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Deterministic threshold models in the theory of epidemics
BOOK REVIEWS
Deterministic Threshold Models in the Theory of Epidemics, by Paul Waltman.
Lecture Notes in Biomathematics 1, Springer-Verlag, Berlin, 1974. 101 pp. DM16. Since the pioneering work of Kermack and McKendrick, mathematicians have come to realise that the theory of epidemics is yet another field in which it is possible to pursue ever more esoteric mathematical studies whilst apparently being engaged in problems of practical importance. Waltman’s book is typical of this approach. It is concerned, not with epidemic theory, but with proving the existence and uniqueness, under appropriate initial conditions, of the solutions to some particular systems of coupled non-linear integral equations. Much of the work presented comes from papers written by the author and by F. Hoppensteadt between 1970 and 1974 and published in Mathematical Biosciences. The Kermack-McKendrick model for the spread of an epidemic is taken as a starting point, from which, in Chapters 2 to 6, more complicated models incorporating latent periods, variable or fixed length infectious periods, infection thresholds, temporary immunity and multiple populations are successively developed. An age-dependent model with an open population is described in Chapter 7 and Chapter 8 consists of a brief discussion of optimalvaccination policies based on the Kermack-McKendrick model. The latter problem is approximated in terms of a finite horizon discrete step dynamic programming recursion and some numerical solutions are presented. As the models become more complicated, the mathematical discussion rapidly reduces to the questions of existence and uniqueness of solutions to the equations; the qualitative insights of the Kermack-McKendrick model are lost and numerical solution becomes the principal recourse. The practical justification advanced for such complication is that the extra features represent observed phenomena-as in fact they do. No attempt, however, is made to point out deficiencies in the real life predictions of the simpler models which make the more complicated models necessary, and one is left with the feeling that the real reason for the developments is that they lead to more sophisticated mathematical problems. Indeed, the only actual epidemic data provided are those used by Kermack and McKendrick, which show a reasonable fit with their original model. The book is printed in the usual Lecture Notes format, by a photographic process direct from the typescript, and shows some of the other marks of a hurried production; such as, for instance, an inaccurate reference for Kermack and McKendrick on page 7, some confusing phrases at the top of page 12 and an 151 In?. J. Bio-Medical Computing Printed in Great Britain
(8) (1977)-o
Applied
Science Publishers
Ltd, England,
1977
152
BOOKREVIEWS
incorrect interchange of limits of integration on page 14. However, the material is, on the whole, clearly explained and easy to read, and gives a sensible presentation of the actual-as opposed to the apparent-topic. It is, nonetheless, a pity that the connection with practical epidemic theory is so tenuous. It would be nice to see rather more Bio in Biomathematics. ANDREWD.
Biomedical Engineering
BARBOUR
Principles, by D. 0. Cooney. Marcel Dekker Inc., New
York and Basel, 1976. In the words of the author’s preface, whether Biomedical Engineering is an entirely ‘new’ discipline or not is not clear; what is clear-and this is quickly established by just flicking through the pages of this book-is that this is a most welcome amalgamation of the life sciences and engineering. The human body is, after all, an assembly of tubes, capillaries, membranes and pumps in which liquids, gases and solids flow, evaporate, react, clot, dialyse, filter, etc. At first sight one might conclude that a medical doctor well versed in traditional physics and chemistry is admirably equipped for understanding human mechanics. I, in common with the author of this book, would dispute this. Clearly, the human body is an intricate and diverse plant, far more complex in function and interaction than many processes found in chemical factories. If, say, a petroleum refining plant breaks down, would anyone expect a classically trained medic to put it right? And yet, this is precisely what one does when calling the doctor for a tummy ache. The point is that an introduction of the human body, seen as an engineering plant, to students in physiology, anatomy and medicine, and their consequent appreciation of some rudiments of engineering, can only be to the good. This book is not only aimed at an audience of medics, but also at engineers who are likely to work in collaboration with medics. It effectively starts with a potted description of the human body (anatomy, composition, mass balances, etc.). It continues with clear statements of the physicochemical properties of blood and the dynamics of the circulatory system leading to a consideration of heat flows. An interesting level of engineering technique is reached in Chapter 6, where simple modelling techniques are used to characterise, among other topics, chemical kinetic behaviour and blood/tissue interactions. After an analysis of transport mechanisms through cell membranes, the author enters yet another important area of potential contribution from the biomedical engineer: artificial aids such as kidney devices and heart-lung devices.
