Mathematics is a useful tool in studying the growth of infections in a population, such as what occurs in epidemics. A simple model is given by a first-order differential equation, the logistic equation, $\frac{dx}{dy}=\beta x(1-x)$ which is discussed in almost any textbook on differential equations. It can be found, for instance, in Chapter 2 of Boyce and DiPrima’s book Elementary Differential Equations and Boundary Value Problems. (See MR0179403 for a short review of the first edition, from 1965.) This is a rudimentary model, but mathematicians have built on it to create more realistic, hence more useful models. There is an informative explanation of how to use a mathematical model for epidemics, including the importance of determining the reproductive number $R_0$ of an infectious disease, in this video made by Tom Britton, a professor of mathematical statistics at Stockholm University. Britton is one of the authors of

MR3015083

Diekmann, Odo(NL-UTRE-NDM); Heesterbeek, Hans(NL-UTRE-NDM); Britton, Tom(S-STOC-NDM)

Mathematical tools for understanding infectious disease dynamics.

Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2013. xiv+502 pp. ISBN: 978-0-691-15539-5

92-01 (62P10 92D30)

the review of which is copied below.

If you are interested in exploring some of the mathematics used in modeling epidemics, you can search MathSciNet using the MSC 92D30, which is the five-digit class for epidemiology, in particular, in the context of population dynamics. Besides the review of Britton’s book, some other reviews are also copied below, to help give a sense of the mathematics used in epidemiology.

MR3015083

Diekmann, Odo(NL-UTRE-NDM); Heesterbeek, Hans(NL-UTRE-NDM); Britton, Tom(S-STOC-NDM)

Mathematical tools for understanding infectious disease dynamics.

Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2013. xiv+502 pp. ISBN: 978-0-691-15539-5

92-01 (62P10 92D30)

This textbook emerged from two other books, both published over a decade ago by (roughly) the same set of authors. The first of these books, written by O. Diekmann and J. A. P. Heesterbeek, was called Mathematical epidemiology of infectious diseases [Wiley Ser. Math. Comput. Biol., Wiley, Chichester, 2000; MR1882991] and focused on deterministic epidemiological systems. The second book, written by H. Andersson and T. Britton, was called Stochastic epidemic models and their statistical analysis [Lecture Notes in Statist., 151, Springer, New York, 2000; MR1784822] and focused on stochastic epidemiological systems. The current book subsumes both of these earlier publications, integrating and expanding on their content. While the earlier books had divergent styles (one was a textbook, one was a monograph), the current book is a textbook, replete with numerous exercises and solutions.

In the preface to the text the authors describe what they feel are the merits of the book, and how these merits connect to its mathematical and biological content: “The value of our book $\dots$ is not in doing rigorous mathematics in `theorem-proof style’ $\dots$. 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.” For the would-be reader of this text who happens to be a mathematician, this transparency provides a useful reminder of the fact that, although this book is a math book, mathematics per se plays only a secondary role. Indeed, one of the standing challenges in any applied mathematics is to subordinate the beauty of the abstract to the pursuit of the practical, and this subordination is generally neither easy nor straightforward. It is important for the mathematical reader to understand that even though the authors characterize the mathematics in this text as a mere “tool”, this status in no sense renders the mathematics trivial, or the work of the mathematician superfluous. The mathematics in this book is interesting, complex, and subtle, and its productive application to issues of disease transmission is a difficult art. This textbook does an outstanding job of illustrating both the nuance and the power of this art.

The book is divided into three main sections. The first is entitled “The bare bones: Basic issues in the simplest context” and illustrates key concepts without the complications entailed by dividing populations into classes. The authors start out by giving a descriptive definition of the basic reproductive number $R_0$, a quantity they characterize in the second part of the book as “arguably the most important quantity in infectious disease epidemiology.” They illustrate the significance of $R_0$ in several settings, including those of both deterministic and stochastic disease dynamics. They briefly discuss the issue of population level heterogeneity, though they defer a serious discussion of this topic to the second part of the book. The authors also introduce the ideas of inference, and specifically maximum likelihood estimators, though again, a detailed discussion of this matter is deferred until later.

