Lyapunov Function for a Dengue Transmission Model where two Species of Mosquitoes are Present: Global Stability

Abstract

In this study, we are interested in the newly observed endemic due to the ZIKA virus. In the absence of a full knowledge of the dynamics of the infection due to this virus, we consider a model of the closely related dengue virus when there are two species of mosquitoes, A. aegypti and A. albopictus mosquitoes who are both capable of transmitting the two viruses. We use a Susceptible-Infected-Recovered (SIR) model to describe the infection of the human populations. We obtain the basic reproduction ratio (R0) and show that if R0 is less than 1, the disease free equilibrium state is global asymptotically stable. If R0 is greater than 1, we use the Lyapunov function approach to find the conditions for the unique dengue endemic equilibrium state to be globally stable. We then point out the insights into the global stability of the ZIKA epidemic that can be gained by looking at the global stability of a model for the dengue infection in the presence of two species of mosquitoes that can transmit the disease.