Deterministic Threshold Models in the Theory of Epidemics, by Paul Waltman.
Lecture Notes in Biomathematics 1, Springer-Verlag, Berlin, 1974. 101 pp. DM16. Since the pioneering work of Kermack and McKendrick, mathematicians have come to realise that the theory of epidemics is yet another field in which it is possible to pursue ever more esoteric mathematical studies whilst apparently being engaged in problems of practical importance. Waltman’s book is typical of this approach. It is concerned, not with epidemic theory, but with proving the existence and uniqueness, under appropriate initial conditions, of the solutions to some particular systems of coupled non-linear integral equations. Much of the work presented comes from papers written by the author and by F. Hoppensteadt between 1970 and 1974 and published in Mathematical Biosciences. The Kermack-McKendrick model for the spread of an epidemic is taken as a starting point, from which, in Chapters 2 to 6, more complicated models incorporating latent periods, variable or fixed length infectious periods, infection thresholds, temporary immunity and multiple populations are successively developed. An age-dependent model with an open population is described in Chapter 7 and Chapter 8 consists of a brief discussion of optimalvaccination policies based on the Kermack-McKendrick model. The latter problem is approximated in terms of a finite horizon discrete step dynamic programming recursion and some numerical solutions are presented. As the models become more complicated, the mathematical discussion rapidly reduces to the questions of existence and uniqueness of solutions to the equations; the qualitative insights of the Kermack-McKendrick model are lost and numerical solution becomes the principal recourse. The practical justification advanced for such complication is that the extra features represent observed phenomena-as in fact they do. No attempt, however, is made to point out deficiencies in the real life predictions of the simpler models which make the more complicated models necessary, and one is left with the feeling that the real reason for the developments is that they lead to more sophisticated mathematical problems. Indeed, the only actual epidemic data provided are those used by Kermack and McKendrick, which show a reasonable fit with their original model. The book is printed in the usual Lecture Notes format, by a photographic process direct from the typescript, and shows some of the other marks of a hurried production; such as, for instance, an inaccurate reference for Kermack and McKendrick on page 7, some confusing phrases at the top of page 12 and an 151 In?. J. Bio-Medical Computing Printed in Great Britain
(8) (1977)-o
Applied
Science Publishers
Ltd, England,
1977
152
BOOKREVIEWS
incorrect interchange of limits of integration on page 14. However, the material is, on the whole, clearly explained and easy to read, and gives a sensible presentation of the actual-as opposed to the apparent-topic. It is, nonetheless, a pity that the connection with practical epidemic theory is so tenuous. It would be nice to see rather more Bio in Biomathematics. ANDREWD.
Biomedical Engineering
BARBOUR
Principles, by D. 0. Cooney. Marcel Dekker Inc., New
York and Basel, 1976. In the words of the author’s preface, whether Biomedical Engineering is an entirely ‘new’ discipline or not is not clear; what is clear-and this is quickly established by just flicking through the pages of this book-is that this is a most welcome amalgamation of the life sciences and engineering. The human body is, after all, an assembly of tubes, capillaries, membranes and pumps in which liquids, gases and solids flow, evaporate, react, clot, dialyse, filter, etc. At first sight one might conclude that a medical doctor well versed in traditional physics and chemistry is admirably equipped for understanding human mechanics. I, in common with the author of this book, would dispute this. Clearly, the human body is an intricate and diverse plant, far more complex in function and interaction than many processes found in chemical factories. If, say, a petroleum refining plant breaks down, would anyone expect a classically trained medic to put it right? And yet, this is precisely what one does when calling the doctor for a tummy ache. The point is that an introduction of the human body, seen as an engineering plant, to students in physiology, anatomy and medicine, and their consequent appreciation of some rudiments of engineering, can only be to the good. This book is not only aimed at an audience of medics, but also at engineers who are likely to work in collaboration with medics. It effectively starts with a potted description of the human body (anatomy, composition, mass balances, etc.). It continues with clear statements of the physicochemical properties of blood and the dynamics of the circulatory system leading to a consideration of heat flows. An interesting level of engineering technique is reached in Chapter 6, where simple modelling techniques are used to characterise, among other topics, chemical kinetic behaviour and blood/tissue interactions. After an analysis of transport mechanisms through cell membranes, the author enters yet another important area of potential contribution from the biomedical engineer: artificial aids such as kidney devices and heart-lung devices.