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Analysis and Research on Some Epidemical Models with Nonlinear Incidence

Author: ZhangZhongHua
Tutor: XingWei
School: Northeastern University
Course: Operational Research and Cybernetics
Keywords: Epidemic model Nonlinear incidence rate Quarantine rate Basic reproduction number Continuous vaccination Pulse vaccination Periodic infection free solution Global stability
CLC: O242.1
Type: Master's thesis
Year: 2008
Downloads: 116
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Abstract


The presence of epidemic has always been a very common phenomenon and some results have been achieved by establishing epidemical models, analyzing these models qualitatively and quantitatively and predicting the trends of the epidemic. Compared with the former epidemic models with nonlinear incidence rate, this paper introduces the population dynamics factors and described the transmitting laws accurately. We discuss the positive invariant set, the existence and the stability of the equilibrium by using the stability theory of ordinary differential equation and obtain the sufficient conditions of the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium. We can attain aim to control the diffusion of epidemic by the means of isolating infected individuals and vaccinating susceptible. The main contents are as follows:Chapter one introduces the original background of the problem studied, development status, the research work and prior knowledge in the paper.Chapter two introduces the SIRS epidemic model with constant input and nonlinear transmission rate. In immunodeficiency conditions, constant control and linear state-feedback control are imposed on the model when the control parameters meet certain conditions, endemic can be eliminated and conditions of global asymptotic stability of the equilibrium. Simulation verifies the correctness of the results.Chapter three introduces the properties of solution of SIRS with density restriction and nonlinear incidence rate, the existence of equilibriums and the local stability of the positive equilibriums of the model. Simulation verifies the correctness of the results.Chapter four introduces the SIRS epidemic model with a nonlinear incidence rate and Smith Growth in susceptible populations, the positive invariant set, the equilibriums and the stability of the equilibriums of the model.Chapter five introduces the SIS epidemic model and the SIQS epidemic model with a nonlinear incidence rate. Basic reproduction numbers of two models are obtained, and stabilities of two models are discussed separately when basic reproduction numbers satisfy certain conditions. Simulation verifies the correctness of the results.Chapter six introduces the bilinear incidence SIRS epidemic models with continuous vaccination and pulse vaccination. The basic reproduction numbers of two SIRS epidemic models are given respectively. The global stability of the infection free equilibrium and positive equilibrium with continuous vaccination are proven by using Lyapunov function and LaSalle invariability principle; dynamic properties in this model with pulse vaccination are studied by using Floquet multipler theory, comparison theorem and nonlinear analysis method of the impulsive differential equation. Sufficient condition of global stability of the disease-free periodic solution is obtained and the strategies of two vaccinations are compared in the end.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Mathematical modeling, approximate calculation > Mathematical modeling
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