The heart of this book deals with structured populations, the subject that comprises the second main section of the text. The “structure” can be of various sorts, including age, location, infectivity, or other characteristics. In order to keep track of these characteristics, the authors introduce the concept of $i$-states and $p$-states, the former consisting of the current values of individual characteristics, the latter the distribution of these characteristics over the entire population. They note that changes on the individual level induce changes on the population level, and that “the equation for $p$-state change is obtained basically by bookkeeping, once the $i$-state dynamics have been described.” Within this framework, the authors define the basic reproductive number $R_0$ as the largest eigenvalue of a “next-generation matrix”, i.e. a matrix that specifies the pattern of infectivity across generations within a heterogenous population. Clearly the nature of this pattern depends on the specific sorts of heterogeneity under consideration: the authors work out details for age and spatially structured populations, in addition to other more involved notions of heterogeneity.

The third section of the text focuses on inference and, as an amalgam of concrete case studies, can be considered in some sense as the most “practical” part of the book. The authors discuss ways of estimating $R_0$ from mechanistic models and give a case study involving hospital infections that illustrates an efficient and effective use of data. They conclude this section with a brief guide to computational statistics. The entire third section is less than one third the length of the first two sections and has the feeling of a coda.

One of the outstanding features of this text is the way it challenges readers to work out solutions on their own. Interleaved within the body of the text are numerous exercises, many of which ask the reader to prove core concepts, or to explore in depth implications of an idea or a definition presented in the text. While not every reader will want to invest the time to work through all the exercises, the authors comment that “the reward is enormous. In literally working through this book the reader acquires modeling skills that are also valuable outside of epidemiology, certainly within population dynamics, but even beyond that.” For the time-limited reader, or perhaps the reader who already has these skills, detailed answers to every question are provided at the back of the book.

This textbook is an excellent addition to a growing body of literature on infectious disease modeling. Other texts that touch on similar themes include the books by N. T. J. Bailey [The mathematical theory of infectious diseases and its applications, second edition, Hafner Press New York, 1975; MR0452809], R. M. Anderson and R. M. May [Infectious diseases of humans: dynamics and control, Oxford Univ. Press, Oxford, 1991], M. J. Keeling and P. Rohani [Modeling infectious diseases in humans and animals, Princeton Univ. Press, Princeton, NJ, 2008; MR2354763], and E. Vynnycky and R. G. White [An introduction to infectious disease modeling, Oxford Univ. Press, Oxford, 2010], while the book edited by F. Brauer, P. van den Driessche and J. Wu [Mathematical epidemiology, Lecture Notes in Math., 1945, Springer, Berlin, 2008; MR2452129] is a collection of individual articles which, collectively, cover a lot of related material. The overtly pedagogical features of this text make it an outstanding choice for someone trying to learn the basic tools of the trade. The mathematician who makes a serious study of this text will be in an excellent position to work fruitfully with biologists or epidemiologists on either theoretical or data-driven problems of disease transmission.

Reviewed by Carl A. Toews

MR1814049

Hethcote, Herbert W.(1-IA)

The mathematics of infectious diseases. (English summary)

SIAM Rev. 42 (2000), no. 4, 599–653.

92D30 (34C60 35B32 35F25 35Q80)

This paper is partly a survey of the mathematics of infectious diseases but with some new twists. The survey part consists of the first two sections, describing the classical epidemic and endemic models and their applications. A new twist, introduced in section three, is that to the usual compartments $S,E,I,R$, of susceptible, exposed, infectious, and removed, is added the class $M$ of infants with maternal antibodies giving passive temporary immunity. So the transfer diagram is $M\to S\to E\to I\to R$. The $MSEIR$ model is considered with standard exponentially distributed waiting times, mass-action contact, but non-constant population size. On dividing through by total population size, equations are obtained for the proportions $m,e,i,r$. The basic reproductive number $R_0$ is determined and it is shown that the usual threshold condition applies: if $R_0\le 1$ the disease-free equilibrium is globally attracting; if $R_0>1$ then the unique endemic equilibrium is locally attracting. An open question is whether this equilibrium attracts all solutions with $e(0)+i(0)>0$ but persistence theory is used to establish that the limit infimum of $i(t)$ exceeds some positive, initial-condition-independent threshold. Consequently, the disease persists when $R_0>1$.

