Disease Evolution - American Mathematical Societyspread of emerging and reemerging diseases like SARS and tuberculosis, predicting their emergence and evaluating methods to control

Disease Evolutio n Models, Concepts , an d Dat a Analyse s

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DIMACS Series i n Discret e Mathematic s

and Theoretica l Compute r Scienc e

Volume 7 1

Disease Evolutio n Models, Concepts , an d Dat a Analyse s

Zhilan Fen g Ulf Dieckman n

Simon Levi n Editors

Center fo r Discret e Mathematic s and Theoretica l Compute r Scienc e

A consortium o f Rutgers University , Princeto n University , AT&T Labs-Research, Bel l Labs (Lucen t Technologies) , NEC Laboratories America , an d Telcordi a Technologie s (with partner s a t Avaya Labs , H P Labs, IB M Research ,

Microsoft Research , an d Steven s Institut e o f Technology)

American Mathematica l Societ y

https://doi.org/10.1090/dimacs/071

2000 Mathematics Subject Classification. P r i m a r y 34C60 , 35Q80 , 62P10 , 92B05 , 92C50 , 92C60, 92D10 , 92D15 , 92D25 , 92D30 .

Library o f Congres s Cataloging-in-Publicat io n D a t a

Disease evolutio n : models , concepts , an d dat a analyses/Zhila n Feng , Ul f Dieckmann , Simo n Levin, editors .

p. cm . — (DIMAC S serie s i n discret e mathematic s an d theoretica l compute r science ; 71 ) Includes bibliographica l reference s ISBN 0-8218-3753- 2 (alk . paper )

1. Communicabl e diseases-Epidemiology . 2 . Molecula r epidemiology . 3 . Evolutionar y genet -ics. I . Feng , Zhilan , 1959 - II . Dieckmann , Ulf , 1966 - III . Levin , Simo n A . IV . Series .

RA652.2.M3.D57 200 6 614.4—dc22 200604297 3

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CONTENTS

Foreword vii

Preface i x

Editors' introductio n xii i

Section I : Mode l Infrastructur e

The implications of spatial structure within populations to the evolution of parasites Mike Boots, Masashi Kamo, and Akira Sasaki 3

Insights fro m Price' s equatio n int o evolutionar y epidemiolog y Troy Day and Sylvain Gandon 2 3

Within-host pathoge n dynamics : Som e ecologica l an d evolutionar y consequence s of transients , dispersa l mode , an d within-hos t spatia l heterogeneit y

Robert D. Holt and Michael Barfield 4 5

Evolutionary an d dynami c model s o f infectio n wit h interna l hos t structur e John K. Kelly 6 7

Basic method s fo r modelin g th e invasio n an d sprea d o f contagious disease s Wayne M. Getz and James O. Lloyd-Smith 8 7

Section II : Application s t o Specifi c Disease s

Modeling th e invasio n an d sprea d o f contagiou s disease s i n heterogeneou s popula -tions

Wayne M. Getz, James 0. Lloyd-Smith, Paul C. Cross, Shirli Bar-David, Philip L. Johnson, Travis C. Porco, and Maria S. Sanchez 11 3

A cophylogeneti c perspectiv e o n host-pathoge n evolutio n Michael A. Charleston and Alison P. Galvani 14 5

The influenc e o f anti-vira l dru g therap y o n th e evolutio n o f HIV-1 pathogen s Zhilan Feng and Libin Rong 16 1

Do rhinoviruses follo w th e neutra l theory ? Th e role of cross-immunity i n maintain -ing th e diversit y o f the commo n col d

William J. Koppelman and Frederick R. Adler 18 1

Drug resistanc e i n acut e vira l infections : Rhinoviru s a s a cas e stud y Alun L. Lloyd and Dominik Wodarz 19 3

Dynamics an d contro l o f antibioti c resistanc e i n structure d metapopulation s David L. Smith, Maciej F. Boni, and Ramanan Laxminarayan 21 3

