Mathematics of Epidemics on Networks: From Exact to Approximate Models PDF
Preview Mathematics of Epidemics on Networks: From Exact to Approximate Models
Interdisciplinary Applied Mathematics 46 István Z. Kiss Joel C. Miller Péter L. Simon Mathematics of Epidemics on Networks From Exact to Approximate Models Mathematics of Epidemics on Networks Interdisciplinary Applied Mathematics Volume 46 Editors S.S.Antman,UniversityofMaryland,CollegePark,MD,USA [email protected] L.Greengard,NewYorkUniversity,NewYork,NY,USA [email protected] P.J.Holmes,PrincetonUniversity,Princeton,NJ,USA [email protected] Advisors R.Durrett,DukeUniversity,Durham,NC,USA L.Glass,McGillUniversity,Montreal,QB,Canada A.Goriely,UniversityofOxford,Oxford,UK R.Kohn,NewYorkUniversity,NewYork,USA P.S.Krishnaprasad,UniversityofMaryland,CollegePark,MD,USA J.D.Murray,UniversityofOxford,Oxford,UK C.Peskin,NewYorkUniversity,NewYork,USA S.S.Sastry,UniversityofCalifornia,Berkeley,CA,USA J.Sneyd,UniversityofAuckland,Auckland,NZ Moreinformationaboutthisseriesathttp://www.springer.com/series/1390 Istva´n Z. Kiss • Joel C. Miller • Pe´ter L. Simon Mathematics of Epidemics on Networks From Exact to Approximate Models 123 Istva´nZ.Kiss JoelC.Miller DepartmentofMathematics AppliedMathematics UniversityofSussex InstituteforDiseaseModeling Falmer,Brighton,UK Bellevue,WA,USA Pe´terL.Simon InstituteofMathematics Eo¨tvo¨sLora´ndUniversity Budapest,Hungary ISSN0939-6047 ISSN2196-9973 (electronic) InterdisciplinaryAppliedMathematics ISBN978-3-319-50804-7 ISBN978-3-319-50806-1 (eBook) DOI10.1007/978-3-319-50806-1 LibraryofCongressControlNumber:2017931653 Mathematics Subject Classification (2010): 00A71, 00A72, 05C82, 34C11, 34C23, 34D20, 35Q84, 37G10, 37N25, 47D06, 47N40, 60J27, 60J28, 60J75, 60J80, 60J85, 60K20, 60K35, 82B43, 91D30, 92C42,92D30 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland In authororder, wededicatethisbookto our wives Katalin,Anja,andNadinka andour childrenLouisa andε;Maaikeand 2 Jojanneke;and Nadinka,Zso´fia,Vikto´ria, Bernadett,and Bru´no´. Preface Over the past decade, the use of networks has led to a new modelling paradigm combiningseveralbranchesofscience,includingphysics,mathematics,biologyand socialsciences. Thespreadof infectiousdiseasesbetweennodesin a networkhas been a central topic of this growing field. The fundamental questions are easily stated,butansweringthemdrawsonobservationsandtechniquesofmanyfields. Thereisalongsuccessfulhistoryofmathematicalmodellinginformingpolicies to mitigate the impact of infectious disease. Typically, models divide the popula- tionintocompartmentsbasedoninfectionstatusandusesimpleassumptionsabout mixingandmovementsbetweenthesecompartments.Overtime,thesemodelshave grown more sophisticated to more accurately incorporate the contact structure of thepopulationandtotakeadvantageofincreasedcomputationalresources.Forex- ample,sexuallytransmitteddiseaseshavebeeninvestigatedusinghigh-dimensional compartmentalmodelsseparatingindividualsbycontactrates,socio-economicsta- tusandmanyotherfactors.However,whenwemaketheadditionalobservationthat partnershipsmaybelong-lasting,anewparadigmisneeded,leadingnaturallytoa networkrepresentationofthepopulationstructure. Progressinmodeldevelopmenthasbeenextremelyfastandhasattractedinter- estfroma diversesetof researchers.The fundamentalobjectiveis to combinethe underlying population contact structure and the properties of the infectious agent toyieldanunderstandingoftheresultingspectrumofepidemicbehaviours.Todo this, researchers translate observed population and disease properties into a well- definedmodel.Inmanycases,themodelsitsattheinterfaceofgraph/networkthe- ory,stochasticprocessesandprobabilitytheory,dynamicalsystems,andstatistical