The age-dependent $MSEIR$ model is also treated with separable contact rate, its basic reproductive number is described, and a formal global stability argument, using the Lyapunov-LaSalle theorem, is given for the disease-free equilibrium when $R_0\le 1$, and an expression is obtained for the endemic equilibrium. Expressions are given for the average age of infection and elaborated for the special cases of exponentially distributed death rate, relevant to developing countries, and where all individuals live to age $L$ and die, relevant for developed countries.

Finally, the $SEIR$ model with age groups is considered. A system of ODEs is obtained on partitioning the age axis into a finite number of intervals, assuming death rates, fertility rates, and all epidemiological rates are constant on sub-intervals. Again, the focus is on the basic reproductive number, stability of disease-free equilibrium when $R_0\le1$, and endemic equilibrium when $R_0>1$. Persistence theory is used to establish persistence of the disease in the population when $R_0>1$. Expressions for the average age of infection are obtained and the model is applied to measles in Niger.

Reviewed by Hal Leslie Smith

MR2821582

Gray, A.(4-STRA-MS); Greenhalgh, D.(4-STRA-MS); Hu, L.(PRC-DHU-AM); Mao, X.(4-STRA-MS); Pan, J.(4-STRA-MS)

A stochastic differential equation SIS epidemic model. (English summary)

SIAM J. Appl. Math. 71 (2011), no. 3, 876–902.

92D30 (34C60 34D05 34F05 60H10 92D25)

The authors study an extension of the classical susceptible-infected-susceptible (SIS) epidemic model from the deterministic framework to a stochastic one. They formulate the model as a stochastic differential equation (SDE) in the number of infectious $I(t)$. They prove that this SDE has a unique global positive solution. As a key parameter, a stochastic variant $R_0^S$ of the classical (deterministic) basis reproduction number $R_0=R_0^D$ is used to establish conditions for the persistence of $I(t)$ and for the extinction of the disease. In the paper it is shown that, for this model, the stochastic basic reproduction number $R_0^S$ is less than $R_0^D$. In the case of persistence, the authors show the existence of a stationary distribution, deriving expressions for its mean and variance. The results are illustrated using examples based on real-life diseases.

Reviewed by Andreas Günter Weber

MR1954511

Ruan, Shigui(1-MIAM); Wang, Wendi(PRC-SWNU)

Dynamical behavior of an epidemic model with a nonlinear incidence rate. (English summary)

J. Differential Equations 188 (2003), no. 1, 135–163.

92D30 (34C23 34C60 37N25)

An ordinary differential equation SIRS epidemic model with susceptible, infective and removed individuals is analyzed in detail. The model assumes constant recruitment into the susceptible class and saturated incidence $kI^2S/(1+\alpha I^2)$ in place of the more usual mass action or standard incidence. With this saturated incidence, the basic reproduction number threshold condition does not occur. Global and bifurcation analyses show that either the disease dies out or that there is a region such that the disease dies out for initial values outside the region but persists for initial values inside the region. In this latter case, a rich variety of dynamics is possible and the model may undergo a Bogdanov-Takens bifurcation. Depending on parameter values, there may exist none, one or two limit cycles. An interesting result found is that the inhibition factor $\alpha>0$ reduces the possibility of disease persistence.

Reviewed by Pauline van den Driessche

MR1857535

Li, Michael Y.(3-AB); Smith, Hal L.(1-AZS); Wang, Liancheng(1-GSO)

Global dynamics an SEIR epidemic model with vertical transmission. (English summary)

SIAM J. Appl. Math. 62 (2001), no. 1, 58–69.

