Impact of the distribution of recovery rates on disease spreading in complex networks

PHYSICAL REVIEW RESEARCH 2, 013046 (2020) Impac o he dis ibu ion o eco e y a es on disease sp eading in complex ne wo ks Guilhe me Fe az de A uda,1Gio anni Pe i,1F ancisco A. Rod igues,2and Yami Mo eno3,1 1ISI Founda ion, Via Chisola 5, 10126 To ino, I aly 2Depa amen o de Ma emá ica Aplicada e Es a ís ica, Ins i u o de Ciências Ma emá icas e de Compu ação, Uni e sidade de São Paulo, Campus de São Ca los, Caixa Pos al 668, 13560-970 São Ca los, São Paulo, B azil 3Ins i u e o Biocompu a ion and Physics o Complex Sys ems and Depa men o Theo e ical Physics, Uni e si y o Za agoza, 50018 Za agoza, Spain (Recei ed 14 Feb ua y 2019; e ised manusc ip ecei ed 12 No embe 2019; published 14 Janua y 2020) We s udy a gene al epidemic model wi h a bi a y eco e y a e dis ibu ions. This simple de ia ion om he s anda d se up is su icien o p o e ha he e ogenei y in he dynamical pa ame e s can be as impo an as he mo e s udied s uc u al he e ogenei y. Ou analy ical solu ion is able o p edic he shi in he c i ical p ope ies induced by he e ogeneous eco e y a es. We ind ha he c i ical alue o in ec i i y ends o be smalle han he one p edic ed by quenched mean- ield app oaches in he homogeneous case and ha i can be linked o he a iance o he eco e y a es. Ou indings also illus a e he ole o dynamical-s uc u al co ela ions, whe e we allow a powe -law ne wo k o dynamically beha e as a homogeneous s uc u e by an app op ia e uning o i s eco e y a es. O e all, ou esul s demons a e ha he e ogenei y in he eco e y a es, e en ually in all dynamical pa ame e s, is as impo an as he s uc u al he e ogenei y. DOI: 10.1103/PhysRe Resea ch.2.013046 I. INTRODUCTION He e ogenei y, whe he in he na u e o he componen s o he pa e n o connec ions, is a key cha ac e is ic o com- plex sys ems. This is pa icula ly e iden in he case o he sp eading o a disease in a ne wo ked popula ion, whe e he inclusion o s uc u al he e ogenei y has long been known o adically change he p ocess’s c i ical beha io [1–6]. As an illus a ion, conside wo classical con agion models, he suscep ible-in ec ed-suscep ible (SIS) and he suscep ible- in ec ed- eco e ed (SIR) models. On a homogeneous ne - wo k, hey bo h p esen a non anishing c i ical poin [5,6]. Howe e , he in oduc ion o s uc u al he e ogenei y, in he o m o b oad deg ee dis ibu ions o he nodes, can esul in a anishing c i ical poin [1,5–7]. Mo e speci ically, in he he - modynamic limi , a di e gence o he second momen o he deg ee dis ibu ion [1,5,6,8] o a di e gence in he maximum deg ee [1,5,6,8] implies a anishing c i ical in ec i i y. This in u n has impo an p ac ical implica ions o eal-wo ld ne wo ks, because many o hem display e y b oad [7–9] (o e en scale- ee) deg ee dis ibu ions [10]. Re e ences [11–18] conside ed, in con as , homogeneous s uc u es bu accoun ed o a bi a y imes in he s a e ansi ions. In e es - ingly, p e ious wo ks s udied models cha ac e ized by a se ies o in ec ed s a es (wi hou a biological in e p e a ion) ha desc ibed he global beha io accu a ely [12–15,17]. No ably, he model p oposed in [12] is gene al, bu each ansi ion has Published by he Ame ican Physical Socie y unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s) and he published a icle’s i le, jou nal ci a ion, and DOI. an assigned unc ion, which complica es he model, making i s p ac ical use limi ed. These wo ks encoun e , howe e , a undamen al limi a ion, because hey desc ibe he popula ion a he mean- ield le el [11–18]. While s uc u al and dynami- cal he e ogenei ies a e independen ly accoun ed o , modeling bo h ypes o he e ogenei ies oge he has ecei ed consid- e ably less a en ion un il ecen ly. Indeed, i was mainly s udied o he SIR model: A message-passing o malism was p oposed in [19,20] and a he e ogeneous mean- ield app oach in [21]. In he la e , he au ho s also pe o med nume ical expe imen s showing ha he popula ion can be mo e ul- ne able in he scena io wi h dynamical he e ogenei y. Mo e ecen ly, his p oblem was in es iga ed on empo al ne wo ks [22] using an SIS p ocess, which was desc ibed wi hin he quenched mean- ield (QMF) o malism and mainly ocusing on sp eading a es [23,24]. A simila app oach was conside ed in [25], whe e howe e he au ho s highligh ed a di e en aspec , i.e., he alloca ion o esou ces du ing an ou b eak. He e we in es iga e a di e en ype o dynamical he