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Foreword

This volum e ha s it s genesi s i n th e activitie s o f th e DIMAC S workin g grou p on Genetic s an d Evolutio n o f Pathogens . Thi s workin g grou p hel d a meetin g o n November 24-25 , 2003 at Rutger s University , an d w e would lik e to expres s ou r ap -preciation t o Zhila n Fen g fo r organizin g an d plannin g thi s successfu l conference . The volum e represent s a n expansio n o f the effort s o f thi s workin g group , an d con -tains papers from expert s in the field who were unable to attend thi s initial meeting . We thank th e thre e editor s Zhila n Feng , Ul f Dieckman n an d Simo n Levi n fo r thei r efforts i n the organization of the volume, and we thank Bruce Levin for his insightfu l Preface an d als o the variou s author s wh o contribute d t o th e volume .

The meetin g wa s par t o f th e 2002-200 7 Specia l Focu s o n Computationa l an d Mathematical Epidemiology , an d wa s organized b y one of a number o f special focu s research groups called "workin g groups" a s part o f the specia l focus. W e extend ou r thanks t o Marti n Fa r ach-Colt on, Sunetr a Gupta , Donal d Hoover , Davi d Krakauer , Simon Levin , Mar c Lipsitch , Davi d Madigan , Mega n Murray , S . Muthukrishnan , David Ozonoff , Fre d Roberts , Burto n Singe r an d Danie l Wartenberg fo r thei r wor k as specia l focu s organizers .

The meetin g brough t togethe r researcher s wh o approac h th e stud y o f epidemi -ology fro m a variet y o f disciplines , som e applie d an d som e theoretical . Thes e in -cluded compute r scientists , mathematicians , statisticians , an d biologist s togethe r with both descriptive and analytica l epidemiologists . Th e goal of the working grou p as wel l a s o f thi s volum e i s the exploratio n o f cross-disciplinar y approache s t o th e study o f topic s relate d t o diseas e evolution , an d ho w the y appl y t o th e stud y o f specific diseases .

DIMACS gratefull y acknowledge s th e generou s suppor t tha t make s thes e pro -grams possible . Specia l thank s g o t o th e Nationa l Scienc e Foundation , th e Jame s S. McDonnel l Foundation , th e Burroughs-Wellcom e Fund , th e Purdu e Universit y Mathematics Departmen t an d t o DIMAC S partner s a t Rutgers , Princeton , AT& T Labs - Research , Bel l Labs , NE C Laboratorie s America , an d Telcordi a Technolo -gies, an d affiliat e partner s Avay a Labs , H P Labs , IB M Research , an d Microsof t Research.

Fred S . Robert s Director

Robert Tarja n Co-Director fo r Princeto n

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Preface

Infections b y microparasite s (bacteria , viruses , protozo a an d singl e celled fungi ) are th e primar y sourc e o f huma n mortalit y i n th e underdevelope d world . And , despite al l o f th e improvemen t i n publi c health , hygiene , nutrition , livin g condi -tions an d medica l interventio n ove r th e pas t century , infection s continu e t o b e a major caus e o f morbidit y an d mortalit y i n th e develope d worl d a s well . Indeed , i f we include peopl e compromise d b y age , cancer s an d othe r disease s wit h immune -suppressing effect s and/o r treatments , coronar y artery , diabetes , an d othe r non -contagious an d degenerativ e diseases , bacteria l infection s (ofte n acquire d i n hos -pitals) ma y wel l b e th e majo r immediat e caus e o f deat h eve n i n overdevelope d countries.

Traditionally, th e stud y o f infectiou s disease s an d thei r preventio n an d treat -ment ha s bee n th e purvie w o f epidemiologists , microbiologists , immunologist s an d clinicians - peopl e who generally have little background in or appreciation for math -ematics beyon d statistics , i f that . Whil e th e importanc e o f quantitativ e reasonin g for studie s o f the epidemiolog y o f infectious disease s ha s bee n recognize d fo r som e time, thi s ha s bee n les s s o fo r investigation s o f th e cours e o f infection s an d thei r treatment withi n individua l patients . Fo r th e mos t part , protocol s fo r prevent -ing th e sprea d o f infection s i n hospital s an d communitie s an d fo r th e treatmen t of individua l patient s ar e base d o n qualitativ e considerations , experienc e an d in -tuition, wit h mone y bein g th e primar y quantitativ e elemen t i n thei r desig n an d implementation.

