DCDS-B
Backward bifurcation and global stability in an epidemic model with treatment and vaccination
Xiaomei Feng Zhidong Teng Kai Wang Fengqin Zhang
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.
keywords: compound matrices. global stability SVEIR epidemic model backward bifurcation
MBE
Analyzing the causes of alpine meadow degradation and the efficiency of restoration strategies through a mathematical modelling exercise
Hanwu Liu Lin Wang Fengqin Zhang Qiuying Li Huakun Zhou

As an important ecosystem, alpine meadow in China has been degraded severely over the past few decades. In order to restore degraded alpine meadows efficiently, the underlying causes of alpine meadow degradation should be identified and the efficiency of restoration strategies should be evaluated. For this purpose, a mathematical modeling exercise is carried out in this paper. Our mathematical analysis shows that the increasing of raptor mortality and the decreasing of livestock mortality (or the increasing of the rate at which livestock increases by consuming forage grass) are the major causes of alpine meadow degradation. We find that controlling the amount of livestock according to the grass yield or ecological migration, together with protecting raptor, is an effective strategy to restore degraded alpine meadows; while meliorating vegetation and controlling rodent population with rodenticide are conducive to restoring degraded alpine meadows. Our analysis also suggests that providing supplementary food to livestock and building greenhouse shelters to protect livestock in winter may contribute to alpine meadow degradation.

keywords: Alpine meadow degradation restoration strategy mathematical modelling stability
MBE
The global stability of an SIRS model with infection age
Yuming Chen Junyuan Yang Fengqin Zhang
Infection age is an important factor affecting the transmission of infectious diseases. In this paper, we consider an SIRS model with infection age, which is described by a mixed system of ordinary differential equations and partial differential equations. The expression of the basic reproduction number $\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then the model only has the disease-free equilibrium, while if $\mathscr{R}_0>1$ then besides the disease-free equilibrium the model also has an endemic equilibrium. Moreover, if $\mathscr{R}_0<1$ then the disease-free equilibrium is globally asymptotically stable otherwise it is unstable; if $\mathscr{R}_0>1$ then the endemic equilibrium is globally asymptotically stable under additional conditions. The local stability is established through linearization. The global stability of the disease-free equilibrium is shown by applying the fluctuation lemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.
keywords: global stability persistence. SIRS model infection age
MBE
Global dynamics of a general class of multi-group epidemic models with latency and relapse
Xiaomei Feng Zhidong Teng Fengqin Zhang
A multi-group model is proposed to describe a general relapse phenomenon of infectious diseases in heterogeneous populations. In each group, the population is divided into susceptible, exposed, infectious, and recovered subclasses. A general nonlinear incidence rate is used in the model. The results show that the global dynamics are completely determined by the basic reproduction number $R_0.$ In particular, a matrix-theoretic method is used to prove the global stability of the disease-free equilibrium when $R_0\leq1,$ while a new combinatorial identity (Theorem 3.3 in Shuai and van den Driessche [29]) in graph theory is applied to prove the global stability of the endemic equilibrium when $R_0>1.$ We would like to mention that by applying the new combinatorial identity, a graph of 3n (or 2n+m) vertices can be converted into a graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.
keywords: global stability Multigroup epidemic model nonlinear incidence Lyapunov function.

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