e o- genei y by cha ac e izing he c i ical p ope ies o an SIS model when eco e y a es a e dis ibu ed he e ogeneously ac oss he popula ion. He e ogeneous eco e y a e dis ibu- ions can be associa ed wi h biological di e ences be ween in- di iduals [26,27], demog aphic cha ac e is ics [28], and social di e ences ha esul in nonhomogeneous access o he heal h sys em [29]. Addi ionally, we also conside he case in which co ela ions a ise be ween he s uc u e and dynamics. We show, bo h analy ically and nume ically, ha such co ela ions can induce opposing and unexpec ed dynamical ou comes, o example, powe -law (PL) ne wo ks displaying non an- ishing c i ical poin s and con e sely homogeneous ne wo ks displaying anishing c i ical poin s. Indeed, we show ha he s anda d QMF p edic ions, which a e a lowe bound in he s anda d scena io, no longe p o ide such a bound in he 2643-1564/2020/2(1)/013046(8) 013046-1 Published by he Ame ican Physical Socie y DE ARRUDA, PETRI, RODRIGUES, AND MORENO PHYSICAL REVIEW RESEARCH 2, 013046 (2020) he e ogeneous a e case. We hen p opose a simple o mu- la ion o o e come his and p o ide a di e en lowe bound o he p ocess. Ou esul s complemen p e ious e idence on he SIR model [21] and imply ha p ope cha ac e iza ion o he dynamical pa ame e s is o u mos impo ance no only o a be e unde s anding o sp eading p ocesses, bu also o many p ac ical applica ions, such as su eillance, o ecas ing, esou ce managemen , and ne wo k econs uc ion, among many o he s. II. THE SIS MODEL WITH HETEROGENEOUS RECOVERIES We s a by conside ing a popula ion composed o N indi iduals wi h an a bi a y pa e n o connec ions in a single connec ed componen . These connec ions can be ep esen ed as a ne wo k and a e desc ibed by he ne wo k’s (usually symme ic) adjacency ma ix A. Each indi idual can be in one o wo s a es: (i) in ec ed (Yi=1) o (ii) suscep ible (Xi=1). Using a Ma ko ian app oach, he epidemic p ocess is modeled as a collec ion o independen Poisson p ocesses. In o de o model he sp eading o he disease h ough he ne wo k o con ac s, wi h each di ec ed edge i∼j, emana ing om he in ec ed indi idual i, we associa e a Poisson p ocess wi h a e λij,Nλij( ) (whose ansi ions a e Yi+Xj→Yi+ Yj). Addi ionally, wi h each in ec ed indi idual, we associa e a Poisson p ocess wi h a e δi,Nδi( ), modeling he eco e y (Yi→Xi). This sys em is s a is ically desc ibed using he o de pa ame e ρand he suscep ibili y χ, de ined as ρ=1 N N  i Yi,χ=nI2−nI2 nI,(1) whe e nIis he numbe o in ec ed indi iduals. Bo h quan- i ies can be di ec ly es ima ed using Mon e Ca lo me hods, in pa icula , he quasis a iona y me hod and he Gillespie algo i hm, whe e each o he p ocesses men ioned abo e is simula ed, and he s a e o he nodes is e alua ed [6]. In he QMF app oach, one implici ly assumes ha XiYj≈ XiYj. Physically, his co esponds o neglec ing dynamical co ela ions. De ining yi=Yi, he p ocess is desc ibed as dyi d =−δiyi+(1 −yi) j λijAijyj.(2) Thus, de ining ii =δi,W=λij, and Q=−1(A◦W), he c i ical poin is gi en as λQMF c=[max(Q)]−1,(3) whe e max(Q) is he leading eigen alue o Q. No e ha he elemen s o Qa e he expec ed numbe o con ac s be o e eco e y. Ob iously, he c i ical poin simpli ies o τQMF,s d c= (λ δ)c=[max(A)]−1in he homogeneous case, i.e., when δi= δand λij =λ. Thus, in he he modynamic limi , he c i ical poin o PL ne wo ks goes o ze o i he maximum deg ee is a g owing unc ion o he ne wo k size [5,6,8]. On he o he hand, we can conside a scena io ha is equi alen o he con ac p ocess (CP) by se ing he sp eading a es as λij =λ ki, which is hus desc ibed by he p obabili y ansi ion ma ix Pij =Aij ki. In his case, he c i ical poin is ini e and 0.00 0.02 0.04 0.06 0.08 0.10 λ 10− 1 100 101 102 χ 2 3 4 5 10 100 1000 α 100101102 α 10−3 10−2 10− 1 λQMF c 0.0 0.1 λc 0.0 0.1 λQMF c FIG. 1. Suscep ibili y cu es ex ac ed om he Mon e Ca lo simula ions o an E d˝ os-Rényi ne wo k wi h N=105and k≈10 conside ing ha he eco e y a e dis ibu ion ollows an in e se- gamma dis ibu ion, whose shape pa ame e αis colo coded. The inse s show he dependence o λQMF con α( op) and a compa ison wi h nume ical esul s (bo om), also o di e en α. τQMF,CP c=1, ega dless o he unde lying s uc u e. No e ha , simila ly o he homogeneous case, his p edic ion [Eq. (3)] is an uppe bound o he he e ogeneous eco e y a e scena io, because i elies on he independence o he andom a iables: I i∼j, hen P(Yi=1|Yj=1) ⩾P(Yi=1) =yi, which im- plies ha he nodal p obabili y is always o e es ima ed (see [2] o a simila a gumen ). F om he e onwa d, we se λij =λ and ocus on he e ec o he eco e y a e dis ibu ion on he c i ical poin . Conside ing an undi ec ed ne wo k, om he ma ix no m we can bound Eq. (3) using he s anda d QMF p edic ions as min(δi)τQMF,s d c⩽λQMF c⩽max(δi)τQMF,s d c.