具有时滞的周期传染病模型的研究及应用

In recent years, many scholars begin to explore the periodic phenomena in the process of disease transmission by using the mathematical models with time-periodic coefficients in terms of ordinary differential equations. However, delay plays an important role in disease transmission for many seasonal infectious diseases. This implies that formulating time-delayed periodic epidemic models is more effective. Little research has been done since there are many difficulties in the theoretical study of nonautonomous system. Just for the purpose, this program will formulate several kinds of nonautonomous epidemic models to characterize the transmission law of seasonal diseases. By using the theory of nonautonomous funtional differential equations and dynamical systems, we give a systematically and originally study for model dynamics, and solve some emerging hotspot and difficulty in epidemiology, such as numerical calculation of basic reproduction number of epidemic models with delay apperaring in the infected equations、the stability of positive periodic solution and the relations between delay and basic reproduction number. Its accomplishment will greatly enrich the theories of epidemic dynamics, and has a broad application prospect.

近年来，许多学者开始利用具有周期系数的常微分方程模型来探讨疾病传播过程中的周期现象。然对许多的季节性疾病，时滞在其传播过程中扮演着重要的角色，故建立非自治时滞微分方程的传染病模型更加合适。但这在数学的理论研究上具有很大地难度，当前国内外的研究很少。鉴于此，本项目将建立几类非自治具有时滞的传染病模型来刻画季节性疾病的传播规律。利用非自治泛函微分方程的理论和动力系统的方法对所建模型的动力学性态给出系统、独创性的研究，解决传染病动力学中目前所关注的一些热点和难点问题，如染病的方程中具有时滞的传染病模型中基本再生数的数值估计、正周期解的稳定性、时滞与基本再生数之间的关系等。项目的完成将极大地丰富传染病动力学的研究理论，有着广泛的应用前景。

本项目针对一些具有季节性的传染病,建立了几类在感染的方程中具有时滞的周期传染病模型。利用算子谱半径的方法定义了模型的基本再生数R_0，分析了模型的阈值动力学行为,给出了估计R_0的数值方法，并表明具有时滞周期传染病模型的R_0与相应周期常微模型的R_0具有很大的不同。同时解决了在文[Gao, Chen, Teng, Bull. Math. Biol., 69 (2007)731-745]所遗留下的一个公开问题：寻找具有时滞脉冲微分方程模型在疾病一致持续和消除之间的阈值。项目的完成为国内外这项近乎空白的研究提供了可供借鉴的方法，极大地丰富了传染病动力学的研究理论。 研究成果:完成论文7篇,发表SCI检索论文5篇(其中2篇接受),另外2篇小修后正在审稿,有望在近期接受。项目执行期间，申请到国家青年自然科学基金一项。