Mathematics of a single-locus model for assessing the impacts of pyrethroid resistance and temperature on population abundance of malaria mosquitoes

This study presents a genetic-ecology modeling framework for assessing the combined impacts of insecticide resistance, temperature variability, and insecticide-based interventions on the population abundance and control of malaria mosquitoes by genotype. Rigorous analyses of the model we developed reveal that the boundary equilibrium with only mosquitoes of homozygous sensitive (resistant) genotype is locally-asymptotically stable whenever a certain ecological threshold, denoted by R0SS(R0RR), is less than one. Furthermore, genotype i drives genotype j to extinction whenever R0j>1 and R0i<1 (where i, j = SS or RR, with i ≠ j). The model exhibits the phenomenon of bistability when both thresholds are less than one. In such a bistable situation, convergence to any of the two boundary equilibria depends on the initial allele distribution in the state variables of the model. Furthermore, in this bistable case, where max{R0SS,R0RR}<1, the basin of attraction of the boundary equilibrium of the mosquito genotype with lower value of the ecological threshold is larger. Specifically, the basin of attraction of the boundary equilibrium for genotype i is larger than that of genotype j if R0i<R0j<1. When both ecological thresholds exceed one (min{R0SS,R0RR}>1), the two boundary equilibria lose their stability, and a coexistence equilibrium (where all three mosquito genotypes coexist) becomes locally-asymptotically stable. Global sensitivity analysis shows that the key parameters that greatly influence the dynamics and population abundance of resistant mosquitoes include the proportion of new adult mosquitoes that are females, the insecticide-induced mortality rate of adult female mosquitoes, the coverage level and efficacy of adulticides used in the community, the oviposition rates for eggs of heterozygous and homozygous resistant genotypes, and the modification parameter accounting for the reduction in insecticide-induced mortality due to resistance. Numerical simulations show that the adult mosquito population increases with increasing temperature until a peak is reached at 31 °C, and declines thereafter. Simulating the model for moderate and high adulticide coverage, together with varying fitness costs of resistance, shows a switch in the dominant genotype at equilibrium as temperature is varied. In other words, this study shows that, for certain combinations of adulticide coverage and fitness costs of insecticide resistance, increases in temperature could result in effective management of resistance (by causing the switch from a stable resistant-only boundary equilibrium (at 18 °C) to a stable sensitive-only boundary equilibrium (at 25 °C)). Finally, this study shows that, for moderate fitness costs of resistance, density-dependent larval mortality suppresses the total population of adult mosquitoes with the resistant allele for all temperature values in the range [18 °C–36 °C].

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.