Modeling pyrethroids repellency and its role on the bifurcation analysis for a bed net malaria model

We develop and analyze a simple mathematical model for malaria transmission where pyrethroids treated nets (PTNs) are used for control purposes, and in which knock-down/lethal, excito-repellent/deterrent effects are incorporated. We explicitly describe the contact rates between mosquitoes and humans by nonlinear functions of bed net usage and repellency rates. Using center manifold theory, we show that our model exhibits, saddle-node, transcritical forward and backward bifurcation when the reproduction number R0 crosses one. Our model reveals that repellency effect plays an important role for the existence of both the endemic equilibrium points and the occurrence of backward bifurcation and a threshold repellency rate is calculated. The epidemiological implication of the backward bifurcation is that, reducing R0 below one alone is not enough to eliminate malaria. We establish that, increasing either the rate of bed nets usage (i.e community protection) or their repellent effect (i.e personal protection) or the combination of both, decreases the contact rates between humans and mosquitoes. As a result, the disease burden metric R0 is reduced. The global asymptotic stability of equilibrium points are proven using the geometric approach and Lyapunov-LaSalle techniques. Furthermore, we show that neglecting repellency underestimates the basic reproduction number and hinders the control of malaria. The disease free equilibrium is shown to be a saddle-node of co-dimension 1 when R0=1. We also observe that R0 is mostly influenced by the bed net coverage rate, repellent effect of pyrethroids and the probability that mosquitoes target human hosts. Our results confirm that PTNs usage is an efficient control strategy to mitigate the malaria ability to spread, and suggest that the utilization of PTNs with high lethal rate, but low repellency rate is better than the use of those with high repellency and low lethal rates.