
Author(s)  Capasso, V. 

Title  Mathematical Structures of Epidemic Systems 
Publisher  SpringerVerlag 
Year of publication  1993 
Reviewed by  Hans Heesterbeek 
Mathematical approaches to the spread in time and space of infectious diseases (of humans, animals and plants) have received increasingly more attention in the last ten years. Initially this renewed interest was due to the onset of the HIVepidemic but in recent years the application of mathematics to epidemiology has started addressing difficult questions, both biologically and mathematically, for a range of infectious diseases as broad as never before. Not only has the range of applications broadened, also the diversity in the type of mathematics used has increased, certainly with the current trends to develop models for diseasehost systems where, for both the infectious agent and the host (and maybe also for the environment), relevant heterogeneity structure is taken into account. The main aim of mathematical epidemiology remains the same as it was at the very beginning of the subject in 1760, when Bernoulli wrote a paper on smallpox, that is to evaluate the effects on the infectious agent and the host population of various control strategies. These effects can, certainly where human diseases are concerned, not be evaluated otherwise.
With the last 'general' book devoted to mathematical epidemiology, Norman Bailey's The Mathematical Theory of Infectious Diseases and its Applications, dating from 1975, and with the mathematical techniques and biological problems becoming more and more involved, there is a definite need for modern books describing large parts of the field. Anderson and May's, 1991, Infectious Diseases of Humans, Dynamics and Control, was the first to fill part of the gap, mainly devoted to the direct use of models and the applied side of the field. The two books under review aim at the deterministic theoretical side of the field; a book devoted to recent stochastic approaches in mathematical epidemiology is still lacking.
The book by Stavros Busenberg, whose death in early 1993 was a great loss to biomathematics, and Ken Cooke, specializes in the mathematical modelling and subsequent analyses of pathogenhost systems where the infection route from mother to unborn child constitutes an important factor in the transmission process. The title of the book however, is too restrictive by only emphasizing vertical transmission, as the contents are much broader. The book is structured according to type of model, with chapters on differential equation models, difference equation models, delay differential equation models and partial differential equation models (agestructured populations). Of the authors' three main objectives, 1) how to formulate models that capture the essentials of given epidemiological situations, 2) how to mathematically analyse these models, and 3) how these models then yield epidemiological inferences, the last objective is not nearly as well represented as the first two. However, the text is interspersed with specific disease examples. The book is well written and could function as a companion volume to the Anderson & May book mentioned above who concentrate on the third objective. Actual proofs are kept to a minimum (certainly in the last chapter on agedependent models, most results are only explained in words and the reader is referred to the literature for details), many relevant mathematical concepts are introduced at the end of chapters and much of the required biological background is introduced as well, to make the book relatively selfcontained.
Vincenzo Capasso's book also deals with the deterministic mathematical approach to infectious disease models, but his aims, exposition and mathematical analysis are very different. Capasso's interest lies not in tailoring models to reflect given epidemiological situations (a union), like the book discussed above, but in finding common structures in a wide range of models (an intersection). The book has four main chapters: linear models (read: bilinear), strongly nonlinear models, quasimonotone systems, and spatial heterogeneity (and an additional review chapter on recent results of agedependent models, which of course greatly overlaps with Busenberg and Cooke's review; as well as a few pages on optimization problems). There are two extended appendices (more than 50 pages in all) about the relevant theory of dynamical systems in finite and in infinite dimensional spaces.
The main argument of the fundamental chapter on bilinear models concentrates on Volterralike systems of the form
dz/dt = diag(z)(e + Az) + b(z)
where b(z) = c + Bz, z, e, c E Rn+ and A, B: Rn+ > Rn+, and Capasso gives a set of criteria that ensure global Lyapunov stability of a steady state (given existence). This an interesting theory about a model which covers a substantial part of models in recent use by modellers, including a large part of the ordinary differential equation models discussed in Busenberg and Cooke (who, in turn, devote a section to Capasso's theory). However, the criteria, based in part on skewsymmetrizability of a certain matrix defined in terms of A, B and the steady state, might for many systems prove very tedious to check. Strangely, an easier to check necessary and sufficient condition for one set of Capasso's conditions, used in Busenberg and Cooke, is not used or mentioned by Capasso in his many examples, even though the appropriate reference is given in the references. In the chapter on highly nonlinear systems, Capasso discusses many variants of the bilinear transmission term that have been studied in the recent literature and the effects these variants have on existence and stability of steady states. The next chapter discusses global stability results for quasimonotone systems. In these three chapters, many illustrations of the theory are given, with varying degree of biological information. In the chapters that follow (spatial and agestructure) however, biological examples are lacking and the text is purely mathematical.
Summarizing, Busenberg and Cooke's book is most suitable as an introductory textbook for mathematically inclined theoretical biologists or biologically inclined applied mathematicians who seek to develop a sense for the results and possibilities of the field. Capasso's book, in contrast, is most suitable for applied mathematicians who already work on epidemiological problems, to gain insight from seeing models put in a larger structure, and to see valuable methods and techniques from mathematics applied to these more general models.
Both books are valuable additions to the literature and together with the book by Anderson and May they provide a badly needed up to date overview of a large part of, both theoretical and applied, deterministic modelling of infectious diseases.