American Institute of Mathematical Sciences

Journals

DCDS-B
Discrete & Continuous Dynamical Systems - B 2014, 19(4): 999-1025 doi: 10.3934/dcdsb.2014.19.999
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:
MBE
Mathematical Biosciences & Engineering 2018, 15(3): 765-773 doi: 10.3934/mbe.2018034

keywords:
MBE
Mathematical Biosciences & Engineering 2014, 11(3): 449-469 doi: 10.3934/mbe.2014.11.449
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:
MBE
Mathematical Biosciences & Engineering 2015, 12(1): 99-115 doi: 10.3934/mbe.2015.12.99
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.
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