Creating models is essentially a way to understand and study the real world. Mathematical models are human conceptual artifacts. They have been developed through the centuries following an evolutionary process. Today, there are many different types and variations of mathematical models, based on divergent principles and built to address a wide variety of problems, so we can comfortably say that they compose an entire universe. Perhaps it is not an exaggeration to consider metaphorically the world of models as an ecosystem that is continuously and dynamically interacting with the real world and evolving as new problems emerge and new theories and technologies are invented, giving birth to new problems and continuing the evolution.

Considering the huge number of mathematical models that have been developed by mathematicians, as well as by scientists from a wide variety of disciplines, their systematic study is imperative. Practitioners and researchers often face new problems that can be addressed very well by existing models (or variations on those models) developed in different fields. Often, it is a waste of time and resources to try to build new models from scratch when the solution can be found in the literature. Therefore, the most important issue in the systematic study of models is the proper taxonomy and concise presentation of problems and methods based on fundamental principles.

This book is in line with the aforementioned considerations. First of all, the title, Mathematical modeling, is quite (and perhaps unnecessarily) generic, given that the book is about modeling using differential (continuous) and difference (discrete) equations. As correctly mentioned in the preface, today there are many books on differential equations and their applications in various sciences. However, according to the author, a global view is lacking, in the sense that there are no books that cover modeling with all types of differential equations. So the author sets a rather ambitious goal: to present in a book of about 250 pages a comprehensive and systematic guide to all types of differential equation models. Of course, the targeted audience should be taken into account when judging the success of such a goal. As the book aims to serve as a textbook for students and not an exhaustive handbook, the length is reasonable.

The book is beneficial in many ways, and never strays from its educational character. The language is generally simple and the material is well organized and structured with several subsections, stimulating examples, solved problems, and exercises. A reader with the appropriate calculus background will not have any problems following the flow of chapters, thus gaining thorough insight into the world of modeling. There are also discussions on advanced topics and research problems.

The material is organized in six chapters, while a seventh chapter includes hints and solutions to exercises. After a brief and informative first chapter on the history, importance, and limitations of mathematical modeling, along with its role in today’s scientific world, the book presents the various types of models. The second chapter discusses various aspects of discrete difference equations, focusing on linear and non-linear discrete models for population growth, change of temperature, bank accounts, drug delivery, economic growth, and other topics. The third chapter presents continuous models with ordinary differential equations. Examples of several real-world problems from physics, epidemics, and other areas are used in this presentation, which includes important concepts for the phenomena described by the models. The fourth chapter proceeds with partial differential equations and their applications to a large number of problems such as fluid flow, motion of strings, traffic problems, and social phenomena. The fifth chapter deals with delay differential equations, that is, equations where the derivative of the unknown function depends on the values of the function in the past. This is a subject that is both advanced and not very common in textbooks on differential equations. The models are described with numerous interesting examples from fields such as biology and medicine. The sixth chapter contains an outline of stochastic differential equations used for problems where randomness is present. The chapter includes a whole section related to the author’s own research on the mathematical modeling of cancer.

Overall, the book is recommended as an educational tool for undergraduate and graduate students with a reasonable calculus background, as well as for their mathematician teachers. It provides a broad perspective of modeling based on differential equations, and will stimulate readers to search beyond the current material for additional application fields. Its main strengths are the global view of different types of differential equations and the plethora of realistic and critical emerging problems from various scientific areas that are used as examples for the presentation and understanding of the mathematical notions.