PROBABILITY AND MATHEMATICAL STATISTICS Vol. 37, Fasc. 2 (2017), pp. 000 000 doi:10.19195/0208-4147.37.2.2 THE SI AND SIR EPIDEMICS ON GENERAL NETWORKS BY DAVID A L D O U S (BERKELEY) Dedicated to Tomasz i hoor of a distiguished career Abstract. Ituitively oe expects that for ay plausible parametric epidemic model, there will be some regio i parameter-space where the epidemic affects (with high probability) oly a small proportio of a large populatio, aother regio where it affects (with high probability) a oegligible proportio, with a lower-dimesioal critical iterface. This dichotomy is certaily true i well-studied specific models, but we kow of o very geeral results. A recet result stated for a bod percolatio model ca be restated as givig weak coditios uder which the dichotomy holds for a SI epidemic model o arbitrary fiite etworks. This result suggests a cojecture for more complex ad more realistic SIR epidemic models, ad the purpose of this article is to record the cojecture. 2010 AMS Mathematics Subject Classificatio: Primary: 60K35; Secodary: 92D60. Key words ad phrases: SI epidemic, SIR epidemic. 1. A BOND PERCOLATION RESULT We start by repeatig almost verbatim the statemet of the mai result of the paper [1]. Take a fiite coected graph (V, E) with edge-weights w = (w e ), where w e > 0 for all e E. To the edges e E attach idepedet expoetial (rate w e ) radom variables ξ e. I the laguage of percolatio theory, say that edge e becomes ope at time ξ e. The set of ope edges at time t costitutes a radom graph G(t), ad i particular determies a radom partitio of V ito the coected compoets of G(t); write C(t) for the largest umber of vertices i ay such coected compoet. Now cosider a sequece (V, E ) of such weighted graphs, Research supported by NSF Grat DMS-1504802. Based o a talk at the workshop Stochastic models of the spread of disease ad iformatio o etworks, ICMS, Ediburgh, July 2016.
2 D. Aldous where both the graph topologies ad the edge-weights are arbitrary subject oly to the coditios that V ad that for some 0 < t 1 < t 2 < (1.1) lim EC (t 1 )/ V = 0, lim if EC (t 2 )/ V > 0. I the laguage of radom graph theory, this coditio says that a giat compoet emerges (with o-vaishig probability) sometime betwee t 1 ad t 2. The followig propositio asserts, iformally, that the icipiet time at which the giat compoet starts to emerge is determiistic to first order. PROPOSITION 1.1 (Aldous [1]). Give a sequece of graphs satisfyig (1.1), there exists a determiistic sequece τ [t 1, t 2 ] such that, for every sequece ε 0 sufficietly slowly, the radom times T := if{t : C (t) ε V } satisfy T τ p 0. 2. REFORMULATION AS AN SI EPIDEMIC MODEL Mathematical modelig of epidemics has a log history ad a large literature, ad relevat issues will be briefly idicated i Sectio 3. A SI model refers to a model i which idividuals are either ifected or susceptible. I our cotext, idividuals are represeted as vertices of a edge-weighted graph, ad the model is the followig: For each edge (vy), if at some time oe idividual (v or y) becomes ifected while the other is susceptible, the the other will later become ifected with some trasmissio probability p vy. These trasmissio evets are idepedet over edges. Regardless of details of the time for such trasmissios to occur, it is clear that this model is closely related to the radom graph model i which edges e = (vy) are preset idepedetly with probabilities p e = p vy, as follows: ( ) The set of ultimately ifected idividuals i the SI model is, i the radom graph model, the uio of the coected compoets which cotai iitially ifected idividuals. I modelig a epidemic withi a populatio with a give graph structure, we regard edge-weights w e = w vy as idicatig relative frequecy of cotact. Itroduce a virulece parameter θ, ad defie trasmissio probabilities (2.1) p e = 1 exp( w e θ).