During th e pas t tw o decades , studie s o f the epidemiology , evolutio n an d within -host biolog y o f infectiou s disease s an d th e developmen t o f method s fo r thei r pre -vention an d treatmen t hav e bee n increasingl y infiltrate d b y quantitativ e method s beyond statistics . A number o f applie d an d not-so-applie d mathematicians , math -ematically traine d an d oriente d epidemiologists , microbiologists , immunologists , ecologists, populatio n an d evolutionar y biologist s an d eve n rea l doctor s hav e bee n using mathematica l an d numerica l model s (compute r simulations ) t o stud y th e epidemiology, evolutio n an d within-hos t dynamic s o f infectious disease s an d t o de -velop an d evaluat e protocol s fo r thei r preventio n an d treatment . Thi s collectio n i s an impressive sampling of the nature an d diversity of this epidemic of mathematica l and numerica l modellin g fo r th e studie s o f infectiou s diseases . I t illustrate s som e of the deliciou s problem s an d opportunitie s fo r mathematician s an d mathematica l biologists tha t infectiou s disease s pos e - problem s tha t hav e th e virtue s o f bein g important t o human healt h an d well-being and , a t the same time, being challengin g and intriguin g eve n fro m th e preciou s height s o f academe .

For generality , tractabilit y an d th e aestheti c appea l o f closed-form mathematica l analysis, traditional model s of the epidemiology an d evolution of infectious disease s have bee n deterministi c an d giv e little o r n o consideratio n t o th e spatial , tempora l and othe r heterogeneitie s o f huma n an d othe r hos t communitie s an d thos e o f th e microbes tha t infec t them . I n recen t years , increasin g number s o f modeler s hav e been confrontin g thes e inconvenience s an d th e unfortunat e finiteness o f th e rea l world an d explorin g ho w the y affec t th e inference s abou t th e epidemiolog y an d evolution o f infectiou s disease s draw n fro m simple r models . Thre e o f th e chapter s in this collection illustrate this trend. Mik e Boots and his collaborators consider th e

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effects o f the spatia l structur e o f host population s o n the evolutio n o f the virulenc e of th e microparasite s tha t infec t them . Th e tw o chapter s b y Wayn e Get z an d hi s cohorts examine, in a pedagogically usefu l a s well as scholarly way, the consequence s of spatia l structur e an d stochasti c processe s o n th e sprea d o f infectiou s diseases . Their chapter s illustrat e th e utilit y o f modellin g t o understandin g th e ascen t an d spread o f emerging an d reemergin g disease s like SARS and tuberculosis , predictin g their emergenc e an d evaluatin g method s t o contro l thei r dissemination .

In a commentar y wit h a titl e tha t shoul d appea l t o thi s audience , "I n Theory" , Sidney Brenne r referre d t o molecula r biolog y a s th e "grea t leveler " an d suggeste d that fo r man y i t ha s mad e thinkin g unnecessary , a position I do not challenge . O n the othe r side , th e eas e wit h whic h dat a an d particularl y thos e o n th e nucleotid e sequences o f DN A ca n b e an d hav e bee n gathere d ha s als o provide d a n opportu -nity fo r th e quantitativ e stud y o f evolutionar y histor y throug h phylogenies . Go d is n o longe r th e onl y on e wh o ca n mak e a tree . Phylogenie s generate d fro m DN A sequence dat a - molecula r phylogenie s - hav e bee n th e larges t growt h industr y i n the evolution busines s for the past decade . And , the developmen t (i f not alway s th e application) o f methods fo r generating , analyzin g an d interpretin g thes e tree s i s a n activity tha t require s serious thinking. Th e chapte r b y Charlesto n an d Galvan i i s a fine example o f thi s thinkin g applie d t o ascertainin g th e evolutionar y relationshi p between interactin g organism s lik e parasite s an d thei r hosts . "Co-phylogenetic " methods o f the sor t the y ar e developin g ar e o f practica l a s wel l a s academi c inter -est. The y ca n b e use d t o determin e th e origin s (origina l hosts ) o f newl y emergin g infectious disease s like HIV/AIDS an d in that wa y better understan d th e condition s responsible fo r thei r emergence .

