V5 Epidemics on networks

1. V5 Epidemics on networks Epidemic models attempt to capture the dynamics in the spreading of a disease (or of an idea, a computer virus, or the adoption of a product). Central questions the epidemic models try to answer are: • How do contagions (dt.ansteckendeKrankheiten) spread in populations? • Will a disease become an epidemic? • Who are the best people to vaccinate? (What do you think?) • Will a given YouTube video go viral? • What individuals should we market for maximizing product penetration? http://www.lsi.upc.edu/~CSN/slides/11epidemic.pdf Mathematics of Biological Networks

2. Different typesofdiseaseepidemics Infectiousdiseasesspreadovernetworksofcontactsbetweenindividuals. Airbournediseaseslike influenzaortuberculosisarecommunicated when 2 peoplebreathetheair in the same room. Contagiousdiseasesandparasitescanbecommunicatedwhenpeopletouch. HIV andothersexuallytransmitteddiseasesarecommunicatedwhenpeople have sex. The patternsof such contactscanberepresentedas (social) networks. Mathematics of Biological Networks

3. Classic epidemicmodels = „fullymixed“ Beforewe will discussthemodellingofepidemics in socialnetworks, wewill introducesomeclassic mathematicalmodelsofepidemics. Mathematicalmodelingofepidemicshasstartedmuchearlierthanstudyingnetworktopologies! The traditional approachesmakeuseof • a fullymixedapproximation(„meanfield“ in thephysicsworld) whereevery individual has an equalchance per unit time ofcomingintocontactwitheveryother. Mathematics of Biological Networks

4. The SI model (susceptible / infected) In thesimplestmathematicalrepresentationof an epidemic, thereare just 2 states, susceptibleandinfected. An individual in thesusceptiblestatedoes not havethediseaseyet but could catch itonce he/shegets in contactwith an infectedperson. An infected individual hasthediseaseandcanpotentially pass it on toothersusceptiblepersonsoncetheygetintocontact. Mathematics of Biological Networks

5. The SI model Letusconsider a diseasespreadingthrough a populationofindividuals. S(t) : average (orexpected) numberofsusceptibleindividualsat time t X(t) : average (orexpected) numberofinfectedpeopleat time t. In thefollowing, wedropthe explicit time-dependenceofS(t)andX(t) . Assumethateach individual has, on average,  contactswithrandomly chosenotherpeopleper unit time. Note thatthediseaseisonlytransmittedwhen an infectedpersonhascontactwith a susceptibleperson. Mathematics of Biological Networks

6. The SI model Let the total populationconsistofnpeople, n = S + X Average probabilitythat a personyoumeet at randomissusceptibleisS / n . → an infectedpersonhascontactwithon average S / nsusceptiblepeople per unit time. On average, thereareXinfectedindividuals in total. → theaverage rate ofnewinfectionsis S X / n. The rate ofchangeofXisthus The numberofsusceptiblepeople goes down at the same rate: Mathematics of Biological Networks

7. The SI model It isconvenienttodefine variables representingthefractionsofsusceptible andinfectedindividuals Then, the differential equationsbecome SinceS + X = n andthuss + x = 1, wedon‘tneedbothequations. Wecan e.g. eliminatesfromtheequationsbyreplacings = 1 – x This gives Mathematics of Biological Networks

8. The SI model - solution This equationoccurs in manyplaces in biology, physicsandelsewhere. Itiscalledthelogisticgrowthequation. Itssolutionis wherex0isthevalueofx at time t = 0. Generally, thisproduces an S-shaped „logisticgrowthcurve“ forthefraction ofinfectedindividualsat time t. The SI modelisthesimplestpossible modelofinfection. An initial burstphaseisfollowedby saturationwhen all peopleareinfected. Mathematics of Biological Networks

9. The SIR model: susceptible – infected – recovered/removed Therearemanywaystoextendthe SI modeltomakeitmore realisticormoreappropriateas a modelof a specificdisease. Onecommonextensionincludesrecoveryfromdisease. In the SI model, infectedindividualsremaininfected (andinfectious) forever. Formany real diseases, however, peoplerecoverfrominfection after a certain time becausetheir immune systemfights off theagentcausingthedisease. Mathematics of Biological Networks

10. The SIR model: susceptible – infected – recovered/removed Furthermore, peopleoftenretaintheirimmunitytothedisease after such a recovery such thattheycannot catch itagain. Forthis, weneed a thirdstate, therecoveredstateR. Forsomeotherdiseases, people do not recover but dieinstead. In epidemiologicalterms, such „removal“ isthe „same thing“ as „recovery“. (This issarcastic …) Wecantreatbothscenarioswiththesame S – I – R model. Mathematics of Biological Networks

11. The SIR model: 2 stages The SIR model was introduced in 1927 by W. O. Kermackand A. G. McKendrick The dynamicsofthefullymixed SIR modelhas 2 stages. In stage 1, susceptibleindividualsbecomeinfectedwhentheyhavecontactwithinfectedindividuals. Contacts happen at an average rate asbefore. In stage 2, infectedindividualsrecover (or die) at someconstantaverage rate .  : time that an infected individual islikelytoremaininfectedbeforetheyrecover. Probabilityofrecovering in any time interval  is  . Probabilityof not recovering in the same interval: 1 -   Mathematics of Biological Networks

