Bifurcation Analysis of Multiscale Malaria Model with Serratia AS1 Bacteria and Saturated Treatment

Abstract

Recent experimental results have implied that Serratia AS1 bacteria efficiently suppress the development of P. falciparum, and rapidly disseminate via vertical transmission and sexual transmission throughout mosquito population. In this paper, we propose a new malaria model to investigate the roles of saturated treatment and AS1 bacteria in malaria transmission. One feature of the model is that human and mosquito populations dynamics run on disparate temporal scales. Singular perturbation techniques are used for the separation of fast and slow dynamics. It also enables us to overcome the difficulties of exploring bifurcations in higher dimensional system. We demonstrate that the reduced systems may undergo backward bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation of codimension 2, which indicates that the delayed effect for treatment has important implications for malaria control. Numerical simulations substantiate these analytical results and show epidemiological insights. Finally, the model without AS1 bacteria transmission is applied to simulate the accumulative malaria cases in Burundi. We observe that malaria may not be eliminated by a single control in Burundi. A variety of available control measures should be adopted to eradicate malaria, such as improving treatment rate, reducing the delayed effect for treatment, and possibly releasing mosquitoes that carry AS1 bacteria.