Backward bifurcation and global stability inan epidemic model with treatment and vaccination

Xiaomei Feng; Zhidong Teng; Kai Wang; Fengqin Zhang

doi:10.3934/dcdsb.2014.19.999

Article Contents

2014, Volume 19, Issue 4: 999-1025. Doi: 10.3934/dcdsb.2014.19.999

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Backward bifurcation and global stability in an epidemic model with treatment and vaccination

1.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046
2.
Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011
3.
Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi

Received: June 2013

Revised: January 2014

Published: June 2014

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Abstract

In this paper, we consider a class of epidemic models described by five nonlinear ordinary differential equations. The population is divided into susceptible, vaccinated, exposed, infectious, and recovered subclasses. One main feature of this kind of models is that treatment and vaccination are introduced to control and prevent infectious diseases. The existence and local stability of the endemic equilibria are studied. The occurrence of backward bifurcation is established by using center manifold theory. Moveover, global dynamics are studied by applying the geometric approach. We would like to mention that in the case of bistability, global results are difficult to obtain since there is no compact absorbing set. It is the first time that higher (greater than or equal to four) dimensional systems are discussed. We give sufficient conditions in terms of the system parameters by extending the method in Arino et al. [2]. Numerical simulations are also provided to support our theoretical results. By carrying out sensitivity analysis of the basic reproduction number in terms of some parameters, some effective measures to control infectious diseases are analyzed.

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Mathematics Subject Classification: Primary: 92D30, 34D23; Secondary: 34C23.

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