aa r X i v : . [ m a t h . P R ] S e p IMS Lecture Notes–Monograph SeriesAsymptotics: Particles, Processes and Inverse Problems
Vol. 55 (2007) 167–178c (cid:13)
Institute of Mathematical Statistics, 2007DOI:
Critical scaling of stochasticepidemic models ∗ Steven P. Lalley1
University of Chicago
To Piet Groeneboom, on the occasion of his 39th birthday.
Abstract:
In the simple mean-field
SIS and
SIR epidemic models, infectionis transmitted from infectious to susceptible members of a finite populationby independent p − coin tosses. Spatial variants of these models are proposed,in which finite populations of size N are situated at the sites of a latticeand infectious contacts are limited to individuals at neighboring sites. Scalinglaws for both the mean-field and spatial models are given when the infectionparameter p is such that the epidemics are critical . It is shown that in allcases there is a critical threshold for the numbers initially infected: below thethreshold, the epidemic evolves in essentially the same manner as its branchingenvelope , but at the threshold evolves like a branching process with a size-dependent drift.
1. Stochastic epidemic models
The simplest and most thoroughly studied stochastic models of epidemics are mean-field models, in which all individuals of a finite population interact in the samemanner. In these models, a contagious disease is transmitted among individuals ofa homogeneous population of size N . In the simple SIS epidemic , individuals areat any time either infected or susceptible ; infected individuals remain infected forone unit of time and then become susceptible. In the simple SIR epidemic (morecommonly known as the
Reed-Frost model), individuals are either infected, suscep-tible, or recovered ; infected individuals remain infected for one unit of time, afterwhich they recover and acquire permanent immunity from future infection. In bothmodels, the mechanism by which infection occurs is random: At each time, for anypair ( i, s ) of an infected and a susceptible individual, the disease is transmittedfrom i to s with probability p = p N . These transmission events are mutually in-dependent. Thus, in both the SIR and the
SIS model, the number J t +1 = J Nt ofinfected individuals at time t + 1 is given by(1) J t +1 = S t X s =1 ξ s , ∗ Supported by NSF Grant DMS-04-05102. University of Chicago, Department of Statistics, 5734 S. University Avenue, Eckhart 118,Chicago, Illinois 60637, USA, e-mail: [email protected]
AMS 2000 subject classifications:
Keywords and phrases: stochastic epidemic model, spatial epidemic, Feller diffusion, branchingrandom walk, Dawson-Watanabe process, critical scaling.16768
S. P. Lalley where S t = S Nt is the number of susceptibles at time t and the random variables ξ s are, conditional on the history of the epidemic to time t , independent, identicallydistributed Bernoulli-1 − (1 − p ) J t . In the SIR model, R t +1 = R t + J t and(2) S t +1 = S t − J t +1 , where R t is the number of recovered individuals at time t , while in the SIS model,(3) S t +1 = S t + J t − J t +1 . In either model, the epidemic ends at the first time T when J T = 0. The most basicand interesting questions concerning these models have to do with the duration T and size P t ≤ T J t of the epidemic and their dependence on the infection parameter p N and the initial conditions. In the simple
SIS and
SIR epidemics, no allowance is made for geographic or socialstratifications of the population, nor for variability in susceptibility or degree ofcontagiousness. Following are descriptions of simple stochastic models that incor-porate a geographic stratification of a population. We shall call these the (spatial)
SIS − d and SIR − d epidemics, with d denoting the spatial dimension.Assume that at each lattice point x ∈ Z d is a homogeneous population of N x in-dividuals, each of whom may at any time be either susceptible or infected, or (in the SIR variants) recovered. These populations may be thought of as “villages”. As inthe mean-field models, infected individuals remain contagious for one unit of time,after which they recover with immunity from future infection (in the
SIR variants)or once again become susceptible (in the
SIS models). At each time t = 0 , , , . . . ,for each pair ( i x , s y ) of an infected individual located at x and a susceptible indi-vidual at y , the disease spreads from i x to s y with probability α ( x, y ).The simple Reed-Frost and stochastic logistic epidemics described in section 1.1terminate with probability one, regardless of the value of the infection parameter p , because the population is finite. For the spatial SIS and
SIR models this isno longer necessarily the case: If P x ∈ Z d N x = ∞ then, depending on the valueof the parameter p and the dimension d , the epidemic may persist forever withpositive probability. (For instance, if N x = 1 for all x and α ( x, y ) = p for nearestneighbor pairs x, y but α ( x, y ) = 0 otherwise, then the SIS − d epidemic is justoriented percolation on Z d +1 , which is known to survive with positive probabilityif p exceeds a critical value p c < SIS- N x = 20224 and infection parameter 1 / All of the models described above have equivalent descriptions as structured ran-dom graphs, that is, percolation processes. Consider for definiteness the simple
SIR ritical Scaling of Stochastic Epidemic Models
Fig 1 . (Reed-Frost) epidemic. In this model, no individual may be infected more than once;furthermore, for any pair x, y of individuals, there will be at most one opportunityfor infection to pass from x to y or from y to x during the course of the epidemic.Thus, one could simulate the epidemic by first tossing a p − coin for every pair x, y ,drawing an edge between x and y for each coin toss resulting in a Head, and thenusing the resulting (Erd¨os-Renyi) random graph determined by these edges to de-termine the course of infection in the epidemic. In detail: If Y is the set of infectedindividuals at time 0, then the set Y of individuals infected at time 1 consists ofall x / ∈ Y that are connected to individuals in Y , and for any subsequent time n ,the set Y n +1 of individuals infected at time n + 1 consists of all x / ∈ ∪ j ≤ n Y j who areconnected to individuals in Y n . Note that the set of individuals ultimately infectedduring the course of the epidemic is the union of those connected components ofthe random graph containing at least one vertex in Y .Similar random graph descriptions may be given for the simple SIS and thespatial
SIS and
SIR epidemic models.
