Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China

• Studied a malaria transmission model with standard incidence and treatment.
• Showed the existence of a backward bifurcation when R0 < 1.
• Proved global stability of the endemic equilibrium when R0 > 1.
• Simulated the malaria data reported by the China CDC from 2002 to 2013.
• Predicted the elimination of malaria in China is practical.

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<1R0<1 and a unique endemic equilibrium exists if R0>1.R0>1. The presence of a backward bifurcation implies that it is possible for malaria to persist even if R0<1.R0<1. Secondly, using geometric method, some sufficient conditions for global stability of the unique endemic equilibrium are obtained when R0>1.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≈0.0161R0≈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.