Applied Mathematical Ecology

Volume 18 of the series Biomathematics pp 119-144

Three Basic Epidemiological Models

  • Herbert W. Hethcote

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There are three basic types of deterministic models for infectious diseases which are spread by direct person-to-person contact in a population. Here these simplest models are formulated as initial value problems for systems of ordinary differential equations and are analysed mathematically. Theorems are stated regarding the asymptotic stability regions for the equilibrium points and phase plane portraits of solution paths are presented. Parameters are estimated for various diseases and are used to compare the vaccination levels necessary for herd immunity for these diseases. Although the three models presented are simple and their mathematical analyses are elementary, these models provide notation, concepts, intuition and foundation for considering more refined models. Some possible refinements are disease-related factors such as the infectious agent, mode of transmission, latent period, infectious period, susceptibility and resistance, but also social, cultural, Ecology by providing a sound intuitive understanding and complete proofs for the three most basic epidemiological models for microparasitic infections.