92D30 (34C60 34D20)

This is an interesting paper that gives a complete characterization of the global dynamical structure for an SEIR type of epidemic model with vertical transmission. To summarize, the paper shows that the dynamics of the disease transmission are governed by a reproductive number $R_0(p,q)$, where $p$ and $q$ are fractions of infected newborns from the exposed and infectious class. If $R_0(p,q)\le 1$, then the disease-free equilibrium is globally stable; if $R_0(p,q)$ exceeds $1$, then there exists a unique endemic equilibrium that attracts all positive solutions.

The above type of dynamical behavior has been known for the SIS (susceptible-infective-susceptible) models which generate monotone flows and the SIR (susceptible-infective-removable) models where the models can be reduced to a two-dimensional system and hence the qualitative analysis can be efficiently carried out. The SEIR type of model considers the possible time-delayed disease transmission that can be caused by a vertical disease transmission, etc. Therefore an additional class, the hosts infected but not yet infectious, is added to the model. An SEIR model can be reduced to a three-dimensional system of differential equations under the assumption of constant total population. However, the global analysis for a three-dimensional system is far from trivial. The main approach of the paper is to exclude the existence of a nontrivial periodic solution by showing that the second compound equation with respect to any solution of the original system is uniformly asymptotically stable, a technique developed earlier by M. Y. Li and J. S. Muldowney [SIAM J. Math. Anal. 27 (1996), no. 4, 1070–1083; MR1393426]. With the further use of the local $C^1$ closing lemma due to Pugh the authors are able to conclude that the $\omega$-limit set corresponding to each nonnegative solution must be an equilibrium. The technique introduced in this paper is quite efficient to study the SEIR type of equations and has the potential to be applied to other differential equations.

Reviewed by Wenzhang Huang

MR1770939

van den Driessche, P.(3-VCTR-MS); Watmough, James(3-VCTR-MS)

A simple SIS epidemic model with a backward bifurcation. (English summary)

J. Math. Biol. 40 (2000), no. 6, 525–540.

92D30 (34K18 34K60)

The authors consider an SIS epidemic model with a non-constant contact rate. The model is based on a Volterra integral equation, which includes ordinary differential equations and delay differential equations as special cases. The main focus of the paper is to investigate dependence of the number of equilibria and their stability on the reproductive number. The reproduction number is the expected number of new infections produced by a single infective individual introduced into a disease-free population. A classical disease transmission model with constant contact rate has a unique stable equilibrium. The authors show that if the contact rate is not constant, but rather a function of the fraction of infective individuals, then an SIS epidemic model may have multiple stable equilibria, a backward bifurcation and hysteresis. The authors investigate both local and global stability of equilibria.

Reviewed by Curtis C. Travis

MR2086967

Brauer, Fred(3-BC)

Backward bifurcations in simple vaccination models. (English summary)

J. Math. Anal. Appl. 298 (2004), no. 2, 418–431.

92D30 (34C23 37N25)

This paper discusses certain simple models of disease transmission with vaccination. Conditions for the existence of multiple endemic equilibria and for backward bifurcations are given. The qualitative analysis of the models relies on the center manifold theorem and examination of normal forms. In particular, it is pointed out that models with vaccination may exhibit backward bifurcation, making the behavior of the model more complicated than the corresponding model without vaccination. It has been argued in some previous works that a partially effective vaccination program, applied to only part of the population at risk, may increase the severity of outbreaks of such diseases as HIV/AIDS. The author found conditions under which the system has backward bifurcation at the point where the basic reproductive number equals 1. It is also shown that the endemic equilibrium of the system is locally asymptotically stable if and only if it corresponds to a point on the bifurcation curve where the slope is positive. By extending the results to models in which the parameters may depend on the level of infection, the paper is able to arrive at some biologically useful conclusions, one of which is that, although the introduction of a vaccination policy may lead to backward bifurcations, it always decreases infective population size. The unforeseen backward bifurcation which may arise as a danger of a vaccination policy can be counteracted by a larger vaccination fraction in order to control a disease. Thus, it is possible to control the disease if a vaccination program can be developed which is completely effective, and decreases the contact rate, so that backward bifurcation does not arise. It is also argued that even a vaccination program which is not fully effective may be a useful approach in controlling infections.

