On a Reproductive Stage-Structured Model for the Population Dynamics of the Malaria Vector

Abstract

A reproductive stage-structured deterministic differential equation model for the population dynamics of the human malaria vector is derived and analysed. The model captures the gonotrophic and behavioural life characteristics of the female Anopheles sp. mosquito and takes into consideration the fact that for the purposes of reproduction, the female Anopheles sp. mosquito must visit and bite humans (or animals) to harvest necessary proteins from blood that it needs for the development of its eggs. Focusing on mosquitoes that feed exclusively on humans, our results indicate the existence of a threshold parameter, the vectorial reproduction number, whose size increases with increasing number of gonotrophic cycles, and is also affected by the female mosquito's birth rate, its attraction and visitation rate to human residences, and its contact rate with humans. A stability analysis of the model indicates that the mosquito can establish itself in the environment if and only if the value of the vectorial reproduction number exceeds unity and that mosquito eradication is possible if the vectorial reproduction number is less than unity, since, then, the trivial steady state which always exist is unique and is globally and asymptotically stable. When a persistent vector population steady state exists, it is locally and asymptotically stable for a range of reproduction numbers, but can also be driven to instability via a Hopf bifurcation as the reproduction number increases further away from unity. The model derivation identifies and characterizes control parameters relating to activities such as human-mosquito contact and the mosquito's survival chances between blood meals and egg laying. Our results show that the total mosquito population size increases with increasing number of gonotrophic cycles. Therefore understanding the fundamental aspects of the mosquito's behaviour provides a pathway for the study of human-mosquito contact and mosquito population control. Control of the mosquito population densities would ultimately lead to malaria control.