12. The SIR model: susceptible – infected – recovered/removed → probabilitythatthe individual is still infected after a total time : (rememberthatn→ isthe same thingas →0) The probabilityp()dthatthe individual remainsinfectedthislongandthenrecovers in theintervalbetween and  +d isthisquantitytimes  d: This is a standardexponentialdistribution. Thus an infectedpersonismostlikelytorecoverdirectlyafter becominginfected, but might in theoryremain in theinfectedstateforquite a longtime. Mathematics of Biological Networks

13. The SIR model: comparisonto real diseases This behavioris not veryrealisticformost real diseaseswheremostvictims remaininfectedforaboutthe same lengthof time (oneoreseveralweeks). Fewstay in theinfectedstate formuchlongerorshorterthan theaverage. Distribution oftimesforwhich an individual remainsinfectedistypicallynarrowlypeaked aroundsomeaveragevalue (darkcurve) for real diseases, quiteunliketheexponential distributionassumedbythe SIR model (greycurve). This isonethingthatwe will improvewhenwereturntolook at networkmodelsofepidemics. Mathematics of Biological Networks

14. The SIR model: mathematicalsolution In termsofthefractionss, x, androfindividuals in the 3 states, theequationsforthe SIR modelare: In addition, the 3 variables satisfys + x + r = 1. Tosolvetheseequations, weeliminatexbyinsertingthe 3rd eq. intothe 1st one Mathematics of Biological Networks

15. The SIR model: mathematicalsolution Tosolvethis, weintegratebothsideswithrespecttot: Heres0isthevalueofs at t = 0 andwehavechosentheconstantofintegration so thattherearenoindividuals in therecoveredstate at t = 0. Mathematics of Biological Networks

16. The SIR model: numericalsolution Now weputx = 1 - s – r intoandusetoget or The solutionofthisis. This integral canbeevaluatednumerically. Fromrwethengetsandx. The figureshows an examplecase. Mathematics of Biological Networks

17. The SIR model: initial behavior In themostcommoncase, thediseasestartseitherwith a singleinfected individual orwith a smallnumbercofindividuals. → the initial valuesofthe variables are In thelimitof large populationsizesn → , wecanwrites0 1. Then, the final valueofrsatisfies If  there will benoepidemic. In thatcase, infectedindividualsrecoverfasterthansusceptibleindividualsbecomeinfected. Mathematics of Biological Networks

18. The SIR model: epidemictransition The transitionbetweentheepidemicand non-epidemicregimeshappens at thepoint =  andiscalledtheepidemictransition. An importantquantity in thestudyofepidemics isthebasicreproductionnumberR0. This isdefinedastheaveragenumberof additional susceptiblepeople towhich an infectedpersonpassesthediseasebeforethepersonrecovers. Ifeachpersoncatchingthediseasepassesitonto 2 others on average, thenR0 = 2. If half ofthem pass it on to just onepersonandtheresttononethenR0 = 0.5 Mathematics of Biological Networks

19. The SIR model: epidemicthreshold IfwehadR0 = 2, thenumberofnewcaseswould double at eachround, thusgrowexponentially. ConverselyifR0 = 0.5 thediseasewould die out exponentially. The pointR0 = 1 separates thegrowingandshrinkingbehavors. This istheepidemicthreshold. WecancalculateR0straightforwardlyforthe SIR model. If an individual remainsinfectiousfor a time , thentheexpectednumberofothersthey will havecontactwithduringthat time is . Mathematics of Biological Networks

20. The SIR model: epidemicthreshold In a „naive population“ at thestartof a disease (whereonly a fewindividualsareinfectedandtheothersusceptible) all ofthepeoplewithwhomonehascontact will besusceptible. Thenweaverageoverthedistributionof  togettheaveragenumberR0 : after usingsomeintegration tricks. The epidemicthresholdofthe SIR modelisthus =  aswehavederivedbefore. Mathematics of Biological Networks

21. The SIS model A different extensionofthe SI modelisonethatallowsforreinfection. Fordiseasesthat do not conferimmunitytotheirvictims after recovery, individualscanbeinfectedmorethanonce. The simplest such modelistheSIS model. Ithasthe2statessusceptibleandinfected. Infectedindividualsmove back intothesusceptiblestate after recovery. withs + x = 1 Mathematics of Biological Networks

22. The SIS model Putting s = 1 - xgives Whichhasthesolution In thecaseof a large populationand a smallnumberof initial carriers, and • >  thisproduces a logisticgrowthcurve. In thismodel, weneverhave thewholepopulationinfected withthedisease. Mathematics of Biological Networks

23. The SIRS model In the SIRS model, individualsrecoverfrominfectionandgaintemporaryimmunity. After a certain time theybecomesusceptibleagainwith an average rate . and This modelcannotbesolvedanalytically. Mathematics of Biological Networks