A prominen t approac h t o drawin g inference s abou t th e natur e an d directio n o f evolution o f parasites an d thei r virulenc e ha s bee n t o stud y thei r ecolog y (popula -tion dynamic s an d demography ) withi n individua l host s o r communitie s o f hosts . In thi s perspective , th e fitnes s o f the microparasite s i s proportional t o thei r repro -ductive number, Ro ~ the number o f secondary infection s i n a largely (o r better yet , wholly) uninfecte d populatio n o f cell s o r tissue s i n a n infecte d hos t o r amon g in -dividuals i n a communit y o f hosts . Whil e thi s ecologica l approac h t o evolutionar y inferences i s explici t abou t natur e an d functiona l for m o f th e selectio n pressure s responsible fo r evolution , i t doe s no t conside r th e geneti c basi s o f th e variabilit y upon whic h tha t selectio n i s operating . I n differen t way s an d wit h differen t foci , two chapter s conside r way s t o mel d th e ecologica l approac h t o th e stud y o f th e evolution o f microparasites an d thei r virulenc e wit h thos e o f population an d quan -titative genetic s i n situation s wher e th e natur e o f inheritance i s explici t bu t wher e the ecologica l basi s o f selectio n i s not . I n thei r chapter , Tro y Da y an d Sylvai n Gandon conside r ho w to appl y class y populatio n geneti c approaches , lik e the Pric e equation, t o studie s o f the evolutio n o f microparasite s an d thei r virulenc e i n com -munities o f hosts . I n hi s chapter , Joh n Kell y use s a combinatio n o f ecologica l an d population geneti c method s t o explor e th e contributio n o f tissu e heterogeneit y t o the evolutio n o f viruses i n within infecte d hosts .

The contributio n o f the heterogeneit y o f th e within-hos t habita t t o th e popula -tion an d evolutionar y dynamic s o f microparasite s i s als o th e focu s o f thre e othe r chapters in this collection. Whil e modelers a s well as experimentalists have the con -venience of separately studyin g microparasit e ecolog y and evolutio n within infecte d host an d i n communitie s o f hosts , i n th e rea l worl d microbe s hav e n o choic e bu t

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to dea l wit h bot h o f these element s o f thei r ecolog y an d evolution . Althoug h thei r models focu s primaril y o n th e within-hos t populatio n an d evolutionar y dynam -ics o f viruse s an d th e contributio n o f within-hos t heterogeneit y t o tha t evolution , in thei r chapte r Rober t Hol t an d Michae l Barfiel d conside r ho w th e within-hos t biology o f microparasite s contribute s t o thei r evolutio n i n communitie s o f hosts . In th e chapte r b y Zhila n Fen g an d Libi n Rong , th e within-hos t heterogeneit y o f concern i s comprised o f the selective environments impose d b y treatment wit h mul -tiple drugs tha t ac t a t differen t stage s in the microparasit e replicatio n cycle . Usin g an age-structure d model , the y conside r th e treatmen t o f HIV/AID S wit h reverse -transcript ase - an d protease - inhibitors , and ho w this treatment contribute s t o th e evolution o f resistance an d rate s o f vira l replication . Antimicrobia l chemotherapy , heterogeneity an d resistanc e ar e als o th e subject s o f th e chapte r b y Davi d Smit h and hi s collaborators . I n thei r case , the drug s ar e antibacteria l (antibiotics ) rathe r than antiviral ; th e heterogeneit y i s bot h spatia l an d i n th e exten t t o whic h th e drugs ar e employed ; an d th e focu s i s the epidemiolog y an d evolutio n o f resistanc e in communitie s o f host s rathe r tha n i n individua l treate d patients .

