Modelling and control of Mendelian and maternal inheritance for biological control of dengue vectors

Mosquitoes are vectors of viral diseases with epidemic potential in many regions of the world, and in absence of vaccines or therapies, their control is the main alternative. Chemical control through insecticides has been one of the conventional strategies, but induces insecticide resistance, which may affect other insects and cause ecological damage. Biological control, through the release of mosquitoes infected by the maternally inherited bacterium Wolbachia, which inhibits their vector competence, has been proposed as an alternative. The effects of both techniques may be intermingled in practice: prior insecticide spraying may debilitate wild population, so facilitating subsequent invasion by the bacterium; but the latter may also be hindered by the release of susceptible mosquitoes in an environment where the wild population became resistant, as a result of preexisting undesired exposition to insecticide. To tackle such situations, we propose here a unifying model allowing to account for the cross effects of both control techniques, and based on the latter, design release strategies able to infect a wild population. The latter are feedback laws, whose stabilizing properties are studied.