# Mathematical Tools for Understanding Infectious Diseases Dynamics

## Synopsis

Mathematical modeling is critical to our understanding of how infectious diseases spread at the individual and population levels. This book gives readers the necessary skills to correctly formulate and analyze mathematical models in infectious disease epidemiology, and is the first treatment of the subject to integrate deterministic and stochastic models and methods.

Mathematical Tools for Understanding Infectious Disease Dynamics fully explains how to translate biological assumptions into mathematics to construct useful and consistent models, and how to use the biological interpretation and mathematical reasoning to analyze these models. It shows how to relate models to data through statistical inference, and how to gain important insights into infectious disease dynamics by translating mathematical results back to biology. This comprehensive and accessible book also features numerous detailed exercises throughout; full elaborations to all exercises are provided.

• Covers the latest research in mathematical modeling of infectious disease epidemiology

• Integrates deterministic and stochastic approaches

• Teaches skills in model construction, analysis, inference, and interpretation

• Features numerous exercises and their detailed elaborations

• Motivated by real-world applications throughout

## Excerpt

This book builds on two previous books on the same topic by the same set of authors (plus one). We feel it is important, right from the start, to make clear how the new book and the old books are related. Both the previous books appeared more than 10 years ago: Mathematical Epidemiology of Infectious Diseases: model building, analysis and interpretation, Diekmann and Heesterbeek, John Wiley & Sons, 2000; and: Stochastic Epidemic Models and their Statistical Analysis, H. Andersson and Britton Springer-Verlag, 2000. The first took a textbook approach to predominantly deterministic modeling — at least deterministic at the population level, but allowing for stochasticity at the level of individuals. The second had more a monograph-like approach to predominantly mathematical and statistical analysis of stochastic epidemic systems — concentrating on analysis, rather than model building. The present book is based on these two earlier volumes, and in fact makes both of them obsolete. It replaces them with a textbook in the spirit of ‘Diekmann and Heesterbeek,’ and the result is, in our (admittedly biased, but humble) opinion, more valuable than the sum of its parts. The new book integrates the deterministic and stochastic theory and approaches, rather than merely merging the old versions, treating both deterministic and stochastic modeling and analysis of infectious disease dynamics. New topics have been added, and for most topics already treated in one of the predecessors the text has been updated, or revised to improve exposition or integration.

We do not see our book as a mathematics monograph in the sense of instilling in the reader the beauty of the mathematical subject and prove theorems. The value of our book, in our view, is not in doing rigorous mathematics in ‘theorem-proof style,’ and also not in highlighting ‘deep problems’ from a mathematical point of view. The value of the book lies in showing how to be very precise in modeling phenomena in infectious disease dynamics, using mathematical reasoning and analysis. Mathematics is the tool, not the aim. We feel that for our aim the narrative style of doing mathematics is much more efficient in getting the message across. Our aim is to be very rigorous in the modeling. If we are being ‘missionary’ at all, it is in trying to get across what (often hidden) assumptions lie behind choices and concepts in modeling, what the consequences are of these choices, and how superficially different concepts are related. The book is about translating assumptions concerning biological (behavioral, immunological, demographical, medical) aspects into mathematics, about mathematical analysis of certain classes of equations aided by interpretation, about inference from data (measurements, observations), and finally about the drawing of conclusions where results from the mathematical and statistical analysis are translated back into biology. We try to offer insight into the relation between assumed mechanisms at the individual level and the resulting phenomena at the population level, both for ‘small’ and ‘large’ populations, and the grey area that lies in between.

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