一类具有隔离项的随机SIQS传染病模型全局正解的渐近行为

发布时间：2018-06-23 13:41

本文选题：随机SIQS传染病模型 + 李雅普诺夫函数　；参考：《暨南大学》2015年硕士论文

【摘要】：传染病动力学的研究目的是探寻疾病流行的内在原因及影响因素,掌握疾病的传播原理和规律,预测疾病的发展趋势,从而能对流行病的防范与消除提供理论依据。众所周知,利用常微分方程建立传染病的确定性模型已经在国内外研究了很多年,得到了大量的研究成果。但确定性传染病模型没有考虑到自然界中存在的随机因素,加入白噪声的随机传染病模型更能反映实际情况,得到更加准确的结果。目前大部分的重点都在SIR,SIRS,HIV等模型的研究上,涉及到将具有隔离项的确定型SIRS模型随机化的研究还很少。所以本文在原有SIQS模型基础上,考虑当系统受到随机因素影响时,建立随机SIQS模型并对模型进行动力学行为研究。本文在确定性SIRS传染病模型中加入随机扰动项得到了随机传染病模型,重点研究了随机SIQS传染病模型的动力学行为。论文首先介绍了一类具有隔离项的SIQS传染病模型,得出了模型的疾病基本再生数??0、无病平衡点E0及地方病平衡点E?的数学表达式。再此基础上研究了系统受噪声影响的随机SIQS传染病模型解的渐近行为。通过构造合理的李雅普诺夫函数证明了该随机模型全局正解的存在惟一性。通常确定性SIRS传染病模型存在无病平衡点和地方病平衡点,但相应的随机SIQS传染病模型不再具有上述平衡点。因此,论文在一定的假设条件下进一步证明,当基本再生数R01时,随机SIQS传染病模型的全局正解关于确定性SIQS模型的无病平衡点具有渐近性质,该性质表明疾病将最终消失;当E01时,随机SIQS传染病模型的全局正解关于确定性模型地方病平衡点具有渐近性质,该性质意味着疾病将流行且最终形成地方病。论文最后通过数值模拟仿真例子验证了本文结论的正确性。
[Abstract]:The purpose of the study on dynamics of infectious diseases is to explore the internal causes and influencing factors of disease prevalence, to master the principles and laws of disease transmission, to predict the trend of disease development, and to provide theoretical basis for the prevention and elimination of epidemic diseases. It is well known that the deterministic model of infectious diseases established by ordinary differential equations has been studied for many years and a great deal of research results have been obtained. But the deterministic infectious disease model does not take into account the random factors that exist in nature. The stochastic infectious disease model with white noise can reflect the actual situation and get more accurate results. At present, most of the emphasis is on the research of SIRSIRSU HIV and other models, and there are few researches on randomization of deterministic Sirs models with isolated terms. Therefore, based on the original SIQS model, the stochastic SIQS model is established and the dynamic behavior of the model is studied when the system is affected by random factors. In this paper, the stochastic infectious disease model is obtained by adding the stochastic perturbation term into the deterministic Sirs infectious disease model, and the dynamic behavior of the stochastic SIQS infectious disease model is studied emphatically. In this paper, we first introduce a class of SIQS infectious disease models with isolation term, and obtain the disease basic regeneration number of the model, the disease-free equilibrium point E0 and the endemic equilibrium point E0? The mathematical expression of. On this basis, the asymptotic behavior of the solution of stochastic SIQS infectious disease model affected by noise is studied. The existence and uniqueness of the global positive solution of the stochastic model are proved by constructing a reasonable Lyapunov function. Usually, the deterministic Sirs epidemic model has disease-free equilibrium and endemic equilibrium, but the corresponding stochastic SIQS infectious disease model no longer has the above equilibrium. Therefore, under certain assumptions, it is further proved that the global positive solution of stochastic SIQS infectious disease model has asymptotic property on deterministic SIQS model when the basic reproduction number is R01, which indicates that the disease will eventually disappear. When E01, the global positive solution of stochastic SIQS infectious disease model has asymptotic property about the endemic equilibrium point of deterministic model, which means that the disease will be prevalent and eventually form endemic disease. Finally, a numerical simulation example is given to verify the correctness of the conclusion.
【学位授予单位】：暨南大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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