24. Review The SIR model is appropriate for infectious diseases that confer lifelong immunity, such as measles or whooping cough. The SIS model is predominantly used for sexually transmitted diseases (STDs), such as chlamydia or gonorrhoea, where repeated infections are common. Keeling & Eames, J R Soc Interface (2005) 2: 295–307. Mathematics of Biological Networks

25. Epidemic Models on Networks Sofar all approachesintroducedhaveassumed „fullmixing“ ofthepopulation. In thiscaseeach individual canpotentiallyhavecontactwithanyotherat a levelsufficienttotransmitthedisease. In the real world, however, thesetof a person‘scontactscanberepresentedas a network. The structureofthatnetworkcanhave a strong effect on theway a diseasespreadsthroughthepopulation. Mathematics of Biological Networks

26. Epidemic Models on Networks Wewill definethetransmission rate  (orinfection rate) astheprobability per unit time that an infected individual will transmitthediseaseto a susceptible individual towhom he/sheisconnectedby an edge in theappropriatenetwork. The transmission rate is a propertyof a particulardisease but also a propertyofthesocialandbehavioralparametersofthepopulation. Mathematics of Biological Networks

27. Epidemics on idealizednetworks Shown are 5 distinct network types containing 100 individuals. These are from left to right: random, lattice, small world (top row), spatial and scale-free (bottom row). In all 5 graphs, the average number of contacts per individual is approximately 4. Keeling & Eames, J R Soc Interface (2005) 2: 295–307. Mathematics of Biological Networks

28. Dynamic spreading on different networkarchitectures Typical SIR epidemics on the 5network types. → the square lattice (top, middle) shows the slowest dynamics → highly connected “hub” nodes accelerate spreading of disease Keeling & Eames, J R Soc Interface (2005) 2: 295–307. Mathematics of Biological Networks

29. Time-dependentpropertiesofEpidemic Networks An SI outbreakstartingwith a singlerandomlychosenvertexsomewhereeventuallyspreadsto all membersofthecomponentcontainingthatvertex. Letusassumethatvertexibelongstothegiantcomponent. Withprobabilitysi , vertexiissusceptible. Tobecomeinfected an individual must catch thediseasefrom a neighboring individual jthatisalreadyinfected. The probabilityforjbeinginfectedis The transmissionofthediseaseduringthe time intervaltandt + dt occurswithprobability. Mathematics of Biological Networks

30. Time-dependentpropertiesofEpidemic Networks Multiplying theseprobabilitiesandthensummingover all neighborsofi , yieldsthe total probabilityofibecominginfected: whereAijis an elementoftheadjacencymatrix. Thus, thesiobey a setofn non-linear differential equations =- Fromwegetthecomplementaryequationforxi. = We will assumeagainthatthediseasestartseitherwith a singlevertex ora smallrandomlyselectednumbercofvertices. Thus xi = c/n→ 0, si = 1 - c/n→ 1 in thelimitfor large n. Mathematics of Biological Networks

31. Time-dependentpropertiesofEpidemic Networks The equation=- is not solvable in closed form forgeneralAij. Byconsideringsuitablelimits, wecancalculatesomefeaturesofitsbehavior. Letus e.g. considerthebehaviorofthesystem at earlytimes. For large nandthegiven initial conditions, xi will besmall in thisregime. Byignoringtermsofquadraticorder, wecanapproximate or in matrix form wherexisthevectorwithelementsxi . Mathematics of Biological Networks

32. Time-dependentpropertiesofEpidemic Networks Write xas a linear combinationoftheeigenvectorsoftheadjacencymatrix wherevristheeigenvectorwitheigenvaluer . Then Bycomparingterms in vrweget This hasthesolution Substitutingthisexpression back gives The fastest growingterm in thisexpressionisthetermcorresponding tothelargesteigenvalue1. Mathematics of Biological Networks

33. Time-dependentpropertiesofEpidemic Networks Assuming thistermdominatesovertheotherswe will get So weexpectthenumberofinfectedindividualstogrowexponentially, just as in thefullymixedversionofthe SI model, but nowwith an exponentialconstantthatdepends not only on  but also on theleadingeigenvalueoftheadjacencymatrix. The probabilityofinfection in thisearlyperiodvariesfromvertextovertex roughlyasthecorrespondingelementoftheleadingeigenvectorv1 . In lecture V1, theelementsoftheleadingeigenvectoroftheadjacencymatrixweretermedtheeigenvectorcentrality. Thus eigenvectorcentralityis a crudemeasureoftheprobabilityofearlyinfectionof a vertex in an SI epidemicoutbreak. Mathematics of Biological Networks

34. Review (V1): Eigenvector Centrality This limitingvector of the eigenvector centralities is simply proportional to the leading eigenvector of the adjacency matrix. Equivalently, we could say that the centrality x satisfies A x = k1x This is the eigenvector centrality first proposed by Bonacich (1987). The centrality xi of vertex i is proportional to the sum of the centralities of its neighbors: This has the nice property that the centrality can be large either because a vertex has many neighbors or because it has important neighbors (or both). Mathematics of Biological Networks