(4) We can see ha Eq. (4) sugges s ha he s anda d QMF p edic ions migh no be a lowe bound o he al e na i e p ocess. No e ha he unce ain y assuming he s anda d QMF p edic ion inc eases as he a iance also inc eases. III. SYNTHETIC NETWORKS To u he cha ac e ize he c i ical beha io o ou model, we i s conside an E d˝ os-Rényi (ER) ne wo k wi h N=105 and k≈10 ( he e o e τQMF,s d c≈0.1). This g aph has a homogeneous s uc u e and allows us o analyze he s uc- u al and dynamical e ec s independen ly. Moun ing e idence shows ha in ec ious imes in eal epidemics ollow a gamma dis ibu ion [17,30,31]. Consequen ly, he a e dis ibu ion mus ollow an in e se-gamma dis ibu ion. We impose he e- o e he eco e y a es o ha e an in e se-gamma dis ibu ion δ∼−1(α,β), whe e αand βa e he shape and scale pa am- e e s, espec i ely. I s mean is δi= β α−1and i s a iance is Va (δi)=β2 (α−1)2(α−2) o α>2. To acili a e he compa ison be ween di e en dis ibu ions, we es ic he dis ibu ions o uni a y mean. In Fig. 1we p esen he c i ical beha io o an ER ne wo k o di e en shapes α. The op inse emphasizes he beha io o he p edic ed c i ical poin as a unc ion o αand i s compa ison wi h es ima ions om Mon e Ca lo 013046-2 IMPACT OF THE DISTRIBUTION OF RECOVERY RATES … PHYSICAL REVIEW RESEARCH 2, 013046 (2020) simula ions. As expec ed, o su icien ly la ge alues o α, he dynamics beha e simila ly o he s anda d SIS model wi h uni o m δ, whe e he p edic ed h eshold coincides (dashed line in he op inse ). The ag eemen be ween analy ical and simula ed c i ical poin s is excellen , as i can be seen in he bo om inse o Fig. 1. Fu he mo e, as αdec eases, he a iance o δand consequen ly he unce ain y o he bounds a e also enla ged. Howe e , we obse ed ha he c i ical poin sys ema ically mo ed owa ds ze o in ou simula ions, which is also consis en wi h Eq. (4). No e ha he in e se-gamma dis ibu ion is asymme ic wi h espec o i s a e age and cen- e ed a δi⩽δi. This implies ha he bounds in Eq. (4)a e also asymme ic. Since he eco e y a es a e sampled om his dis ibu ion, we expec ha he numbe o indi iduals ha ake longe o eco e is g ea e han he numbe o indi iduals ha eco e as . This sugges s ha , ia in ec ion/ ein ec ion mechanisms, he disease can su i e o lowe alues o λ as compa ed o he s anda d QMF p edic ions. In addi ion o he c i ical p ope ies, we also show a di e en supe c i ical beha io ; o example, he low-α egime o a ne wo k can be simila o, and e en in some egions i can be mis aken o , he high-α egime on op o a ne wo k wi h a di e en s uc u e. This mul iplici y o supe c i ical beha io s also aises ques ions in ne wo k econs uc ion models based on disease dynamics. IV. REAL-WORLD NETWORKS We con i med ha ou esul s hold in eal-wo ld ne wo ks. As be o e, we conside an in e se-gamma dis ibu ion. Fig- u e 2shows esul s o simula ions in wo eal ne wo ks: he UC I ine messages social ne wo k [Fig. 2(a)][32,33] and he OpenFligh s ne wo k [Fig. 2(b)][33,34]. These ne wo ks ep- esen di e en spa ial scales o a simila sp eading p ocess: The social ne wo k co esponds o smalle scales and spa ially localized sys ems, while he OpenFligh s ne wo k cap u es a wide spa ial scale. In he op inse o Fig. 2(b) we show ha he c i ical poin p edic ions a e ema kably good o in e se-gamma eco e y a es, e en o hese eal ne wo ks. Figu es 1and 2 e eal ha he c i ical poin dec eases as we inc ease he a iance o he eco e y a e dis ibu ion. Thus, i we conside he e ogeneous ne wo ks, assuming an a e age eco e y a e in he QMF is no enough o p o ide an ade- qua e cha ac e iza ion o he p ocess. I is no a lowe bound anymo e, as can be seen in Fig. 1, whe e he nume ically es ima ed c i ical poin is always sligh ly lowe han he alue p edic ed by he s anda d QMF p edic ions (τQMF,s d c≈0.1). The p ope co ec ion o he QMF p edic ions is gi en by Eq. (3), which is a lowe bound o he unde lying p ocess (see he bo om inse in Fig. 1). V. EFFECTS OF DYNAMICS-STRUCTURE CORRELATIONS The bounds in Eq. (4) implici ly assume ha he e a e no co ela ions be ween he s uc u e and dynamics. F om he Ge shgo in ci cle