physics. The diversity of researcher backgrounds and the variety of applications considered have led to the developmentof many different modelling approaches. Asthefieldmatures,thereisaneedtoincreaseunderstandingofhowthesediffer- entmodelsfit together,how theyrelate to the underlyingassumptionsandhow to developanappropriatemathematicalframeworktounifydifferentapproaches. vii viii Preface This booksets outto make a contributionto modellingepidemicson networks bysynthesisingalargepoolofmodels,rangingfromexactandstochastictoapprox- imatedifferentialequationmodels,sothatwemay: 1. recogniseunderlyingmodelassumptionsandtheresultingmodelcomplexity; 2. provideamathematicalframeworkwithwhichwecandescribeobservedphe- nomenaandpredictfuturescenarios; 3. permitdirectcomparisonofthemainmodelsandprovidetheirhierarchy;and 4. identify research gaps and opportunities for further rigorous mathematical exploration. Chapter1introducesthereadertothefundamentalsofdiseasetransmissionmod- elsandtheunderlyingnetworks.Chapter2takesarigorousprobabilisticviewand frames disease transmission on a network as a continuous-time Markov chain. In contrast, Chapter 3 builds a hierarchy of models starting at the node level which depend on the node–neighbour pairs, which in turn depend on triples formed by consideringthenext-nearestneighbours.Chapter4focusesonmean-fieldandpair- wise models and their analysis on homogeneousnetworks. Chapter 5 extends ap- proaches of Chapter 4 to heterogeneousnetworks and introduces effective degree models. In Chapter 6, the focus is primarily on SIR epidemics, and percolation theory methods are used to derive the low-dimensional edge-based compartmen- talmodel.Chapter7 bringsthedifferentSIRmodelstogether,showingthatunder reasonableassumptions,thehigh-dimensionalmodelsofearlierchaptersreduceto the low-dimensionalmodel of Chapter 6. Chapter 8 extends the earlier models to accountforthesimultaneousspreadofthediseaseandchangeinthenetwork,con- sideringseveralscenariosforhownetworksvaryintime.Chapter9generalisesthe pairwiseandedge-basedcompartmentalmodelstonon-Markovianepidemics,lead- ing to integro-differentialand delay differentialequations. Chapter 10 starts from a Markov chain to derive the Fokker–Planck equation for the distribution of the numberof infected individualsas a functionof time and uses the resultingpartial differentialequation (PDE) to investigateepidemic processes. Finally, Chapter 11 shows that our models can perform surprisingly well even in networks, including empirically observed networks, for which the assumptions they are based on do notappearto be satisfied. TheAppendixgivesefficientsimulationalgorithmsand discussesissuesencounteredinsimulatingepidemicsonnetworks. With more space, we would have liked to make a stronger emphasis on prob- abilistic models. Moreover, we would have examined epidemic control measures suchas vaccinationandcontacttracing,as wellashouseholdmodels.Manyother topics,forexample,multilayernetworks(networkswithmultipletypesofconnec- tions),areleftout,althoughmanyofthetechniqueswediscussapplytothem.An additionaltopic,deservingofabookonitsown,wouldbetheuseofreal-worlddata toparametrisenetworkmodels. This book contains a number of rigorous mathematical arguments and proofs. However, a guiding principle throughout is to appeal to and be useful for audi- ences in fields outside of mathematics. Some quantitative sophistication will be Preface ix necessary; in particular, previous exposure to linear algebra, calculus, differential equations, dynamical systems and basics of probability and stochastic processes