For each of the stochastic epidemic models discussed above there is an associatedbranching process that serves, in a certain sense, as a “tangent” to the epidemic.We shall refer to this branching process as the branching envelope of the epidemic.The branching envelopes of the simple mean-field epidemics are ordinary Galton-Watson processes; the envelopes of the spatial epidemics are branching randomwalks. There is a natural coupling of each epidemic with its branching envelope inwhich the set of infected individuals in the epidemic is at each time (and in thespatial models, at each location) dominated by the corresponding set of individualsin the branching envelope. S. P. Lalley
Fig 2 . Following is a detailed description of the natural coupling of the simple
SIS epidemic with its branching envelope. The branching envelope is a Galton-Watsonprocess Z n with offspring distribution Binomial-( N, p ), where p is the infectionparameter of the epidemic, and whose initial generation Z coincides with the setof individuals who are infected at time 0. Particles in the Galton-Watson processare marked red or blue : red particles represent infected individuals in the coupledepidemic, while blue offspring of red parents represent attempted infections that arenot allowed because the attempt is made on an individual who is not susceptible,or has already been infected by another contagious individual. Colors are assignedas follows: (1) Offspring of blue particles are always blue. (2) Each red particlereproduces by tossing a p − coin N times, once for each individual i in the population.Each Head counts as an offspring, and each represents an attempted infection. Ifseveral red particles attempt to infect the same individual i , exactly one of these ismarked as a success (red), and the others are marked as failures (blue). Also, if anattempt is made to infect an individual who is not susceptible, the correspondingparticle is colored blue. Clearly, the collection of all particles (red and blue) evolvesas a Galton-Watson process, while the collection of red particles evolves as theinfected set in the SIS epidemic. See the figure below for a typical evolution of thecoupling in a population of size N = 80 ,
000 with p = 1 /
2. Critical behavior: mean-field case
When studying the behavior of the simple
SIR and
SIS epidemics in large pop-ulations, it is natural to consider the scaling p = p N = λ N /N for the infectionparameter p . In this scaling, λ = λ N is the mean of the offspring distribution inthe branching envelope. If λ < λ > N behavior of the size of the SIR epidemic in this case is ritical Scaling of Stochastic Epidemic Models well understood: see for example [12] and [14].
