Abstract
The dynamics of the transmission and spread of infectious diseases are known to be highly complex largely due to the heterogeneity of the host population and the ecology of the pathogens that causes the disease. Factors contributing to the heterogeneity of the host population include age distributions, social and ethnical groups, and spatial distributions, all of which can create complex contact patterns among hosts. Ecological factors for disease pathogens include life cycles, disease vectors, multiple hosts, and environmental influences due to local seasonal changes and large-scale climate changes. Mathematical models that incorporate these factors of heterogeneity often result in a large-scale system of nonlinear differential or difference equations that has a high dimension, multi-components and multi-parameters. While these type of models are more realistic than the classical SIR or SEIR models, its mathematical analysis is highly nontrivial because of the high-dimensionality and their validation from data for reliable predictions is often problematic because of the large number of model parameters. In this chapter, I present a graph-theoretic approach to the construction of Lyapunov functions for establishing the global dynamics of large-scale epidemic models. I will start in Sect. 3.1 with an introduction to epidemic modeling and give two examples of large-scale epidemic models. I will also show two methods for computing the basic reproduction number \(\mathcal {R}_0\): the method of van den Driessche and Watmough and the method of using Lyapunov functions. Both methods are based on local stability analysis of the disease-free equilibrium P 0. In Sect. 3.2, I will introduce the notion of dynamical systems on networks as a mathematical framework for large-scale epidemic models and explain the graph-theoretic approach to constructing Lyapunov functions in this general framework. In Sect. 3.3, I will present applications of the graph-theoretic approach to various large-scale models in epidemiology and ecology.
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Acknowledgements
Much of the work presented in this lecture note is based on joint work with Drs. Hongbin Guo and Zhisheng Shuai. I acknowledge the support of Natural Science and Engineering Research Council of Canada (NSERC) and Canada Foundation for Innovation (CFI) for my research.
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Li, M.Y. (2019). Large-Scale Epidemic Models and a Graph-Theoretic Method for Constructing Lyapunov Functions. In: Bianchi, A., Hillen, T., Lewis, M., Yi, Y. (eds) The Dynamics of Biological Systems. Mathematics of Planet Earth, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-030-22583-4_3
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