Investigators studyin g th e evolutio n o f infectiou s diseas e ar e almos t invariabl y adapt at ionists; the y assum e tha t selectio n i n th e host , parasit e o r bot h popula -tions i s responsibl e fo r th e virulenc e o f th e parasit e an d fo r maintainin g geneti c diversity i n th e parasit e population . I n thei r articl e o n th e serologica l diversit y o f the rhinoviruse s responsibl e fo r th e commo n cold , Willia m Koppelma n an d Fred -erick Adle r conside r th e neutral , nul l hypothesi s alternativ e - tha t th e 10 0 o r s o serotypes o f Rhinoviruses responsibl e fo r cold s are consequences o f a high mutatio n rate an d geneti c drif t rathe r tha n immune-mediate d selection . Rhinoviruse s an d the cross-immunit y the y engende r ar e als o stag e cente r i n th e chapte r b y Alu n L . Lloyd an d Domini k Wodarz , bu t th e focus o f their investigatio n o f these ubiquitou s and annoying , albei t rarel y lethal , viruse s i s chemotherap y an d th e contributio n of th e hos t immun e respons e t o th e evolutio n o f resistanc e t o th e antivira l drug s employed.

This collectio n ca n b e an d I believ e shoul d b e see n a s a testimon y t o th e wor k of Roy Anderso n an d Rober t May . Whil e they ar e no t th e discoverer s o f infectiou s diseases (a t leas t I don't think they are) or even the first t o use mathematical model s to investigat e them , thei r researc h mor e tha n tha t o f an y othe r investigator s ha s been responsibl e fo r th e renaissanc e (epidemic ) i n th e us e o f model s fo r studyin g infectious disease s an d thei r control . Th e researc h reporte d i n almos t al l o f th e chapters in this volume have antecedents i n Anderson an d May' s work. Whil e ther e is no formal dedicatio n t o the m i n the fron t matte r o f this volume , tha t dedicatio n is wher e i t reall y counts . Th e contribution s o f eithe r Rober t Ma y and/o r Ro y Anderson ar e acknowledge d a t leas t onc e i n ever y chapte r an d now , appropriately , in thi s Preface .

Enjoy, Bruce R . Levi n Atlanta, Septembe r 200 5

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Editors' introductio n

The goa l of this volume i s to show how to use mathematical tool s to understan d the evolutio n o f infectiou s diseases . Inspiratio n fo r thi s projec t come s fro m wor k of th e DIMAC S Workin g Grou p o n Genetic s an d Evolutio n o f Pathogens , whic h is organized unde r th e auspice s o f DIMACS ' Specia l Focu s o n Computationa l an d Mathematical Epidemiology .

This volume i s divided int o two sections: Mode l Infrastructure an d Application s to Specifi c Diseases . Sectio n I discusse s th e impac t o n diseas e evolutio n o f vari -ous factors, includin g spatia l structure , transien t dynamics , couplin g o f within-hos t and between-host dynamics , heterogeneity i n host populations , an d drug resistance . Section I I i s concerne d wit h investigation s associate d wit h specifi c infectiou s dis -eases suc h a s rhinovirus , HIV/AIDS , tuberculosis , an d malaria .

We than k Bruc e Levi n fo r hi s excellen t Preface . W e als o expres s ou r gratitud e to members o f DIMACS' staf f wh o kindly helped wit h the suppor t o f the worksho p and th e preparatio n o f thi s volume . Th e leadershi p o f Fre d Robert s i n developin g the multi-yea r epidemiolog y progra m ha s been a n inspiratio n t o man y researchers , and thi s volum e owe s it s existenc e t o hi s efforts . W e als o than k al l th e Purdu e Mathematics Departmen t fo r providin g technica l support . Finally , w e than k th e authors fo r thei r outstandin g contributions .

The workshop and the preparation o f this volume were partially supporte d b y a n NSF gran t t o DIMACS , an d b y NS F an d Jame s S . McDonnel l Foundatio n grant s to ZF .

