兰州大学机构库 >数学与统计学院
两种群非局部扩散SIR传染病模型的行波解
Alternative TitleTraveling wave solutions in a two-group SIR epidemic model with nonlocal dispersal
申梅
Thesis Advisor李万同
2018-04-01
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword非局部扩散 行波解 潜伏期 Schauder's不动点定理 Perron-Frobenius定理
Abstract

本文主要研究了两种群的非局部扩散SIR传染病模型行波解的存在性与不存在性, 以及最小波速c^{*}关于参数的连续依赖性.首先,利用Fourier变换的方法, 导出本文所研究的具体模型, 即一类两种群非局部扩散SIR传染病模型.其次, 在考虑潜伏期的情况下, 研究两种群非局部扩散SIR传染病模型行波解的存在性与不存在性. 由于非局部算子的引入, 导致系统本身的解没有足够的正则性, 从而已有的讨论行波解存在性的方法不能直接应用. 为了克服此困难, 利用截断的方法研究当基本再生数R_{0}(S^{0}_{1},S^{0}_{2})>1且c>c^{*}时行波解的存在性, 这里c^{*}为临界波速. 具体通过构造有界区域上的闭锥, 运用Schauder's不动点定理及逼近方法得到行波解的存在性. 进一步, 讨论行波解的边界渐近行为. 因为解没有足够的正则性, 所以患病个体的有界性及易感个体+\infty处的渐近行为很难得到. 尽管如此, 本文通过细致的分析在大波速条件下克服了这一困难. 另一方面, 对R_{0}(S^{0}_{1},S^{0}_{2})>1且c\in(0,c^{*})的情况, 利用双边Laplace变换研究了行波解的不存在性. 同时, 采用Perron-Frobenius定理讨论了当R_{0}(S^{0}_{1},S^{0}_{2}\leq1时行波解的不存在性.最后, 考虑了最小波速c^{*}关于参数的连续依赖性. 得到已感染个体的扩散率D_{i}, 潜伏个体的扩散率D_{L_i}和感染率\beta_{ij}可以加快疾病的传播速度; 已感染个体的移出率r_{i},潜伏个体的移出率M_i和潜伏期\tau可以减慢疾病的传播速度.

Other Abstract

In this paper, we study the existence and nonexistence of traveling wave solutions in a two-group epidemic model with nonlocal dispersal, and the continuous dependence of minimum wave speed on parameters.Firstly, the Fourier transformation method is used to derive the specific model studied in this paper. That is, a class of a two-group SIR epidemic model with nonlocal dispersal.Secondly, the existence and nonexistence of traveling wave solutions in a two-group SIR epidemic model with nonlocal dispersal are studied under the consideration of latent period. Due to the introduction of nonlocal operators, the solution of the system itself does not have enough regularity, so the existing methods of the existence of the traveling wave solutions can not be applied directly. In order to overcome this difficulty, we use truncation method to study the existence of wave solutions when the basic reproduction number R_{0} (S^{0}_{1}, S^{0}_{2}) >1 and c>c^{*}, where c^{*} is the critical wave velocity. The existence of traveling wave solutions is obtained by constructing closed cones on bounded regions and using Schauder's fixed point theorem and approximation method. Furthermore, the boundary asymptotic behavior of the traveling wave solution is discussed. Because the solution does not have sufficient regularity, the boundedness of the affected individual and the asymptotic behavior of the susceptible individual at +\infty are difficult to obtain. Nevertheless, this paper overcomes this difficulty by careful analysis under the condition of high wave velocity. On the other hand, for the case of R_{0} (S^{0}_{1}, S^{0}_{2}) >1 and c\in (0, c^{*}), the nonexistence of traveling wave solutions is studied by using the bilateral laplace transform.At the same time, the nonexistence of traveling wave solutions is discussed by Perron-Frobenius theorem when R_{0} (S^{0}_{1}, S^{0}_{2})\leq1.Finally, we consider the continuous dependence of the minimum wave velocity c^{*} on the parameters. The diffusion rate D_{i} of the infected individuals, the diffusion rate D_{L_i} of latent individual, and the infection rate \beta_{ij} can accelerate the spread of the disease. Emigration rate r_{i} of the infected individuals, the migration rate M_i of latent individuals and latent period \tau can slow the spread of the disease.

URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224874
Collection数学与统计学院
Recommended Citation
GB/T 7714
申梅. 两种群非局部扩散SIR传染病模型的行波解[D]. 兰州. 兰州大学,2018.
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