heo em we know ha e e y eigen- alue o Qlies a leas in one o he disks D(Qii,Ri) cen- e ed in Qii wi h a adius gi en as Ri=i= j|Qij|. The e- o e, conside ing a symme ic ma ix, |k|⩽Qii +Rk, hence 0.000 0.005 0.010 0.015 0.020 0.025 0.030 λ 10−1 100 101 χ (a) 0.000 0.005 0.010 0.015 0.020 λ 10−1 100 101 χ (b) 0.00 0.02 λ 0.0 0.1 0.2 ρ 0.00 0.02 λ 0.00 0.05 0.10 ρ 2 3 4 5 10 100 α 0.01 0.02 λc 0.01 0.02 λQMF c FIG. 2. Resul s o he suscep ibili y when he dynamics a e on op o eal ne wo ks. We use (a) he UC I ine messages social ne wo k [32,33](•) and (b) he OpenFligh s ne wo k [33,34](). In bo h cases, we conside ed he undi ec ed e sion o he gian compo- nen . In he main igu e o each panel, we p esen he suscep ibili y o di e en alues o αand λ. In he bo om igh inse s, we p esen he o de pa ame e . In he op inse o (b), we p esen a compa ison be ween he QMF es ima ed and p edic ed c i ical poin s. max ⩽Q∞, whe e he in ini y no m is de ined as Q∞=max 1⩽i⩽N⎛ ⎝ N  j=1 Aij δi⎞ ⎠=max 1⩽i⩽Nki δi.(5) I he s uc u e and he dynamics a e co ela ed, Eq. (5) migh gi e us u he insigh . Fo ins ance, o he PL case, he leading eigen alue o Adi e ges in he he modynamic limi , leading o a anishing c i ical poin . Con e sely, using Eq. (5) and a p ope choice o δi, we can change his beha io . In ac , assuming ha δi(ki)∝kiin he he modynamic limi , we ha e lim N→∞ Q∞=lim N→∞ max 1⩽i⩽Nki δi=c,(6) whe e c<∞is a ini e eal cons an . This adically changes he c i ical beha io o he dynamics. No e ha bo h he CP (λij =λ ki) and he δi=kicases a e desc ibed, a i s o de , by he p obabili y ansi ion ma ix P, yielding τQMF,CP c= λQMF c=1. 013046-3 DE ARRUDA, PETRI, RODRIGUES, AND MORENO PHYSICAL REVIEW RESEARCH 2, 013046 (2020) 1.0 1.2 1.4 1.6 1.8 λ 100 101 102 χ (a) γ=3.5γ=2.7γ=2.1 N=10 3 N=10 4 N=10 5 N=10 6 0.1 0.2 0.3 0.4 0.5 λ 10−1 100 101 102 χ (b) γ=3.5γ=2.7γ=2.1 103104105106 N 0.2 0.3 0.4 λc FIG. 3. Fini e-size analysis conside ing s uc u e-dynamics co - ela ions. We p esen he suscep ibili y (colo s ep esen he sizes) in (a) he he e ogeneous eco e y a es, whe e δi=kion op o he powe -law ne wo ks, whose exponen s a e γ=2.1,2.7,3.5, and (b) he SIS model wi h he e ogeneous eco e y a es, conside ing an E d˝ os-Rényi ne wo k and he eco e y a es as δi=ki kPL ,whe ekPL is a disc e e powe law, mo e speci ically, he deg ee sequence o he ne wo ks used in (a). In he inse we p esen he c i ical poin as a unc ion o he sys em size in log-log scale. In Fig. 3(a) we pe o m a ini e-size analysis when he dynamics occu s on op o PL ne wo ks and eco e y a es a e δi=ki, inding e idence o a ini e c i ical poin , i.e., he alue o λco esponding o he peak o χdoes no anish. Fo compa ison, he esul s o he CP a e epo ed in Appendix E. We ema k ha he con e gence owa ds he c i ical alue o g owing Nseems o be slowe in he CP case. In summa y, ou esul s show ha a ne wo k wi h a powe -law deg ee dis ibu ion may show a ini e c i ical poin o he SIS dynamics i deg ees and eco e y a es a e app op ia ely co ela ed. Nex we show ha he in e se scena io is also possible. Conside an ER ne wo k wi h k≈10 and δi=ki kPL , whe e kPL is a disc e e powe law, i.e., P(kPL)∼k−γ PL . Tha is, we now ha e a homogeneous s uc u e and a he e ogeneous eco e y a e dis ibu ion. In Fig. 3(b) we show a ini e-size analysis o his con igu a ion wi h a ying γ=2.1,2.7,3.5. We obse e ha o γ=2.7 and 3.5 he unde lying s uc u e plays an impo an ole, sugges ing a non anishing c i ical poin o conside able educ ion in he scaling exponen [see Fig. 3(b) inse , whe e bo h cu es ha e a slope close o ze o]. Howe e , o γ=2.1 ou esul s indica e he exis ence o a anishing c i ical poin [see Fig. 3(b) inse ]. I seems easonable o hypo hesize ha he scena io obse ed when γ=2.1 is due o he ac ha , in he s eady s a e, he in- ec ion p obabili ies a e in e sely p opo ional o he nodal eco e y a es and hus ha he e alua ion o he eco e y ime a bo h ends o e e y edge enables an in ec ion- ein ec ion mechanism. Unde s anding wha he necessa y and su icien condi ions o obse e such a mechanism a e and he in e play be ween s uc u e and dynamics equi es, howe e , u he explo a ion. In [8,35,36] i was shown ha , depending on he ne wo k s uc u e, he ansi ion can be igge ed by di e en ac i a ion mechanisms, namely, (i) collec i e (e.g., ER ne wo ks o he CP), (ii) kco e (unco ela ed powe law wi h 2 <γ <2.5), o (iii) hub (unco ela ed powe law wi h γ>2.5). No e ha o unco ela ed powe