for the control of dengue disease.