Epidemics o geeral etworks 3 Note this allows completely arbitrary values of (p e ), by appropriate choice of (w e ). Now the poit of the parametrizatio (2.1) is that the radom graph i ( ) above is exactly the same as the radom graph G(θ) i Sectio 1. So we ca study how to traslate Propositio 1.1 ito a statemet about the SI epidemic model. It is importat to ote a coceptual shift i this traslatio. Propositio 1.1 is most aturally iterpreted as a result about a radom graph process evolvig with time t, ad the proof i [1] relies o this beig a Markov process o graph-space. However, i the SI model we retai o otio of time ; we use (2.1) as a device to defie a oeparameter family (with parameter θ) of edge-trasmissio probabilities, desiged to pass through a arbitrary give set (p e ), ad our results cocer how the size of the epidemic varies with θ. The traslatio rests upo a simple observatio leadig to (2.2) below. For a graph with vertex-set V ad trasmissio probabilities (p e ), write C for the size of the largest coected compoet i the radom graph model, ad write C k for the umber of ultimately ifected idividuals i the SI epidemic model started with k uiformly radom ifected idividuals. From relatio ( ) we clearly have C k kc ad P(C k C C) 1 (1 C/ V )k. These iequalities imply P(C k ε V ) P(C k 1 ε V ) P(C k ε V ) ( 1 (1 ε) k) P(C ε V ). Cosiderig edge-weighted graphs V ad trasmissio probabilities of form (2.1), we see that the relatio betwee the largest compoet size C (θ) ad the umber of ultimately ifected idividuals C,k (θ) is of the form ( 1 (1 ε) k (2.2) ) P ( C (θ) ε V ) P ( C,k (θ) ε V ) P ( C (θ) k 1 ε V ). But we ca apply Propositio 1.1 to the ( C (θ) ), uder coditio (1.1), ad write its coclusio as follows: there exist determiistic τ such that, for every sequece ε 0 sufficietly slowly, for each fixed δ > 0 P ( C (τ δ) ε V ) 0, P ( C (τ + δ) ε V ) 1. It is ow straightforward to use (2.2) to traslate this ito a result for the SI epidemic, which we state carefully as follows. Say a sequece of o-egative radom variables (Y ) is bouded away from zero i probability if ad write this as Y p 0. lim lim sup P(Y δ) = 0, δ 0
4 D. Aldous PROPOSITION 2.1. Take edge-weighted graphs with V, cosider the SI epidemics with trasmissio probabilities of form (2.1), ad write C,k (θ) for the umber of ultimately ifected idividuals i the epidemic started with k uiformly radom ifected idividuals. Suppose there exist some 0 < θ 1 < θ 2 < such that, for all k sufficietly slowly, (2.3) lim EC,k (θ 1 )/ V = 0, lim if EC,k (θ 2 )/ V > 0. The there exist determiistic τ [θ 1, θ 2 ] such that, for all k sufficietly slowly, C,k (τ δ)/ V p 0, C,k (τ + δ)/ V p 0 for all fixed δ > 0. Propositio 2.1 provides a subcritical/supercritical dichotomy for the SI epidemics uder cosideratio. The coceptual poit is that, for virulece parameter θ ot close to the critical value τ, either almost all or almost oe of the realizatios of the epidemic affect a o-egligible proportio of the populatio. 3. EPIDEMIC MODELS ON NETWORKS Classical results o epidemic models ca be foud i textbooks such as [2], ad a more recet extesive accout is [4]. Sice aroud 2000 there has bee itesive study of models with explicit etwork structure; recet surveys are [5] from the statistical physics viewpoit ad [3] from the epidemiology/applied probability viewpoit. But all this literature focuses o the aalysis of specific models. Ituitively oe expects that for ay plausible parametric epidemic model, there will be some regio i parameter-space where the epidemic affects (with high probability) oly a small proportio of a large populatio, aother regio where it affects (with high probability) a o-egligible proportio, with a lower-dimesioal critical iterface. This dichotomy is certaily true i well-studied specific models, but we kow of o attempt at very geeral results. Ideed, discussio i the survey papers cited above ad i [6] metios the difficulty i modelig populatio heterogeeity realistically i a specific model, whereas our settig allows arbitrary heterogeeity. Note also that the classical way of viewig the sub/supercritical dichotomy is via a effective growth rate R 0, the umber of ew ifectives arisig from a typical ifective, with the sub/supercritical dichotomy determied by R 0 < 1 or R 0 > 1. But this does ot apply to typical spatial models with short-rage iteractio, so is ot helpful for the very geeral results we seek. I fact, the R 0 > 1 coditio is better iterpreted tha the coditio for order ifectives to occur i O(log ) time.
