Differential equation models for infectious diseases: Mathematical modeling, qualitative analysis, numerical methods and applications

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Abstract
Mathematical epidemiology has a long history of origin and development. In particular, mathematical modeling and analysis of infectious diseases has become a fundamental and indispensable approach to discovering the characteristics and mechanisms of the transmission dynamics of epidemics, thereby effectively predicting possible scenarios in reality, as well as controlling and preventing diseases. In recent decades, differential equations have been widely used to model many important infectious diseases. The study of these differential equation models is very useful in both theory and practice, especially in proposing appropriate strategies for disease control and prevention. This is of great benefit to public health and health care. In this survey article, we review many recent developments and real-world applications of deterministic ordinary and partial differential equations (ODEs and PDEs) in modeling major infectious diseases, particularly focusing on the following aspects: mathematical modeling, qualitative analysis, numerical methods, and real-world applications. We also present and discuss some open problems and future directions that research in differential equation models for infectious diseases can take. This article provides a comprehensive introduction to epidemic modeling and insights into nonstandard finite difference methods.
Funding Information
  • Vietnam Institute for Advanced Study in Mathematics
  • Bergische Universität Wuppertal

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