Zhilan Fen g (Purdu e University ) Ulf Dieckman n (Internationa l Institut e fo r Applie d System s Analysis , Austria ) Simon Levi n (Princeto n University )

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Titles i n Thi s Serie s

71 Zhila n Feng , Ul f Dieckmann , an d Simo n Levin , Editors , Diseas e Evolution : Models ,

Concepts, an d Dat a Analyse s

70 Jame s Abel l o an d Graha m Cormode , Editors , Discret e Method s i n Epidemiolog y

69 S iemio n Fajtlowicz , Patric k W . Fowler , Pierr e Hansen , Melv i n F . Janowitz ,

and Fre d S . Roberts , Editors , Graph s an d Discover y

68 A . Ashikhmi n an d A . Barg , Editors , Algebrai c Codin g Theor y an d Informatio n Theor y

67 Rav i Janardan , Michie l Smid , an d Debas i s h D u t t a , Editors , Geometri c an d

Algorithmic Aspect s o f Computer-Aide d Desig n an d Manufacturin g

66 P i y u s h Gupta , Gerhar d Kramer , an d Adriaa n J . va n Wijngaarden , Editors ,

Advances i n Networ k Informatio n Theor y

65 Santos h S . Vempala , Th e Rando m Projectio n Metho d

64 M e l v y n B . Nathanson , Editor , Unusua l Application s o f Numbe r Theor y

63 J . Neset f i l an d P . Winkler , Editors , Graphs , Morphism s an d Statistica l Physic s

62 Gerar d J . Foschin i an d Sergi o Verdu , Editors , Multiantenn a Channels : Capacity ,

Coding an d Signa l Processin g

61 M . F . Janowitz , F.-J . Lapointe , F . R . McMorris , B . Mirkin , an d F . S . Rober t s ,

Editors, Bioconsensu s

60 Saugat a B a s u an d Laurean o Gonzalez-Vega , Editors , Algorithmi c an d Quantitativ e

Real Algebrai c Geometr y

59 Michae l H . Goldwasser , Davi d S . Johnson , an d Catherin e C . McGeoch , Editors , Data Structures , Nea r Neighbo r Searches , an d Methodology : Fift h an d Sixt h DIMAC S