wouldbeuseful.Wedonotassumeknowledgeofgraphtheory. Advancedundergraduateandgraduatestudentscanusethebookasafoundation forlearningthemainmodellingandanalysistechniques.Therearemanyexercises designedtodevelopadeeperunderstandingofthetopic.Modelsandresultsofim- mediateapplicabilityaresignpostedthroughtheuseofgreyboxes. Weusethisformattohighlightreadilyimplementablemodelsortosummarise modeloutcomes,suchassteadystates,finalepidemicsize,basicreproductive ratioR ,probabilityofanepidemic,etc. 0 Doctoralstudents,researchersandexpertsinthisareacanusethebooknotonly as a reference guide or synthesis of the major modelling frameworks and model analysis tools but also to (i) confirm the validity and optimal range of applicabil- ity of models, (ii) understand how mathematical tools have been and are used in networkmodellingand(iii)identifyfurthersynergiesbetweenmainstreammathe- maticalmethodsandproblemsarisinginnetworkmodelling. To enhancethe flow of the presentation,citationsto previousresearchare con- centratedeitherat thebeginningor endofchapters.Thisallowsusto (a)build up modelsfromthegroundupbyunifyingdifferentapproachesleadingtosynthesised modelsand(b)citefurthernewdevelopmentsthatwecouldnotcover. Pseudocode for efficient epidemic simulation algorithms is given in the Appendix,andready-to-runsourcecodeisavailableatthefollowingwebsite: https://springer-math.github.io/Mathematics-of-Epidemics-on-Networks/ These include stochastic simulation of SIS and SIR on networks and numerical solutions of many differentialequation models we present in the book. An exten- sivePythonpackageusingNetworkX[130]isprovided,andmanyofthesearealso availablein Matlab.We hopeto addadditionallanguages.These willhelpreaders tocompletemanyofthesimulation-basedexercisesproposedinthebookandmay assist other researcherswith their own projects. Other resources are available; for example, a useful package in C++ is EpiFire [143]. Solutions to exercises will be madeavailableforinstructorswhousethebook.Inevitably,smallerrorscreepinto anybook.Pleasecontactusdirectlyforsolutionsortoreporterrors. Acknowledgements: The authors wish to thank their former and current co- workers and collaborators, research students and their current and former insti- tutions. IZ Kiss thanks the University of Leeds, the University of Oxford and the University of Sussex; JC Miller thanks Pennsylvania State University (Penn State), Monash University (in particular MAXIMA) and the Institute for Disease Modeling and PL Simon thanks the Department of Applied Analysis and Com- putational Mathematics and the whole Institute of Mathematics at the Eo¨tvo¨s Lora´nd University in Budapest, Hungary. The community at tex.stackexchange. com hashelped with intricaciesof LATEX. The authorsthank Dr. JohnHaigh from x Preface the University of Sussex for reading early drafts with great care and attention to detail. Finally, the authors thank their families for providingconstant supportand encouragement. Final thoughts: We would like to end with a memorable summary of our book aboutepidemicsonnetworks.Wehopethisepidemicsonnetworks: Whenpartnershipsenduresolongthatto Diseasetheyarelikefrozentiesthatbind, Massactionfailsustillnewparadigms Emerge;andnetworksthenareusefultools. Equationcountsareexponentialtill Reduced—throughautomorphicsymmetries Orcaref’lycuttingoutsomevertices. Butyetcomplexityistoohighstill. Andsoourmod’lermustapproximate Andcloseequations—butnottoosimply. Forshemustdoublycountahighdegree. Or,shemaywatchdiseasespercolate. Withthesetechniquesourmod’lerhasnewkeys Tolearnhowpartnershipsaffectdisease. Brighton,UK Istva´nZ.Kiss Bellevue,WA,USA JoelC.Miller Budapest,Hungary Pe´terL.Simon November2016
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