Mathematical structures of epidemic systems.

*(English)*Zbl 0798.92024
Lecture Notes in Biomathematics. 97. Berlin: Springer-Verlag. xiv, 283 p. (1993).

Scientists and mathematicians have been studying models for the dynamics of diseases in biological populations for nearly a century. As a result there exists a huge number of mathematical models in the literature. The diversity of models arises because simplifying assumptions must be made in what are naturally complex biological systems in order to derive tractable models. Thus, there are models that account for various kinds of subpopulations (susceptibles, infecteds, immunes, etc.); models assuming constant total population size and models with population dynamics; models with age structure; models with periodic forcing; models with spatial heterogeneity, and so on.

The author offers some organization to this vast field by identifying classes of models with common mathematical structure and/or classes of models that are amenable to similar mathematical analysis. He first classifies models in which the “force of infection” is linear. For these models he presents general theorems concerning the existence and stability of equilibria. He describes a long list of models that have been used in the literature to study one epidemiological situation or another, all of which are of this type. A shorter chapter is devoted to nonlinear models in which a general SEIR model is analyzed. The main mathematical tool used in the analysis of these models is that of Lyapunov functions. The author also categorizes epidemiological models that are of a type that is amenable to another powerful mathematical method of analysis, namely quasimonotone systems. Both of these methods are applied to models with spatial heterogeneity in which populations are allowed to diffuse. Virtually all of the mathematical results concern threshold phenomena, that is, the study of parameter regions in which the disease free state is globally asymptotically stable and regions in which there is a positive globally stable equilibrium, implying that the disease is asymptotically established.

This readable and well thought out account of epidemiological models provides the interested reader with a very nice introduction to this vast and diverse discipline. Several appendices are given in which the mathematical theories of Lyapunov functions and quasimonotone flows (in both finite and infinite dimensions) are concisely, but rigorously, presented (without proofs). Most of the fundamental models of the subject are described and a long list of references is provided for further study.

The author offers some organization to this vast field by identifying classes of models with common mathematical structure and/or classes of models that are amenable to similar mathematical analysis. He first classifies models in which the “force of infection” is linear. For these models he presents general theorems concerning the existence and stability of equilibria. He describes a long list of models that have been used in the literature to study one epidemiological situation or another, all of which are of this type. A shorter chapter is devoted to nonlinear models in which a general SEIR model is analyzed. The main mathematical tool used in the analysis of these models is that of Lyapunov functions. The author also categorizes epidemiological models that are of a type that is amenable to another powerful mathematical method of analysis, namely quasimonotone systems. Both of these methods are applied to models with spatial heterogeneity in which populations are allowed to diffuse. Virtually all of the mathematical results concern threshold phenomena, that is, the study of parameter regions in which the disease free state is globally asymptotically stable and regions in which there is a positive globally stable equilibrium, implying that the disease is asymptotically established.

This readable and well thought out account of epidemiological models provides the interested reader with a very nice introduction to this vast and diverse discipline. Several appendices are given in which the mathematical theories of Lyapunov functions and quasimonotone flows (in both finite and infinite dimensions) are concisely, but rigorously, presented (without proofs). Most of the fundamental models of the subject are described and a long list of references is provided for further study.

Reviewer: J.M.Cushing (Tucson)

##### MSC:

92D30 | Epidemiology |

92-02 | Research exposition (monographs, survey articles) pertaining to biology |