Implementation Challenge s

58 S imo n Thomas , Editor , Se t Theory : Th e Hajna l Conferenc e

57 Eugen e C . Freude r an d Richar d J . Wallace , Editors , Constrain t Programmin g an d

Large Scal e Discret e Optimizatio n

56 Alexande r Bar g an d Simo n Litsyn , Editors , Code s an d Associatio n Scheme s

55 Ding-Zh u D u , Pano s M . Pardalos , an d Ji e Wang , Editors , Discret e Mathematica l

Problems wit h Medica l Application s

54 Eri k Winfre e an d Davi d K . Gifford , Editors , DN A Base d Computer s V

53 Nathanie l D e a n , D . Fran k Hsu , an d R . Ravi , Editors , Robus t Communicatio n

Networks: Interconnectio n an d Survivabilit y

52 Sangutheva r Rajasekaran , Pano s Pardalos , an d D . Fran k Hsu , Editors , Mobil e

Networks an d Computin g

51 Pierr e Hansen , Patric k Fowler , an d Maol i n Zheng , Editors , Discret e Mathematica l

Chemistry

50 Jame s M . Abe l l o an d Jeffre y Scot t Vi t ter , Editors , Externa l Memor y Algorithm s

49 Ronal d L . Graham , Ja n Kratochvil , Jarosla v Neset f i l , an d Fre d S . Roberts ,

Editors, Contemporar y Trend s i n Discret e Mathematic s

48 Harve y Rubi n an d Davi d Harla n W o o d , Editors , DN A Base d Computer s II I

47 Mart i n Farach-Colton , Fre d S . Roberts , Mart i n Vingron , an d Michae l

Waterman, Editors , Mathematica l Suppor t fo r Molecula r Biolog y

46 Peng-Ju n Wan , Ding-Zh u D u , an d Pano s M . Pardalos , Editors , Multichanne l

Optical Networks : Theor y an d Practic e

45 Mario s Mavronicolas , Michae l Merrit t , an d Ni r Shavit , Editors , Network s i n

Distributed Computin g

44 Laur a F . Landwebe r an d Eri c B . B a u m , Editors , DN A Base d Computer s I I

43 Pano s Pardalos , Sangutheva r Rajasekaran , an d Jos e Rol im , Editors ,

Randomization Method s i n Algorith m Desig n

42 Ding-Zh u D u an d Fran k K . Hwang , Editors , Advance s i n Switchin g Network s

TITLES I N THI S SERIE S

41 Davi d Aldou s an d Jame s Propp , Editors , Microsurvey s i n Discret e Probabilit y

40 Pano s M . Pardalo s an d Dingzh u D u , Editors , Networ k Design : Connectivit y an d

Facilities Locatio n

39 Pau l W . Beam e an d Samue l R Buss , Editors , Proo f Complexit y an d Feasibl e

Arithmetics

38 Rebecc a N . Wrigh t an d Pete r G . N e u m a n n , Editors , Networ k Threat s

37 Bori s Mirkin , F . R . McMorris , Fre d S . Roberts , an d Andre y Rzhetsky , Editors ,

Mathematical Hierarchie s an d Biolog y

36 Josep h G . Rosenste in , Debora h S . Franzblau , an d Fre d S . Roberts , Editors ,

Discrete Mathematic s i n th e School s

35 Dingzh u D u , Ju n Gu , an d Pano s M . Pardalos , Editors , Satisfiabilit y Problem :

Theory an d Application s

34 Nathanie l Dean , Editor , Africa n American s i n Mathematic s

33 Rav i B . Boppan a an d Jame s F . Lynch , Editors , Logi c an d Rando m Structure s

32 Jean-Charle s Gregoire , Gerar d J . Holzmann , an d Doro n A . Pe led , Editors , Th e

SPIN Verificatio n Syste m

31 Nei l Immerma n an d Phokio n G . Kolait is , Editors , Descriptiv e Complexit y an d

Finite Model s

30 Sandee p N . Bhat t , Editor , Paralle l Algorithms : Thir d DIMAC S Implementatio n

Challenge

29 Doro n A . Peled , Vaugha n R . Prat t , an d Gerar d J . Holzmann , Editors , Partia l

Order Method s i n Verificatio n

28 Larr y Finkelstei n an d Wil l ia m M . Kantor , Editors , Group s an d Computatio n I I

27 Richar d J . Lipto n an d Eri c B . B a u m , Editors , DN A Base d Computer s

26 Davi d S . Johnso n an d Michae l A . Trick , Editors , Cliques , Coloring , an d

Satisfiability: Secon d DIMAC S Implementatio n Challeng e

25 Gilber t Baumslag , Davi d Epste in , Rober t Gilman , Hamis h Short , an d Charle s

Sims, Editors , Geometri c an d Computationa l Perspective s o n Infinit e Group s

24 Loui s J . Billera , Curti s Greene , Rodic a Simion , an d Richar d P . Stanley , Editors , Formal Powe r Serie s an d Algebrai c Combinatorics/Serie s Formelle s e t Combinatoir e Algebrique, 199 4

23 Pano s M . Pardalos , Davi d I . Shalloway , an d Guolian g Xue , Editors , Globa l

Minimization o f Nonconve x Energ y Functions : Molecula r Conformatio n an d Protei n Foldin g

22 Pano s M . Pardalos , Maurici o G . C . Resende , an d K . G . Ramakrishnan , Editors ,

Parallel Processin g o f Discret e Optimizatio n Problem s

21 D . Fran k Hsu , Arnol d L . Rosenberg , an d Dominiqu e Sot teau , Editors ,

Interconnection Network s an d Mappin g an d Schedulin g Paralle l Computation s

20 Wil l ia m Cook , Laszl o Lovasz , an d Pau l Seymour , Editors , Combinatoria l

Optimization

19 Ingema r J . Cox , Pierr e Hansen , an d Bel a Julesz , Editors , Partitionin g Dat a Set s

18 G u y E . Blel loch , K . Man i Chandy , an d Sures h Jagannathan , Editors ,

Specification o f Paralle l Algorithm s

17 Eri c Sve n Ristad , Editor , Languag e Computation s

16 Pano s M . Pardalo s an d Henr y Wolkowicz , Editors , Quadrati c Assignmen t an d Related Problem s

For a complet e lis t o f t i t le s i n thi s series , visi t th e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .