DIMACS Series in
Discrete Mathematics and Theoretical Computer Science

VOLUME Seventy One
TITLE: "Disease Evolution: Models, Concepts, and Data Analyses"
EDITORS: Zhilan Feng, Ulf Dieckmann and Simon Levin

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Infections by microparasites (bacteria, viruses, protozoa and single celled fungi) are the primary source of human mortality in the underdeveloped world. And, despite all of the improvement in public health, hygiene, nutrition, living conditions and medical intervention over the past century, infections continue to be a major cause of morbidity and mortality in the developed world as well. Indeed, if we include people compromised by age, cancers and other diseases with immune-suppressing effects and/or treatments, coronary artery, diabetes, and other noncontagious and degenerative diseases, bacterial infections (often acquired in hospitals) may well be the major immediate cause of death even in overdeveloped countries.

Traditionally, the study of infectious diseases and their prevention and treatment has been the purview of epidemiologists, microbiologists, immunologists and clinicians - people who generally have little background in or appreciation for mathematics beyond statistics, if that. While the importance of quantitative reasoning for studies of the epidemiology of infectious diseases has been recognized for some time, this has been less so for investigations of the course of infections and their treatment within individual patients. For the most part, protocols for preventing the spread of infections in hospitals and communities and for the treatment of individual patients are based on qualitative considerations, experience and intuition, with money being the primary quantitative element in their design and implementation.

During the past two decades, studies of the epidemiology, evolution and withinhost biology of infectious diseases and the development of methods for their prevention and treatment have been increasingly infiltrated by quantitative methods beyond statistics. A number of applied and not-so-applied mathematicians, mathematically trained and oriented epidemiologists, microbiologists, immunologists, ecologists, population and evolutionary biologists and even real doctors have been using mathematical and numerical models (computer simulations) to study the epidemiology, evolution and within-host dynamics of infectious diseases and to develop and evaluate protocols for their prevention and treatment. This collection is an impressive sampling of the nature and diversity of this epidemic of mathematical and numerical modelling for the studies of infectious diseases. It illustrates some of the delicious problems and opportunities for mathematicians and mathematical biologists that infectious diseases pose - problems that have the virtues of being important to human health and well-being and, at the same time, being challenging and intriguing even from the precious heights of academe.

For generality, tractability and the aesthetic appeal of closed-form mathematical analysis, traditional models of the epidemiology and evolution of infectious diseases have been deterministic and give little or no consideration to the spatial, temporal and other heterogeneities of human and other host communities and those of the microbes that infect them. In recent years, increasing numbers of modelers have been confronting these inconveniences and the unfortunate finiteness of the real world and exploring how they affect the inferences about the epidemiology and evolution of infectious diseases drawn from simpler models. Three of the chapters in this collection illustrate this trend. Mike Boots and his collaborators consider the effects of the spatial structure of host populations on the evolution of the virulence of the microparasites that infect them. The two chapters by Wayne Getz and his cohorts examine, in a pedagogically useful as well as scholarly way, the consequences of spatial structure and stochastic processes on the spread of infectious diseases. Their chapters illustrate the utility of modelling to understanding the ascent and spread of emerging and reemerging diseases like SARS and tuberculosis, predicting their emergence and evaluating methods to control their dissemination.

In a commentary with a title that should appeal to this audience, "In Theory", Sidney Brenner referred to molecular biology as the "great leveler" and suggested that for many it has made thinking unnecessary, a position I do not challenge. On the other side, the ease with which data and particularly those on the nucleotide sequences of DNA can be and have been gathered has also provided an opportunity for the quantitative study of evolutionary history through phylogenies. God is no longer the only one who can make a tree. Phylogenies generated from DNA sequence data - molecular phylogenies - have been the largest growth industry in the evolution business for the past decade. And, the development (if not always the application) of methods for generating, analyzing and interpreting these trees is an activity that requires serious thinking. The chapter by Charleston and Galvani is a fine example of this thinking applied to ascertaining the evolutionary relationship between interacting organisms like parasites and their hosts. "Co-phylogenetic" methods of the sort they are developing are of practical as well as academic interest. They can be used to determine the origins (original hosts) of newly emerging infectious diseases like HIV/AIDS and in that way better understand the conditions responsible for their emergence.

A prominent approach to drawing inferences about the nature and direction of evolution of parasites and their virulence has been to study their ecology (population dynamics and demography) within individual hosts or communities of hosts. In this perspective, the fitness of the microparasites is proportional to their reproductive number, R0 - the number of secondary infections in a largely (or better yet, wholly) uninfected population of cells or tissues in an infected host or among individuals in a community of hosts. While this ecological approach to evolutionary inferences is explicit about nature and functional form of the selection pressures responsible for evolution, it does not consider the genetic basis of the variability upon which that selection is operating. In different ways and with different foci, two chapters consider ways to meld the ecological approach to the study of the evolution of microparasites and their virulence with those of population and quantitative genetics in situations where the nature of inheritance is explicit but where the ecological basis of selection is not. In their chapter, Troy Day and Sylvain Gandon consider how to apply classy population genetic approaches, like the Price equation, to studies of the evolution of microparasites and their virulence in communities of hosts. In his chapter, John Kelly uses a combination of ecological and population genetic methods to explore the contribution of tissue heterogeneity to the evolution of viruses in within infected hosts.