Reviewed by Yongwimon Lenbury

MR0840096 (87k:92016) Reviewed

Ball, Frank(4-NOTT)

A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models.

Adv. in Appl. Probab. 18 (1986), no. 2, 289–310.

92A15

The present paper provides a unified probabilistic approach to the distribution of total size and total area under the trajectory of infectives for a general stochastic epidemic having any specified distribution of the infectious period. Generalization of the author’s results to epidemics spreading amongst a heterogeneous population is straightforward.

Bailey’s (1975) general stochastic epidemic uses a Markov model for the spread of infection in a closed homogeneously mixing population. Initially there are $a$ infectives and $N$ susceptibles. Let $X(t)$, $Y(t)$ and $Z(t)$ denote the numbers of susceptible, infective and removed individuals at time $t\ge0$. The epidemic is completely specified by the process $\{X(t),Y(t),t\ge0\}$. Transitions cease as soon as the number of infectives becomes 0. Let $N^\ast=N-X(\infty)$ be the total size of the epidemic, i.e., the number of initially susceptible individuals who ultimately contract the disease, and let $P_w=P(N^\ast=w)$ for $w =0,1,2,\cdots,N$ be the distribution of $N^\ast$. Whittle (1955) has studied the distribution $\{P_w\}$.

The total area under the trajectory of infectives is the statistic $T_A=\int^\infty_0Y(t)\,dt$ and has received considerable attention in the literature. In other words, $T_A$ is the total person time units of infection during the course of the epidemic. The distribution of $T_A$ was studied by Gani and Jerwood (1972) and is an important component of the cost of an epidemic. $T_A$ is also closely related to the probabilistic structure of the ultimate spread of an epidemic.

Three assumptions are made in a general stochastic epidemic that are unlikely even approximately to be met in many real-life epidemics: (1) there is no latent period, i.e., newly infected susceptibles are immediately able to infect further susceptibles, (2) the infectious period is exponentially distributed and (3) the population is homogeneously mixing. The present author shows that the distribution of $(N^\ast,T_A)$ is invariant relative to quite general assumptions regarding a latent period. The author further presents a unified approach to the distributions of $N^\ast$ and $T_A$ for a model in which the infectious period can have any given distribution. This result is generalized to multipopulation epidemic models which incorporate homogeneous mixing. The probabilistic structure of the epidemic is directly exploited in the present paper and this, in turn, enhances our insight into the mechanisms underlying the spread of an epidemic.

A fundamental result in the paper is the proof of Wald’s identity for epidemics: Theorem: For all $t\ge0$, $$E[\exp(-tT_a)/\varphi(t) ^{N^\ast+a}]=1,$$ where $\varphi(t)=E[\exp(-tT_I)]$ is the moment generating function of $T_I$.

By differentiating this result with respect to $t$, the author obtains the result $E(T_A)=(E[N^\ast]+a)\cdot E(T_I)$, which generalizes Downton’s (1972) result for the general stochastic epidemic.

These results are generalized to multipopulation epidemic models. Suppose that a population of individuals is split into $m$ groups. Initially there are $N_i$ susceptibles and $a_i$ infectives in group $i$ for $i=1,2,\cdots,m$. If $i\ge1$ and $j\le m$, it is assumed that a given individual in group $j$ makes adequate contact with a given individual in group $i$ at the points of a homogeneous Poisson process of rate $\beta_{ij}$. Note that $\beta_{ij}

e \beta_{ji}$. The contact processes between different pairs of individuals are mutually independent. By choosing particular forms for the infection rates $\beta_{ij}$ and infectious periods, many of the epidemic models in the literature are obtained. If the infectious periods are assumed to be exponentially distributed, one obtains Watson’s (1972) model. If it is assumed that $\beta _{ij}=\beta_i$ for $i=1,2,\cdots,m$ and $j=1,2,\cdots,m$, one obtains the Gart epidemic, discussed by the author (1985). If the infectious periods are all assumed as constant, the epidemic process will have the same total size distribution as that of a multipopulation Reed-Frost chain-binomial epidemic. The host-vector and spatial models are also special cases of the author’s general model.

Reviewed by B. Raja-Rao