The behavior of both the
SIS and
SIR epidemics is more interesting in the criticalcase λ N ≈
1. When the set of individuals initially infected is sufficiently small rel-ative to the population size, the epidemic can be expected to evolve in much thesame manner as a critical Galton-Watson process with Poisson-1 offspring distri-bution. However, when the size of the initially infected set passes a certain criticalthreshold , then the epidemic will begin to deviate substantially from the branchingenvelope. For the
SIR case, the critical threshold was (implicitly) shown by [11] and[1] (see also [12]) to be at N / , and that the critical scaling window is of width N − / : Theorem 1 ([11], [1]) . Assume that p N = 1 /N + a/N / + o ( n − / ) , and that thenumber J N of initially infected individuals is such that J N /N / → b as the popu-lation size N → ∞ . Then as N → ∞ , the size U N := P t J t obeys the asymptoticlaw (4) U N /N / D −→ T b where T b is the first passage time to the level b by W t + t / at , and W t is astandard Wiener process. The distribution of the first passage time T b can be given in closed form: See[11], also [8], [13].For the critical SIS epidemic, the critical threshold is at N / , and the criticalscaling window is of width N − / : Theorem 2 ([4]) . Assume that p N = 1 /N + a/N / + o ( n − / ) , and that theinitial number of infected individuals satisfies J N ∼ bN / as N → ∞ . Then thetotal number of infections U N := P t J t during the course of the epidemic obeys (5) U N /N D −→ τ ( b − a ; − a ) where τ ( x ; y ) is the time of first passage to y by a standard Ornstein-Uhlenbeckprocess started at x . For both the
SIR and
SIS epidemics, if the number of individuals initially infectedis much below the critical threshold then the evolution of the epidemic will not differnoticeably from that of its branching envelope. It was observed by [7] (and provedby [9]) that a (near-) critical Galton-Watson process initiated by a large number M of individuals behaves, after appropriate rescaling, approximately as a Fellerdiffusion : In particular, if Z Mn is the size of the n th generation of a Galton-Watsonwith Z M ∼ bM with offspring distribution Poisson(1 + a/M )then as M → ∞ ,(6) Z M [ Mt ] /M D −→ Y t where Y t satisfies the stochastic differential equation dY t = aY t dt + p Y t dW t , (7) Y = b. S. P. Lalley
What happens at the critical threshold, in both the
SIR and
SIS epidemics, is thatthe deviation from the branching envelope exhibits itself as a size-dependent driftin the limiting diffusion:
Theorem 3 ([4]) . Let J N ( n ) = J N [ n ] be the number infected in the n th generationof a simple SIS epidemic in a population of size N . Then under the hypotheses ofTheorem 2, (8) J N ( √ N t ) / √ N D −→ Y t where Y = b and Y t obeys the stochastic differential equation (9) dY t = ( aY t − Y t ) dt + p Y t dW t Note that the diffusion (9) has an entrance boundary at ∞ , so that it is possibleto define a version Y t of the process with initial condition Y = 0. When the SIS epidemic is begun with J N ≫ √ N initially infected, the number J Nt infected willrapidly drop (over the first ε √ N generations) until reaching a level of order √ N , andthen evolve as predicted by (8). The following figure depicts a typical evolution in apopulation of size N = 80 , p = 1 /N and I = 10 , Theorem 4 ([4]) . Let J N ( n ) = J N [ n ] and R N ( n ) = R N [ n ] be the numbers of infectedand recovered individuals in the n th generation of a simple SIR epidemic in apopulation of size N . Then under the hypotheses of Theorem 1, (10) (cid:18) N − / J N ( N / t ) N − / R N ( N / t ) (cid:19) D −→ (cid:18) J ( t ) R ( t ) (cid:19) where J = b , R = 0 , and dJ ( t ) = ( aJ ( t ) − J ( t ) R ( t )) dt + p J ( t ) dW t , (11) dR ( t ) = J ( t ) dt. Theorems 1–2 can be deduced from Theorems 3–4 by simple time-change argu-ments (see [4]).
Fig 3 .ritical Scaling of Stochastic Epidemic Models
The critical thresholds for the
SIS − d and SIR − d epidemics can be guessed by sim-ple comparison arguments using the standard couplings of the epidemics with theirbranching envelopes. Consider first the critical SIS epidemic in a population of size N . Recall (Section 1.4) that the branching envelope is a critical Galton-Watsonprocess whose offspring distribution is Binomial-( N, /N ). The particles of thisGalton-Watson process are marked red or blue, in such a way that in each genera-tion the number of red particles coincides with the number of infected individualsin the SIS epidemic. Offspring of blue particles are always blue, but offspring ofred particles may be either red or blue; the blue offspring of red parents in eachgeneration represent attempted infections that are suppressed.Assume that initially there are N α infected individuals and thus also N α indi-viduals in the zeroth generation of the branching envelope. By Feller’s theorem, wemay expect that the extinction time of the branching envelope will be on the order N α , and that in each generation up to (shortly before) extinction the branchingprocess will have order N α individuals. If α is small enough that the SIS epidemicobeys the same rough asymptotics (that is, stays alive for O ( N α ) generations andhas O ( N α ) infected individuals in each generation), then the number of blue off-spring of red parents in each generation will be on the order N × ( N α /N ) (becausefor each of the N individuals of the population, the chance of a double infectionis about N α /N ). Since the duration of the epidemic will be of the same roughorder of magnitude as the size of the infected set in each generation, there shouldbe at most O (1) blue offspring of red parents in any generation (if there were more,the red population would die out long before the blue). Thus, the critical thresholdmust be at N / .A similar argument applies for the SIR epidemic. The branching envelope of thecritical