The contribution of the heterogeneity of the within-host habitat to the population and evolutionary dynamics of microparasites is also the focus of three other chapters in this collection. While modelers as well as experimentalists have the convenience of separately studying microparasite ecology and evolution within infected host and in communities of hosts, in the real world microbes have no choice but to deal with both of these elements of their ecology and evolution. Although their models focus primarily on the within-host population and evolutionary dynamics of viruses and the contribution of within-host heterogeneity to that evolution, in their chapter Robert Holt and Michael Barfield consider how the within-host biology of microparasites contributes to their evolution in communities of hosts. In the chapter by Zhilan Feng and Libin Rong, the within-host heterogeneity of concern is comprised of the selective environments imposed by treatment with multiple drugs that act at different stages in the microparasite replication cycle. Using an age-structured model, they consider the treatment of HIV/AIDS with reversetranscriptase - and protease - inhibitors, and how this treatment contributes to the evolution of resistance and rates of viral replication. Antimicrobial chemotherapy, heterogeneity and resistance are also the subjects of the chapter by David Smith and his collaborators. In their case, the drugs are antibacterial (antibiotics) rather than antiviral; the heterogeneity is both spatial and in the extent to which the drugs are employed; and the focus is the epidemiology and evolution of resistance in communities of hosts rather than in individual treated patients.

Investigators studying the evolution of infectious disease are almost invariably adaptationists; they assume that selection in the host, parasite or both populations is responsible for the virulence of the parasite and for maintaining genetic diversity in the parasite population. In their article on the serological diversity of the rhinoviruses responsible for the common cold, William Koppelman and Frederick Adler consider the neutral, null hypothesis alternative - that the 100 or so serotypes of Rhinoviruses responsible for colds are consequences of a high mutation rate and genetic drift rather than immune-mediated selection. Rhinoviruses and the cross-immunity they engender are also stage center in the chapter by Alun L. Lloyd and Dominik Wodarz, but the focus of their investigation of these ubiquitous and annoying, albeit rarely lethal, viruses is chemotherapy and the contribution of the host immune response to the evolution of resistance to the antiviral drugs employed.

This collection can be and I believe should be seen as a testimony to the work of Roy Anderson and Robert May. While they are not the discoverers of infectious diseases (at least I don't think they are) or even the first to use mathematical models to investigate them, their research more than that of any other investigators has been responsible for the renaissance (epidemic) in the use of models for studying infectious diseases and their control. The research reported in almost all of the chapters in this volume have antecedents in Anderson and May's work. While there is no formal dedication to them in the front matter of this volume, that dedication is where it really counts. The contributions of either Robert May and/or Roy Anderson are acknowledged at least once in every chapter and now, appropriately, in this Preface.


Bruce R. Levin
Atlanta, September 2005


Foreword                                                vii

Preface                                                  ix

Editor's Introduction                                  xiii

Section I: Model Infrastructure

The implications of spatial structure within
   populations to the evolution of parasites
     M. Boots, M. Kamo, and A. Sasaki                     3

Insights from Price's equation into evolutionary
     T. Day and S. Gandon                                23

Within-host pathogen dynamics: Some ecological and
   evolutionary consequences of transients, dispersal
   mode, and within-host spatial heterogeneity
     R.D. Holt and M. Barfield                           45

Evolutionary and dynamic models of infection with
   internal host structure
     J.K. Kelly                                          67

Basic methods for modeling the invasion and spread of
   contagious diseases
     W.M. Getz and J.O. Lloyd-Smith                      87

Section II: Applications to Specific Diseases

Modeling the invasion and spread of contagious diseases
   in heterogeneous populations
     W.M. Getz, J.O. Lloyd-Smith, P.C. Cross,
       S. Bar-David, P.L. Johnson, T.C. Porco, and
       M.S. Sanchez                                     113

A cophylogenetic perspective on host-pathogen evolution
     M.A. Charleston and A.P. Galvani                   145

The influence of anti-viral drug therapy on the
   evolution of HIV-1 pathogens
     Z. Feng and L. Rong                                161

Do rhinoviruses follow the neutral theory? The role of
   cross-immunity in maintaining the diversity of the
   common cold
     W.J. Koppelman and F.R. Adler                      181

Drug resistance in acute viral infections: Rhinovirus
   as a case study
     A.L. Lloyd and D. Wodarz                           193

Dynamics and control of antibiotic resistance in
   structured metapopulations
     D.L. Smith, M.F. Boni, and R. Laxminarayan         213

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Document last modified on August 31, 2006.