title: Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China.
abstract: In this paper, we consider a deterministic malaria transmission model with standard incidence rate and treatment. Human population is divided into susceptible, infectious and recovered subclasses, and mosquito population is split into susceptible and infectious classes. It is assumed that, among individuals with malaria who are treated or recovered spontaneously, a proportion moves to the recovered class with temporary immunity and the other proportion returns to the susceptible class. Firstly, it is shown that two endemic equilibria may exist when the basic reproduction number R0<1 and a unique endemic equilibrium exists if R0>1. The presence of a backward bifurcation implies that it is possible for malaria to persist even if R0<1. Secondly, using geometric method, some sufficient conditions for global stability of the unique endemic equilibrium are obtained when R0>1. To deal with this problem, the estimate of the Lozinskiı˘ measure of a 6 × 6 matrix is discussed. Finally, numerical simulations are provided to support our theoretical results. The model is also used to simulate the human malaria data reported by the Chinese Ministry of Health from 2002 to 2013. It is estimated that the basic reproduction number R0≈0.0161 for the malaria transmission in China and it is found that the plan of eliminating malaria in China is practical under the current control strategies.
authors.affiliation: Department of Mathematics, Yuncheng University, Yuncheng 044000, People's Republic of China.
authors.forename: Xiaomei
authors.initials: X
authors.lastname: Feng
authors.affiliation: Department of Mathematics, University of Miami Coral Gables, FL 33124-4250, USA.
authors.forename: Shigui
authors.initials: S
authors.lastname: Ruan
authors.affiliation: College of Mathematics and System Sciences, Xinjiang University Urumqi 830046, People's Republic of China. Electronic address: zhidong@xju.edu.cn.
authors.forename: Zhidong
authors.initials: Z
authors.lastname: Teng
authors.affiliation: Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, People's Republic of China.
authors.forename: Kai
authors.initials: K
authors.lastname: Wang
mesh_terms: D002681:China; D003140:Communicable Disease Control; D006801:Humans; D008288:Malaria / Q000635:transmission*; D008962:Models, Theoretical
keywords: Backward bifurcation; Geometric approach; Global stability; Lozinski measure; Malaria