Endemic Equilibrium and Forward Bifurcation in the Mathematical Model for Using Wolbachia to Control Spread of Zika Virus Disease

Abstract

This paper focuses on the use of wolbachia to control the spread of zika virus disease. Zika virus disease is an arboviral disease that spreads through bites of female mosquitoes in the aedes family especially, aedes aegypti. Experimental studies have indicated that wolbachia could be used to prevent the spread of zika virus disease by infecting aedes aegypti with wolbachia in a laboratory and releasing them in the wild to mate with the wild aedes aegypti. A system of nonlinear ordinary differential equations is used to model the use of wolbachia to stop the spread of zika virus disease in the human and mosquito populations. as well as the population of wolbachia-infected aedes aegypti used as control. It is shown through bifurcation analysis that the model exhibits forward bifurcation, which confirms that a unique endemic equilibrium exists in the model when the control reproduction number, $ \mathcal{R}_c>1$. The existence of forward bifurcation in the model means that $ \mathcal{R}_c<1$ is enough to guarantee eradication of zika virus disease using wolbachia as a biocontrol. Hence, the spread of zika virus disease can be controlled irrespective of the initial sizes of infected human and mosquito populations