Article

# Impact of heterogeneity on the dynamics of an SEIR epidemic model

Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada.
(Impact Factor: 0.84). 04/2012; 9(2):393-411. DOI: 10.3934/mbe.2012.9.393
Source: PubMed

ABSTRACT

An SEIR epidemic model with an arbitrarily distributed exposed stage is revisited to study the impact of heterogeneity on the spread of infectious diseases. The heterogeneity may come from age or behavior and disease stages, resulting in multi-group and multi-stage models, respectively. For each model, Lyapunov functionals are used to show that the basic reproduction number R0 gives a sharp threshold. If R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and the disease dies out from all groups or stages. If R0 > 1, then the disease persists in all groups or stages, and the endemic equilibrium is globally asymptotically stable.

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Available from: Zhisheng Shuai, Mar 27, 2014
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• "which have been widely used in many papers, see [6] [28] [40] [42] and references therein . The forms of f kj (S k , I j ) satisfying assumptions (H 1 ) − (H 3 ) include common transmission functions (see [7, 28, 39, 42–44]) such as f kj (S k , I j ) = S k I j , f kj (S k , "
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ABSTRACT: In this paper, we investigate a class of multi-group epidemic models with general exposed distribution and nonlinear incidence rate. Under biologically motivated assumptions, we show that the global dynamics are completely determined by the basic production number $R_0$. The disease-free equilibrium is globally asymptotically stable if $R_0\leq1$, and there exists a unique endemic equilibrium which is globally asymptotically stable if $R_0>1$. The proofs of the main results exploit the persistence theory in dynamical system and a graph-theoretical approach to the method of Lyapunov functionals. A simpler case that assumes an identical natural death rate for all groups and a gamma distribution for exposed distribution is also considered. In addition, two numerical examples are showed to illustrate the results.
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• "The global stability of the unique endemic equilibrium of model (2) was proved by using a graph-theoretical approach and Lyapunov functionals. Global stability results were also obtained for other multi-group epidemic models [4, 5, 6, 7, 8, 9, 10, 11, 12]. Vaccination is one of the commonly used control measures to prevent and reduce the transmission of infectious diseases. "
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• "In the case of í µí± í µí± í µí± (í µí± í µí± , í µí°¼ í µí± ) = í µí± í µí± í µí°¼ í µí± , system (15) will reduce to the system studied in [14] [22]. Here Theorem 4 extends related results in [14] [22] to a result to a more general case allowing a nonlinear incidence rate. Our result also cover the related results of single group model in [13] for the case of í µí±(í µí±, í µí°¼) = í µí±(í µí±)í µí°¼. "
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ABSTRACT: We investigate a class of multigroup epidemic models with general exposed distribution and nonlinear incidence rates. For a simpler case that assumes an identical natural death rate for all groups, and with a gamma distribution for exposed distribution is considered. Some sufficient conditions are obtained to ensure that the global dynamics are completely determined by the basic production number . The proofs of the main results exploit the method of constructing Lyapunov functionals and a graph-theoretical technique in estimating the derivatives of Lyapunov functionals.
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