Epidemics o geeral etworks 5 4. A CONJECTURE FOR A VERY GENERAL SIR MODEL ON NETWORKS The proof of Propositio 1.1 relies o the expoetial distributio assumptio but (ituitively) such results must hold much more geerally. Let us formulate a cojecture for a very geeral SIR model o etworks. Recall R stads for recovered: ifectives will after a time recover ad ot be susceptible i future. We eed to defie a set H of distributio fuctios ot wildly differet from expoetial. Let us tetatively use the followig defiitio. For a costat β > 1 write H (1) β for the set of distributio fuctios for desities f o (0, ) with mea oe ad f(x) β exp( x/β). The write H β for the set of distributios of cy, where Y has distributio fuctio i H (1) β ad 0 < c <. We model a SIR epidemic o populatio size as follows. Itroduce a virulece parameter 0 < θ < ad a differece from expoetial parameter β > 1. Each idividual v, if ifected, remais ifectious for a radom time with some distributio ι(v, θ). For each idividual v ad parameter θ the distributio fuctio for ι(v, θ) is i H β. For each idividual v, the distributios ι(v, θ) are stochastically icreasig as θ icreases. For each ordered pair (vy) where v is ifectious ad y is susceptible, ifectio may spread from v to y at probability rate q vw (θ) per uit time. For each uordered pair (vy), the fuctio θ q vw (θ) is i H β or is the zero fuctio. We wat to cojecture that a aalog of Propositio 2.1 remais true at this level of geerality. Cosider a sequece of such models with, ad write C,k (θ) for the umber of idividuals ever ifected, give k iitial ifectives. As before, suppose this umber is o() for very small θ ad is ot o() for very large θ. That is, we assume that, for k sufficietly slowly, (4.1) lim EC,k (θ 1 )/ = 0, lim EC,k (θ 2 )/ > 0 for some 0 < θ 1 < θ 2 <. CONJECTURE 1. Uder the assumptios above, with fixed β, there exist determiistic θ [θ 1, θ 2 ] such that, for all k sufficietly slowly, for all fixed δ > 0. C,k (θ δ)/ V p 0, C,k (θ + δ)/ V p 0 We have ot attempted to prove the cojecture; a possible start would be to look for a proof of Propositio 1.1 i the case where distributios were i a class such as H β.
6 D. Aldous REFERENCES [1] D. Aldous, The icipiet giat compoet i bod percolatio o geeral fiite weighted graphs, Electro. Commu. Probab. 21 (2016), paper o. 68. [2] D. J. Daley ad J. Gai, Epidemic Modellig: A Itroductio, Cambridge Stud. Math. Biol., Vol. 15, Cambridge Uiversity Press, Cambridge 1999. [3] L. Dao et al., Networks ad the epidemiology of ifectious disease, Iterdiscip. Perspect. Ifect. Dis., Vol. 2011 (2011), article ID 284909. [4] O. Diekma, H. Heesterbeek, ad T. Britto, Mathematical Tools for Uderstadig Ifectious Disease Dyamics, Priceto Ser. Theor. Comput. Biol., Priceto Uiversity Press, Priceto, NJ, 2013. [5] R. Pastor- Satorras, C. Castellao, P. Va Mieghem, ad A. Vespigai, Epidemic processes i complex etworks, Rev. Moder Phys. 87 (3) (2015), pp. 925 979. [6] L. Pellis et al., Eight challeges for etwork epidemic models, Epidemics 10 (2015), pp. 58 62. David Aldous U.C. Berkeley Departmet of Statistics 367 Evas Hall # 3860 U.C. Berkeley CA 94720 E-mail: aldous@stat.berkeley.edu Received o 14.7.2016; revised versio o 23.5.2017