# Mathematics of Epidemics on Networks

###### István Z. Kiss, Joel C. Miller, and Péter L. Simon
Publisher:
Springer
Publication Date:
2017
Number of Pages:
413
Format:
Hardcover
Series:
Interdisciplinary Applied Mathematics 46
Price:
79.99
ISBN:
9783319508047
Category:
Monograph
[Reviewed by
William J. Satzer
, on
09/20/2017
]

Mathematical modeling of infectious disease goes back at least a hundred years. Variations of the pioneering work of Kermack and McKendrick from 1927 are still in use today. Their models are often identified by the acronyms SIS or SIR. SIS models are designed for environments where susceptible individuals become infected at certain rates, recover, and then become susceptible again. In SIR models, those infected recover and become immune to further infection. Such models represent the dynamics of infection and recovery via systems of ordinary differential equations. Over time the models have become more elaborate because they often incorporate separate equations for subgroups of populations. (This occurs, for example, in models for sexually transmitted diseases such as HIV.)

The current book describes an alternative approach that combines aspects of the older models with a network structure. The spread of infectious disease is interpreted as occurring between nodes on a network. Individuals are represented by nodes and the contact pattern among them is encoded via the edges of the network. The book’s main topic is the investigation of continuous-time stochastic epidemic models on networks. While the simplest approach to this is stochastic simulation, the authors choose to focus on mathematical modeling and analysis of stochastic processes. But they don’t ignore simulation.

For most of the book the networks are assumed to be static and unweighted with an SIS or SIR disease spreading along the edges. Typically transmission from an infected node to a susceptible node occurs across an edge as a Poisson process with a certain transmission rate, and an infected node recovers as a Poisson process with a specified recovery rate.

The authors present several models throughout the book, but three kinds receive primary treatment. The first is a continuous-time Markov chain model on a network with $N$ nodes. It has either $2^N$ or $3^N$ states (for SIS or SIR models respectively) and the system that defines the model has a correspondingly large number of equations. By restricting the network to a particular special structure the number of equations can be reduced considerably without making any approximations. The authors call this a top-down model.

For many networks such a reduction is not possible or desirable, and approximate models (either individual-based or population-based) use differential equations formulated in terms of the probability of individual nodes having a given status (S, I or R). These are called bottom-up models.

The third main alternative is to write the differential equations in terms of population-level quantities such as the expected number of susceptible, infected or recovered nodes and the expected number of edges connecting different types of nodes. Such models are called mean-field models.

The authors devote several of the later chapters to special models. For example, they discuss the application of percolation models to SIR epidemics. Another chapter considers how dynamic and adaptive models can be used to analyze the transient nature of more realistic networks that exhibit oscillations and bistability. An appendix focuses on techniques for efficient stochastic simulation and provides associated pseudocode.

This is one of the first books to appear on modeling epidemics on networks. A variety of survey articles have appeared on the subject (all listed in the comprehensive bibliography), but this appears to be the first complete treatment.

This is a comprehensive and well-written text aimed at students with a serious interest in mathematical epidemiology. It is most appropriate for strong advanced undergraduates or graduate students with some background in differential equations, dynamical systems, probability and stochastic processes.

Bill Satzer ([email protected]) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.