Mosquito feeding preference and pyrethroids repellent effect eliminate backward bifurcation in malaria dynamics

Pyrethroid-treated bed-nets (PTNs) protect individuals against malaria by blocking and repelling mosquitoes. We develop and analyze a PTNs malaria model that explicitly includes mosquito host choice (also known as feeding/biting preference) and Pyrethroid repellent effect. Our model reveals that mosquito biting/feeding preference on infectious hosts pi and repellent effect r drive for the existence of both the endemic equilibrium points and the occurrence or elimination of backward bifurcation. The threshold parameters for the mosquito biting preference on infectious hosts pi* and repellent effect r* for the occurrence and elimination of backward bifurcation are computed. Moreover, it is shown that, increasing the mosquito host choice rate or decreasing the repellent effect rate, annihilates backward bifurcation, thus facilitating the control of malaria. Furthermore, we prove that the threshold of mosquito biting preference is a monotone increasing function of the repellent effect r. We show that the model exhibits both trans-critical forward bifurcation and backward bifurcation when either the mosquito host choice pi crosses a threshold value pi(1) or the repellent effect r passes through a threshold repellent rate r(1). Sufficient conditions for the global asymptotic stability of the equilibrium point are derived. On the other hand, it is established that, decreasing the mosquito biting preference or increasing the rate of the repellent effect (i.e personal protection) or the combining both actions, decreases the malaria control reproduction number R-0. Finally, the interplay between the bed-nets treated repellent effect and mosquito host choice and its potential on the dynamics of malaria is investigated and illustrated numerically.