laws, his coincides wi h he di e - en egimes o leading eigen ec o localiza ion [37][ o he analysis o he in e se pa icipa ion a io (IPR) o he leading eigen ec o o Qwe e e he eade o Appendix F]. Thus, we conjec u e ha i may be possible o al e hese s uc u al mechanisms wi h a p ope eco e y a e dis ibu ion, pos- sibly conside ing dynamics-s uc u e co ela ions. No e ha we implici ly showed [Fig. 3(a)] how o ans o m k-co e (γ=2.1) and hub-ac i a ed (γ=2.7,3.5) mechanisms in o a collec i e beha io phenomenology. Impo an ly, ou indings may lead o al e na i e p e en ion/in e en ion echniques ha ake ad an age o he phenomenology epo ed he e. We also highligh ha he consequences o ou indings a e no limi ed o he cases explo ed he e. Fo ins ance, i is na u al o conjec u e he exis ence o G i i h’s phase in ou se up. In his ype o ansi ion, we ha e an ex ended c i ical egion ins ead o a single c i ical poin , which was s udied in complex ne wo ks in [36,38–42]. Mo e speci ically, in [39] he au ho s showed ha slow dynamics on a weighed eelike s uc u e can occu in a con ac p ocess. I s simila i ies o ou scena io hus sugges ha simila phenomenology is also possible in ou case. VI. CONCLUSION We ha e analyzed he impac o in oducing he e ogenei y in he eco e y a es o an SIS disease dynamics. We showed ha dynamical he e ogenei y is as signi ican as s uc u al he e ogenei y and ha i can induce d as ic changes in he SIS c i ical p ope ies. Fu he mo e, ou esul s show ha he s an- da d QMF app oach does no p o ide a lowe bound o he he e ogeneous case anymo e. To sol e his inconsis ency, we p oposed a solu ion ha ela es he s uc u al and dynamical ea u es ia he spec al p ope ies o a di e en ma ix Q. This o mula ion p esen s oppo uni ies o u u e esea ch. Fo example, ou indings aise ques ions abou he conse- quences o he e ogenei ies in sp eading a es and he in e play be ween sp eading and eco e y a es. Fu he mo e, wi h e- ga d o con ol/con ainmen s a egies, he e ogenei y in he eco e y a es can be conside ed o in e en ion s a egies (o e en immuniza ion, δi→0). In con as , he e ogenei y o sp eading a es would be ela ed o p e en ion. In his con ex , 013046-4 IMPACT OF THE DISTRIBUTION OF RECOVERY RATES … PHYSICAL REVIEW RESEARCH 2, 013046 (2020) ou indings migh also be o po en ial in e es . In addi ion o he speci ic conclusions d awn he e, he e a e o he s ha conce n mo e gene al aspec s o disease sp eading p ocesses as well as he cha ac e iza ion o complex sys ems in gene al. Fo ins ance, di e ences in he localiza ion p ope ies induced by dynamical he e ogenei ies migh in luence he p edic abil- i y o complex sys ems, and in pa icula o diseases [43], o in he econs uc ion o ne wo ks om he dynamics as p oposed in [44]. Finally, we also s ess ha ou esul s migh also ha e an impac on in o ma ion sp eading p ocesses since indi iduals ha e di e en ac i i y imescales, which could ul ima ely be ela ed o he eco e y ime dis ibu ion. ACKNOWLEDGMENTS G.F.d.A., G.P., and Y.M. acknowledge pa ial suppo om In esa Sanpaolo Inno a ion Cen e . Y.M. acknowledges pa - ial suppo om he Go e nmen o A agón, Spain h ough G an No. E36-17R and om MINECO and FEDER unds (G an No. FIS2017-87519-P). Resea ch was ca ied ou using he compu a ional esou ces o he Cen e o Ma hema ical Sciences Applied o Indus y unded by FAPESP (G an No. 2013/07375-0). APPENDIX A: EQUATION (4) AS A BOUND FOR EQ. (3) The c i ical poin gi en by Eq. (3) can be bounded using a ma ix no m. Thus, using he 2-no m, we ob ain A2 max()⩽−1A2⩽A2 min(),(A1) whe e A2=max(ATA). Equa ion (A1) he e o e p o- ides bounds on he leading eigen alue o Qgi en by he s uc u e and he a iance o δi. Nex , o undi ec ed ne wo ks we ha e ha A2=max(A)=(τQMF c)−1. Mo eo e , he Pe on-F obenius heo em o non-nega i e ma ices s a es ha e e y non-nega i e ma ix can be w i en as a limi o posi i e ma ices, and he e o e one has he exis ence o an eigen ec o wi h non-nega i e componen s and he co e- sponding eigen alue will be non-nega i e and g ea e han o equal (in absolu e alue) o all o he eigen alues [45]. The e o e, om he no m de ini ion, Q2=σmax(Q)=max i[i(Q)] (A2) since Qis non-nega i e. Finally, in e ing he equa ions and w i ing i in e ms o he c i ical p ope ies, we ob ain Eq. (4). APPENDIX B: ANALYZING EQ. (4) FOR THE INVERSE-GAMMA DISTRIBUTION The bounds ob ained in Eq. (4)o (A1) canno be u he imp o ed wi hou knowledge abou he ela ionship be ween he ma ix Ao Wand −1[ o mo e de ails abou he e ec s o dynamics-s uc u e co ela ions, see Eq. (5)]. Al e na i ely, one can use he asymme ies in he eco e y a e dis ibu ion o unde s and how he lowe and uppe bounds beha e. Mo e speci ically, he cumula i e dis ibu ion o he in e se-gamma 101102103104 0.3 0.4 0.5 0.6 0.7 α δ≤ P ( 1) δ > P ( 1) P FIG. 4. E alua ion o P(δ⩽1) and P(δ>1) as a unc ion o α, he shape pa ame e o an in e se-gamma dis ibu ion wi h β= α−1. dis ibu ion is gi en as P(δ⩽x)=α, β x (α)=Qα, β x,(B1) whe e (a,b) is he uppe incomple e Gamma unc ion, (a) is he Gamma unc ion, and Q(a,b) is he egula ized gamma unc ion. Nex , keeping he a e age ixed, i.e., se ing β= α−1, we can analyze how hese bounds beha e as a unc ion o he shape α. We can calcula e he p obabili y ha δis la ge o smalle han a gi e alue x. Fo mally, P(δ⩽x)=Qα, α−1 x,(B2) P(δ>x)=1−Qα, α−1 x.(B3) Nex , e alua ing (B2) and (B3) o he a e age alue δi= x=1, we ob ain he ac ion o indi iduals ha ha e a eco e y a e ha is lowe and highe , espec i ely, han a e age. Since Q(α, α−1 x)⩾1 2 he in e se-gamma dis ibu ion is peaked below he a e age alue, he e o e sugges ing ha he lowe bound dec eases as e han he uppe bound. In- e es ingly, α→∞implies ha P(δ⩽1) =P(δ>1) =1 2, and hus he unce ain y is symme ical. This is also depic ed in Fig. 4. We ema k ha he discussion o his Appendix conce ns he unce ain y o he bounds and hei asymme y, which only sugges s (bu does no p o e) ha he c i ical poin migh dec ease. This hypo hesis is suppo ed by ou simula ions. APPENDIX C: UNIFORM SYMMETRIC RECOVERY RATE DISTRIBUTION In he main ex we ocused on he in e se-gamma dis i- bu ion. This dis ibu ion is asymme ic. In his Appendix we ocus on a symme ic scena io conside ing he uni o m dis- ibu ion U(a,2−a) whose a e age is δi=1 and a iance is Va (δi)=(1−a)2 3. We ema k ha a no mal dis ibu ion was no conside ed since i is de ined o nega i e numbe s, while a es a e posi i e. Figu e 5shows he c i ical poin es ima ions 013046-5 DE ARRUDA, PETRI, RODRIGUES, AND MORENO PHYSICAL REVIEW RESEARCH 2, 013046 (2020) 0.00 0.02 0.04 0.06 0.08 0.10 λ 10− 1 100 101 102 χ 0.01 0.1 0.2 0.4 0.6 0.8 1.0 0.0 0.1 λc 0.0 0.1 λQMF c a FIG. 5. Suscep ibili y cu es ex ac ed om he Mon e Ca lo simula ions o an E d˝ os-Rényi ne wo k wi h N=105and k≈10 conside ing ha he a e dis ibu ion ollows a uni o m dis ibu ion (symme ic) U(a,2−a), whe e he pa ame e acon ols he a i- ance and is colo coded. in his scena io. The same conclusions we ob ained o Fig. 1 also apply he e. The same end is obse ed, in which he c i ical poin mo es o he le . The main obse ed di e ence is ha , in his case, he c i ical p edic ions a e poo e when he a iance is la ge . Al hough he e o is la ge in hese cases, ou p edic ions a e s ill be e han he s anda d QMF o mula ion. APPENDIX D: SUPERCRITICAL BEHAVIOR Aside om he c i ical beha io , discussed in he main ex , i is also ins uc i e o e alua e he supe c i ical beha io o ou dynamics. Figu e 6shows he phase diag am o E d˝ os- Rényi and PL ne wo ks wi h γ=2.1,2.7,3.5. I is clea ha no only he c i ical bu also he supe c i ical beha io changes o di e en alues o α. No e ha , in some cases, he lowe -α egime is simila o he highe -α egime o a di e en s uc u e. This is pa icula ly clea o he PL case o γ=3.5 and α=2.0, which is e y simila o γ=2.7 and α→∞. A simila e ec is also isible o o he pa ame e s. This is a pa icula ly impo an issue i one uses he dynamical p ocess o in e he ne wo k s uc u e [44]. Fu he mo e, o su icien ly la ge alues o λall he cu es end o con e ge; howe e , his con e gence is a guably slowe o PL ne wo ks, as emphasized in he inse o Fig. 6. APPENDIX E: CONTACT PROCESS In he con inuous o mula ion, he CP is modeled as a se o Poisson p ocesses, whe e each edge has he sp eading a e de ined as λij =λ ki. The eco e y a e is kep ixed o all nodes. In his dynamical p ocess, he a e age numbe o con ac s o each node is he same o all indi iduals in he ne wo k ega dless o hei deg ee. This p ocess is desc ibed by he p obabili y ansi ion ma ix P, hence τQMF,CP c=1. In Fig. 7we can obse e ha he c i ical poin p edic ions a e no as accu a e as o he p e ious case we s udied in he main ex [Fig. 3(a)]. This was an icipa ed in [46,47]. In e es ingly, a be e p edic ion o he c i ical poin in he CP is ob ained using he he e ogeneous mean ield, which (a) (b) FIG. 6. Phase diag am o (a) E d˝ os-Rényi (•)ne wo ksand (b) PL ne wo ks. In (b), om op