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Mathematical Modelling in Health, Social and Applied Sciences

Hemen Dutta, ed.
Publication Date: 
Number of Pages: 
[Reviewed by
Johnna Barnaby
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This book is a collection of papers each of which presents a different mathematical model. They cover various topics in the biological and social sciences. The book consists of 10 chapters, and each chapter presents a different model.  Chapters are not dependent on one another.
The author of each paper does an excellent job of introducing and motivating the problem. You do not need to be a biologist, ecologist, or economist to follow the model derivation. The types of mathematics presented in each model varies from compartment model driven ordinary differential equations (ODEs), to stochastic ODEs, to partial differential equations. Each author presents all the details of model derivation, any simplifications, and parameter estimation, details that are usually reserved for the supplemental materials a paper. It makes it very easy to completely follow the author’s thought process and fully understand the model.
The first five chapters of the book present models of various health topics including virus spread in the body, influenza, Dengue fever, HPV, and prostate cancer. Many of these models use compartmental ordinary differential equation models to capture the dynamics. For each of these ODE models, the authors give in-depth details of model development from biological assumptions. They then go on to analyze, parameterize, make conclusions about each model.  Other models employ Markov chain models to include stochasticity, wavelet approaches to analyze data, and delayed differential equations to capture the time it takes for an uninfected cell to become infected.
In the last three chapter, all three models use fractional order calculus (FOC) to describe the dynamics. The authors describe how FOC can be applied to mathematical models. In Chapter 8, the authors apply FOC to a reaction diffusion equation describing a two and three population predator-prey model to look for diffusion-driven instability in these populations. They implore numerical methods to explore pattern formation. In Chapter 9 the authors apply FOC to an epidemiology model of computer virus spread. In Chapter 10 the authors apply FOC to a three species Predator-Prey model. They give an in depth analytical analysis of the steady states, as well as a numerical analysis of the population dynamics. 
This book is written for graduate students and researchers with a basic knowledge of mathematical modeling. Advanced undergraduate student working on research in the field could work through the chapters with the help of an advisor. This book is an excellent resource for anyone interested in mathematical modeling, especially in the biological and social sciences.


Johnna Barnaby is an assistant professor at Shippensburg University. Her research interests include mathematical models of cancer progression and response to treatment.