On a three-stage structured model for the dynamics of malaria transmission with human treatment, adult vector demographics and one aquatic stage.

A modelling framework that describes the dynamics of populations of the female Anopheles sp mosquitoes is used to develop and analyse a deterministic ordinary differential equation model for dynamics and transmission of malaria amongst humans and varying mosquito populations. The framework includes a characterization of the gonotrophic cycle of the female mosquito. The epidemiological model also captures a novel feature whereby treated human's blood can become mosquitocidal to the questing mosquitoes upon the successful ingestion of the treated human's blood. Analysis of the disease free system, that is the model in the absence of infection in the human and mosquito populations, reveals the presence of a basic offspring number, N, whose size determines the existence and stability of a thriving mosquito population in the sense that when N≤1 we have only the mosquito extinction steady state which is globally asymptotically stable, while for N > 1 we have the persistent mosquito population steady state which is also globally asymptotically stable for these range of values of N. In the presence of disease, N still strongly affects the properties of the epidemiological model in the sense that for N≤1 the only steady state for the system is the mosquito extinction steady state, which is globally and asymptotically stable. As N increases beyond unity in the epidemiological model, we obtained the epidemiological basic reproduction number, R. For R < 1, the disease free equilibrium, with both healthy thriving susceptible human and mosquito populations, is globally asymptotically stable. Both N and R are studied for control purposes and our study highlights that multiple control schemes would have a stronger impact on reducing both N and R to values small enough for a possible disease vector control and disease eradication. Our model further illustrates that newly emerged mosquitoes that are infected with the malaria parasite during their first blood meal play an important and strong role in the malaria disease dynamics. Additionally, mosquitoes at later gonotrophic cycle stages also impact the dynamics but their contributions to the total mosquito population size decreases with increasing number of gonotrophic cycles. The size of the contribution into the young mosquito population is also dependent on the length of the gonotrophic cycles, an important bionomic parameter, as well as on how the mosquitoes at the final gonotrophic cycles are incorporated into the modelling scheme.