Nonautonomous Structured Population Models with Application to Amphibians and Associated Diseases

在这个项目中，研究人员开发了一类一般的结构化模型，它由一阶偏微分方程组和常微分方程组组成。对于这类新的模型，他们将研究解的存在唯一性，并开发一个框架，在其中计算解和估计参数，这是将这些工具与观测数据联系起来所必需的。特别是发展了二阶有限差分方法，这种方法在计算这些模型的解时效率很高，精度也很高。将建立这些方案的收敛结果。以前的工作已经证明，归入这一一般类别的模型成功地描述了绿色树蛙种群的动态。研究人员将证明，这类模型的应用可以扩展到树突巴氏杆菌(Bd)在两栖动物中的传播。最近发现，共生菌变青霉(JL)通过产生抗真菌代谢产物为某些种类的两栖动物提供对这种疾病的免疫力。在这个项目中开发的数值方法将被用来了解疾病Bd和共生菌JL对两栖动物持久性的影响。此外，该项目的一个重要目标是培养数学生物学多学科领域的博士生，以帮助实施拟议的研究。两栖动物种群的减少和灭绝是科学界非常关注的问题。由于两栖动物在陆地和水生生态系统之间旅行，并跨越不同的营养水平，它们往往是气候变化、栖息地丧失、污染和非本地物种造成的大规模环境退化的第一个指标。例如，引入的疾病减少和削弱了两栖动物的数量，这反过来又改变了生态景观。在这个项目中，对两栖动物和相关疾病的研究将导致建立结构模型，以了解人口和疾病动态。这些模型的解的计算将需要新的数值方法，这些方法将被开发出来。在本项目期间对模型进行的广泛的数值模拟将提供关于如何通过明智地引入共生菌JL来对抗BD的计划如何最有效地执行的洞察力。此外，由于开发的模型的一般性和本项目中建立的数学技术的抽象性，这些工具无疑将对其他重要的生物学问题有用。已经开发的属于拟议类别的模型的具体例子包括：(I)与红细胞生成有关的细胞动力学，(Ii)疟疾的宿主内动力学，(Iii)海洋分枝杆菌(一种类似结核的细菌)在鱼类中的传播动力学。

In this project, the investigators develop a general class of structured models which consists of a system of first-order partial differential equations coupled to a system of ordinary differential equations. For this novel class of models, they will study existence-uniqueness of solutions and develop a framework within which to compute solutions and estimate parameters, which is necessary to connect these tools to observed data. In particular, second order finite difference methods will be developed, which are efficient in computing solutions of these models with high accuracy. Convergence results for these schemes will be established. Previous work has demonstrated that models falling under this general class are successful in describing the dynamics of a green tree frog population. The investigators will demonstrate that the application of this class of models can be extended to the spread of Batrachochytrium dendrobatidis (Bd) among amphibians. It has been recently discovered that the symbiont Janthinobacterium lividum (Jl) provides immunity to this disease for some species of amphibians through the production of anti-fungal metabolites. The numerical methods developed during this project will be used to understand the effects the disease Bd and the symbiont Jl have on the persistence of amphibians. Further, an important aim of this project is to train Ph.D. students in the multidisciplinary field of mathematical biology to help carry out the proposed research.The decline and extinction of amphibian populations are of immense concern to the scientific community. Because amphibians travel between terrestrial and aquatic ecosystems, and across different trophic levels, they are often the first indicators of large-scale environmental degradation caused by climate change, habitat loss, pollution, and non-native species. For example, introduced diseases diminish and weaken amphibian populations, which in turn transform the ecological landscape. The study of amphibians and associated diseases in this project will result in structured models to understand population and disease dynamics. The computation of solutions of these models will require novel numerical methods, which will be developed. The extensive numerical simulations of the models conducted during this project will provide insight as to how a program to combat Bd with the judicious introduction of the symbiont Jl might be carried out most effectively. Furthermore, because of the generalities of the models developed and the abstractness of the mathematical techniques established in this project, these tools will undoubtedly be useful to other important biological problems. Specific examples of models that fall under the proposed class that have already been developed include: (i) cell dynamics involved in erythropoiesis, (ii) intra-host dynamics of malaria, (iii) the transmission dynamics of Mycobacterium marinum (a TB-like bacterium) among fish.