o bo om, we ha e γ=2.1(), γ=2.7(), and γ=3.5(). In he inse in (a) we p esen he beha io o la ge λ’s in all he ne wo ks. They comp ise N=104 nodes. The eco e y a e dis ibu ion is gi en as an in e se-gamma dis ibu ion, whose shape pa ame e αis colo coded. p edic s τHMF, CP c=k k−1[47]. Thus, he misma ch be ween p edic ion and es ima ed c i ical poin s, in bo h cases, seems o be ela ed o dynamical co ela ions. Due o i s simila i ies, i is ins uc i e o compa e his model wi h ou dynamic-s uc u e co ela ed model, i.e., δi= ki.InFig.7we show he ini e-size analysis o he CP case on op o powe -law ne wo ks wi h exponen s γ=2.1,2.7,3.5. We ema k ha his p ocess has a ini e c i ical poin . Thus, compa ing wi h Fig. 3(a), we obse e ha bo h p ocesses p esen a simila beha io as a unc ion on N. Howe e , no e ha bo h models ha e sligh ly di e en es ima ed c i ical poin s. Aside om ha , ou esul s also sugges ha he con e gence o he asymp o ic alue seems o be as e in he CP case. We can hus conclude ha al hough bo h p ocesses ha e he same p edic ions o he c i ical poin , he second- o de e ec s a e di e en in bo h o hem. APPENDIX F: LOCALIZATION PROPERTIES On op o some s uc u es and jus abo e he c i ical poin , he mean- ield heo y p edic s ha he disease can be es ic ed (localized) in a subg aph [3,48]. This phenomenon is known as me as able localiza ion. In he s anda d scena io, his p ope y depends only on he s uc u e, mo e speci ically, 013046-6 IMPACT OF THE DISTRIBUTION OF RECOVERY RATES … PHYSICAL REVIEW RESEARCH 2, 013046 (2020) 100 101 102 χ γ=3.5γ=2.7γ=2.1 (a) 1.0 1.2 1.4 1.6 1.8 λ 10−5 10−4 10−3 10−2 10−1 ρ (b) N=10 3 N=10 4 N=10 5 N=10 6 FIG. 7. Fini e-size analysis o he CP o (a) suscep ibili y and (b) he o de pa ame e . The colo s ep esen he sizes and om bo om o op, he cu es a e g ouped by he powe -law exponen s γ=2.1,2.7,3.5, espec i ely. on he leading eigen ec o o A. This p ope y is e y obus wi h espec o he sp eading mechanism. In [48] he au ho s showed ha bu s s o in ec ion a e no su icien o u n a localized sp eading in o a delocalized sp eading. In e es ingly, in ou model, we can show ha he eco e y a es a e enough o al e he localiza ion. We ema k ha he con ol o he lo- caliza ion o diseases is s ill an open and in e es ing p oblem, as also poin ed ou in [48]. As an icipa ed, ou model plays an impo an ole in he localiza ion o he leading eigen ec o . Mo e speci ically, he ma ix Qmigh p esen a di e en localiza ion pa e n i compa ed o he adjacency ma ix. To quan i y localiza ion we use he so-called in e se pa icipa ion a io, ini ially used in epidemics in [3]. Fo mally, i is gi en as IPR = N  i 4 i,(F1) FIG. 8. The IPR e sus N o he E d˝ os-Rényi, PL ne wo ks and he co ela ed case −1A, whe e i=ki kPL o (a) γ=2.1and (b) γ=3.5. Fi y ne wo ks we e conside ed in each case. whe e is he no malized leading eigen ec o , i.e.,  = 1. As shown in [37] and ecen ly o malized in [48], he eigen ec o is localized in a subex ensi e po ion, i.e., IPR ∼ O(N−ν) wi h 0 <ν<1. Thus, in he ully delocalized case, IPR ∼O(N−1), and in he ully localized case, IPR ∼O(1). As expec ed, Fig. 8shows ha in an ER ne wo k he IPR scales wi h ν≈1, ν<1 o a PL ne wo k wi h γ=2.1 [Fig. 8(a)], and a PL ne wo k wi h γ=3.5 is localized [Fig. 8(b)]. Complemen a ily, when δi=ki, he leading eigen- ec o o P=Qis homogeneously dis ibu ed; he e o e IPR ∼O(N−1) and he disease is ully delocalized, which sugges s ha his is an e ec i e s a egy o ull delocaliza- ion. Fu he mo e, Fig. 8also shows ha when he e ogeneous eco e y a es a e conside ed, he IPR migh also change. I is possible ha a ne wo k ha exhibi s ull delocaliza ion in he s anda d case (ligh pink ci cles o an ER ne wo k in Fig. 8) will ins ead exhibi localiza ion when he eco e y a es and s uc u e a e co ela ed (blue iangles in Fig. 8). Thus, by in oducing localiza ion and/o uning, i s scaling exponen νis no i ial. Mo e speci ically, one migh nai ely use he eco e y a es as δi=ki kPL as a s a egy. In his case, kiwould delocalize he ma ix, while kPL would con ol he new IPR owa ds he dis ibu ion kPL. To e alua e his s a egy, we ixed he s uc u e o an ER ne wo k. Nex we de ined kPL as a PL deg ee dis ibu ion and op ed o a disc e e app oach o kPL, which allowed o a compa ison wi h he localiza ion p ope ies o a ne wo k wi h he same deg ee dis ibu ion, i.e., kPL. Howe e , as we obse ed in Fig. 8, his was insu icien o con ol localiza ion. In he e alua ed cases, he esul ing ma ix Qis localized. No e ha kPL ∼k−2.1is pa ially delo- calized [da k pink squa es in Fig. 8(a)], bu i s composi ion wi h he delocalized s uc u e esul s in a localized IPR. The case o kPL ∼k−3.5is less su p ising, bu also emphasizes he ole o he e ogenei y in he eco e y imes in disease localiza ion. The e o e, wi h his simple s a egy, we could no con ol he localiza ion o an a bi a y ν. [1] R. Pas o -Sa o as and A. Vespignani, Phys. Re . Le . 