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 R₀^SS(R₀^RR), is less than one. Furthermore, genotype i drives genotype j to extinction whenever R₀^j>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{R₀^SS,R₀^RR}<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<R₀^j<1. When both ecological thresholds exceed one (min{R₀^SS,R₀^RR}>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].