Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes

Aedes aegypti (Ae. aegypti: mosquito) is a known vector of several viruses including yellow fever, dengue, chikungunya and zika. In the current paper, we present a delayed mathematical model describing the dynamics of Ae. aegypti. Our model is governed by a system of three delay differential equations modeling the interactions between three compartments of the Ae. aegypti life cycle (females, eggs and pupae). By using time delay as a parameter of bifurcation, we prove stability/switch stability of the possible equilibrium points and the existence of bifurcating branch of small amplitude periodic solutions when time delay crosses some critical value. We establish an algorithm determining the direction of bifurcation and stability of bifurcating periodic solutions. In the end, some numerical simulations are carried out to support theoretical results..