86,3200 (2001). [2] P. V. Mieghem, J. Omic, and R. Kooij, IEEE/ACM T ans. Ne w. 17,1(2009). [3] A. V. Gol se , S. N. Do ogo se , J. G. Oli ei a, and J. F. F. Mendes, Phys. Re . Le . 109,128702 (2012). [4] G. F. de A uda, E. Cozzo, T. P. Peixo o, F. A. Rod igues, and Y. Mo eno, Phys. Re . X 7,011014 (2017). 013046-7 DE ARRUDA, PETRI, RODRIGUES, AND MORENO PHYSICAL REVIEW RESEARCH 2, 013046 (2020) [5] R. Pas o -Sa o as, C. Cas ellano, P. Van Mieghem, and A. Vespignani, Re . Mod. Phys. 87,925 (2015). [6] G. F. de A uda, F. A. Rod igues, and Y. Mo eno, Phys. Rep. 756,1(2018). [7] S. Cha e jee and R. Du e , Ann. P obab. 37,2332 (2009). [8] M. Boguñá, C. Cas ellano, and R. Pas o -Sa o as, Phys. Re . Le . 111,068701 (2013). [9] A. Clause , C. Shalizi, and M. Newman, SIAM Re . 51,661 (2009). [10] I. Voi alo , P. an de Hoo n, R. an de Ho s ad, and D. K iouko , Phys. Re . Res. 1,033034 (2019). [11] H. W. He hco e, H. W. S ech, and P. an den D iessche, J. Ma h. Biol. 13,185 (1981). [12] Z. Feng and H. R. Thieme, SIAM J. Appl. Ma h. 61,803 (2000). [13] Z. Feng and H. R. Thieme, SIAM J. Appl. Ma h. 61,983 (2000). [14] A. L. Lloyd, P oc.R.Soc.B268,985 (2001). [15] A. L. Lloyd, Theo . Popul. Biol. 60,59 (2001). [16] Z. Feng, D. Xu, and H. Zhao, Bull. Ma h. Biol. 69,1511 (2007). [17] O. K ylo a and D. J. D. Ea n, J. R. Soc. In e ace 10,20130098 (2013). [18] D. Clancy, S a . P obab. Le . 85,1(2014). [19] B. Ka e and M. E. J. Newman, Phys. Re . E 82,016101 (2010). [20] N. She bo ne, J. C. Mille , K. B. Blyuss, and I. Z. Kiss, J. Ma h. Biol. 76,755 (2018). [21] W. Gou and Z. Jin, In ec . Dis. Model. 2,353 (2017). [22] A. Da bon, D. Colombi, E. Valdano, L. Sa ini, A. Gio annini, and V. Colizza, R. Soc. Open Sci. 6,181404 (2019). [23] B. Qu and H. Wang, IEEE T ans. Ne w. Sci. Eng. 4,177 (2017). [24] B. Qu and H. Wang, Physica A 472,13 (2017). [25] S. O a iano, F. De Pelleg ini, S. Bonacco si, and P. Van Mieghem, J. Complex Ne w. 6,800 (2017). [26] S. Segal and A. V. Hill, T ends Mic obiol. 11,445 (2003). [27] H. R. F ye , J. F a e , A. Duda, M. G. Robe s, The SPARTAC T ial In es iga o s, R. E. Phillips, and A. R. McLean, PLOS Pa hogens 6,e1001196 (2010). [28] S. Do jee, Z. Poljak, C. W. Re ie, J. B idgland, B. McNab, E. Lege , and J. Sanchez, Zoonoses Public Heal h 60,383 (2013). [29] R. M. Ba be , N. Fullman, R. J. D. So ensen, T. Bollyky, M. McKee, E. Nol e, A. A. Abajobi , K. H. Aba e, C. Abba a i, K. M. Abbas e al.,Lance 390,231 (2017). [30] M. Eichne and K. Die z, Am. J. Epidemiol. 158,110 (2003). [31] H. J. Wea ing, P. Rohani, and M. J. Keeling, PLOS Med. 2,e174 (2005). [32] T. Opsahl and P. Panza asa, Social Ne w. 31,155 (2009). [33] J. Kunegis, P oceedings o he 22nd In e na ional Con e ence on Wo ld Wide Web, Rio de Janei o, 2013 (ACM, New Yo k, 2013), pp. 1343–1350. [34] OpenFligh s ne wo k da a se , KONECT, 2016. [35] C. Cas ellano and R. Pas o -Sa o as, Sci. Rep. 2,371 (2012). [36] W. Co a, A. S. Ma a, and S. C. Fe ei a, Phys.Re .E98,012310 (2018). [37] R. Pas o -Sa o as and C. Cas ellano, Sci. Rep. 6,18847 (2016). [38] M. A. Muñoz, R. Juhász, C. Cas ellano, and G. Ódo , Phys. Re . Le . 105,128701 (2010). [39] G. Ódo and R. Pas o -Sa o as, Phys.Re .E86,026117 (2012). [40] R. Juhász, G. Ódo , C. Cas ellano, and M. A. Muñoz, Phys. Re . E85,066125 (2012). [41] H. K. Lee, P.-S. Shim, and J. D. Noh, Phys. Re . E 87,062812 (2013). [42] G. Ódo , R. Dickman, and G. Ódo , Sci. Rep. 5,14451 (2015). [43] S. V. Sca pino and G. Pe i, Na . Commun. 10,898 (2019). [44] T. P. Peixo o, Phys. Re . Le . 123,128301 (2019). [45] Ma ix Analysis and Applied Linea Algeb a,edi edby C. D. Meye (Socie y o Indus ial and Applied Ma hema ics, Philadelphia, 2000). [46] S. C. Fe ei a, R. S. Fe ei a, and R. Pas o -Sa o as, Phys. Re . E83,066113 (2011). [47] A. S. Ma a, R. S. Fe ei a, and S. C. Fe ei a, New J. Phys. 16, 053006 (2014). [48] Q. Liu and P. Van Mieghem, IEEE T ans. Ne w. Sci. Eng. 6,983 (2019). 013046-8