Study of some systems from population dynamics and disease transmission dynamics with heterogeneity

本研究计划由两部分组成。(A)研究可能解释气候变化或工业化引起的环境/资源转移的空间转移异质性下两类系统的时空动态。一类是具有成熟延迟的成熟种群的标量延迟微分方程，另一类是相互作用物种（包括竞争物种和捕食-被捕食物种）的种群动态系统。(B)探索捕食者-猎物系统中的各种恐惧效应，以及传染病传播动力学中恐惧效应的相应概念。对于(A)，重点将放在转移速度和剖面，以及所涉及物种的生物学特征，将如何影响控制系统的时空动态。这些特征包括出生功能的类型、死亡率、成熟延迟、相互作用的类型、扩散率和相互作用的强度。可能的主题包括灭绝/持续，共存模式，传播速度，推动行波及其稳定性，以及初始分布的影响。我们将遇到一些具有特殊类型的具有延迟的时空异质性的新的动力系统。我希望开发一些新的方法，并修改和改进一些现有的方法来实现我们的目标。研究结果为空间动力系统的研究提供了理论和方法，也有助于理解气候变化和工业化的影响。(B)部分是现实世界的问题驱动的。我们将捕食者-猎物（P-P）系统中的“恐惧效应”概念扩展到传染病传播动力学建模中的“易感-感染”（S-I）相互作用，以反映在传染病流行期间，人类的行为（如社会活动和旅行）会发生变化的现实，这种变化是由于对疾病的恐惧，并可能随着流行病而演变。这一部分的重点是猎物的扩散、觅食和生产策略是如何随着对捕食者的感知而进化的；人类的预防水平将如何演变以及这种预防水平的演变对长期和短期疾病动态的影响。本研究结果有望在农业和公共卫生领域具有潜在的应用价值：P-P系统的恐惧效应研究结果可能为一些农业鼠害的防治提供新的环保方法；关于人类行为改变及其影响的“恐惧效应”（例如以媒体报道和公共卫生人员的警告为代表）的结果可能有助于公共卫生领域的决策者了解控制或减轻传染病流行的宝贵指导方针。

This proposed research program consists of two parts. (A) Study spatial temporal dynamics of two classes of systems under spatially shifting heterogeneity that may account for environment/resource shifting caused by climate change or industrialization. One class are scalar delay differential equations for mature populations with maturation delay, and the other class are systems for population dynamics of interacting species including competing species and predator-prey species. (B) Explore various fear effects in predator-prey systems and the corresponding notion of fear effects in transmission dynamics of infectious diseases. For (A), the focus will be on how the shifting speed and the profile, together with the biological characteristics of the species involved, will affect the spatial temporal dynamics of governing systems. These characteristics include the type of birth functions, death rate, maturation delays, and type of interaction, rate of dispersal  and strengths of interaction. The possible topics include extinction/persistence, pattern of coexistence, spreading speed, pushed traveling waves and their stability, and impact of initial distribution. We will encounter some new dynamical systems with a special type of spatial-temporal heterogeneity with delay. I expect to develop some new approaches/methods, abd to modify and improve some existing ones to achieve our objectives. Thus, the results may contribute to the theory and methods for spatial dynamical systems, and they may also help understand the impact of climate change and industrialization. Part (B) is real world problem driven. We extend the notion of "fear effects" in predator-prey (P-P) systems to "susceptible-infectious" (S-I) interactions in modeling transmission dynamics of infectious diseases to reflect the reality that during the epidemic of an infectious disease, human beings behaviors (e.g., social activities and travel) will change and such changes are due to the fear of disease and may evolve with the epidemic. The focus in this part will be on how a prey's dispersal, foraging and production strategies evolve in response to perceiving its predator(s); and how human being's precaution level will evolve and the impact of such an evolution of precaution level on the long term and short term disease dynamics. I expect the results of this proposed research will have potential applications in agriculture and public health: the results on fear effect in P-P system may suggest new and environmentally friendly approaches for controlling some agriculture rodent pests; and the results on "fear effects" (e.g. represented by media coverage and public health agents' warning) on human's behavior change and its impacts may help inform decision makers,  in public health,  of valuable guidelines for controlling or mitigating the epidemics of an infectious disease.