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2017•Journal of Applied Mathematics and PhysicsOpen access

Global Analysis of an SEIR Epidemic Model with a Ratio-Dependent Nonlinear Incidence Rate

Xiaomei Ren, Tiansi Zhang

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Abstract

In this paper, a SEIR model with ratio-dependent transmission rate in the form is studied and the basic reproduction number which determines the disease’s extinction or continued existence is obtained. By constructing the proper Lyapunov function, we prove that if R0 ≤ 1, the disease-free equilibrium point of the model is globally asymptotically stable and the disease always dies out; if R0 > 1, the endemic equilibrium point is globally asymptotically stable and the disease persists.

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What this paper is about

In this paper, a SEIR model with ratio-dependent transmission rate in the form is studied and the basic reproduction number which determines the disease’s extinction or continued existence is obtained. By constructing the proper Lyapunov function, we prove that if R0 ≤ 1, the disease-free equilibrium point of the model is globally asymptotically stable and the disease always dies out; if R0 > 1, the endemic equilibrium point is globally asymptotically stable and the disease persists.

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Available abstract

In this paper, a SEIR model with ratio-dependent transmission rate in the form is studied and the basic reproduction number which determines the disease’s extinction or continued existence is obtained. By constructing the proper Lyapunov function, we prove that if R0 ≤ 1, the disease-free equilibrium point of the model is globally asymptotically stable and the disease always dies out; if R0 > 1, the endemic equilibrium point is globally asymptotically stable and the disease persists.

Key concepts: Epidemic model, Nonlinear system, Mathematics, Incidence (geometry), Applied mathematics, Econometrics, Statistics, Physics

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