SIR is once again a critical Galton-Watson process with offspring distribu-tion Binomial-( N, /N ), with constituent particles again labeled red or blue, redparticles representing infected individuals in the epidemic. The rule by which redparticles reproduce is as follows: Each red particle tosses a p − coin N times oncefor each individual i in the population. Each Head counts as an offspring, and rep-resents an attempted infection. However, if a Head occurs on a toss at individual i where i was infected in an earlier generation, then the Head results in a blue offspring. Similarly, if more than one red particle tosses a Head at an individual i which has not been infected earlier, then one of these is labeled red and the excessare all labeled blue.Assume that initially there are N α infected individuals. As before, we may expectthat the extinction time of the branching envelope will be on the order N α , andthat in each generation up to extinction the branching process will have order N α individuals. If α is small enough, the extinction time and the size of the redpopulation will also be O ( N α ). Consequently, the size of the recovered populationwill be (for all but the first few generations) on order N α . Thus, in each generation,the number of blue offspring of red parents will be on order ( N α /N ) × N α (becausethe chance that a recovered individual is chosen for attempted infection by aninfected individual is O ( N α /N )). Therefore, by similar reasoning as in the SIS case, the critical threshold is at N / , as this is where the the number of blueoffspring of red parents in each generation is O (1). S. P. Lalley
3. Critical behavior:
SIS -1 and
SIR -1 Spatial epidemics
Consider now the spatial
SIS - d and SIR - d epidemic models on the d -dimensionalinteger lattice Z d . Assume that the village size N x = N is the same for all sites x ∈ Z d , and that the infection probabilities α ( x, y ) are nearest neighbor homogeneous,and uniform, that is,(12) α ( x, y ) = ( p = p N , if | x − y | ≤ , otherwise . The branching envelope of a spatial
SIS − d or SIR − d epidemic is a nearest neighbor branching random walk on the integer lattice Z d . This evolves as follows: Any parti-cle located at site x at time t lives for one unit of time and then reproduces, placingrandom numbers ξ y of offspring at the sites y such that | y − x | ≤
1. The randomvariables ξ y are mutually independent, with Binomial-( N, p N ) distributions.The analogue for branching random walks of Feller’s theorem for Galton-Watsonprocesses is Watanabe’s theorem . This asserts that, after suitable rescaling, as theparticle density increases, critical branching random walks converge to a limit,the
Dawson-Watanabe process, also known as super Brownian motion . A precisestatement follows: Consider a sequence of branching random walks, indexed by M = 1 , , . . . , with offspring distribution Binomial-( N, p M ) as above, and(13) p M = p N,M = 1(2 d + 1) N − aN M . (Note: N may depend on M .) The rescaled measure-valued process X Mt associatedwith the M th branching random walk puts mass 1 /M at location x/ √ M and time t for each particle of the branching random walk that is located at site x at time [ M t ].(Note: The branching random walk is a discrete-time process, but the associatedmeasure-valued process runs in continuous time.)
Watanabe’s theorem ([15]) . Assume that the initial values X M converge weakly(as finite Borel measures on R d ) to a limit measure X . Then under the hypothesis (13) the measure-valued processes X Mt converge in law as M → ∞ to a limit X t : (14) X Mt = ⇒ X t . The limit process is the
Dawson-Watanabe process with killing rate a and ini-tial value X . (The term killing rate is used because the process can be obtainedfrom the “standard” Dawson-Watanabe process ( a = 0) by elimination of massat constant rate a .) The Dawson-Watanabe process X t with killing rate a can becharacterized by a martingale property: For each test function φ ∈ C c ( R d ),(15) h X t , φ i − h X , φ i − σ Z t h X s , ∆ φ i ds + a Z t h X s , ϕ i ds is a martingale. Here σ = 2 d/ (2 d + 1) is the variance of the random walk ker-nel naturally associated with the branching random walks. It is known [10] thatin dimension d = 1 the random measure X t is for each t absolutely continuousrelative to Lebesgue measure, and the Radon-Nikodym derivative X ( t, x ) is jointlycontinuous in t, x (for t > d ≥ X t is almost surelysingular, and is supported by a Borel set of Hausdorff dimension 2 [3]. ritical Scaling of Stochastic Epidemic Models SIS - and SIR - epidemics: critical scaling As in the mean-field case, there are critical thresholds for the
SIS -1 and
SIR -1epidemics at which they begin to deviate noticeably from their branching envelopes.These are at N / and N / , respectively: Theorem 5 ([5]) . Fix α > , and let X Nt be the discrete-time measure-valuedprocess obtained from an SIS − or an SIR − epidemic on a one-dimensional gridof size- N villages by attaching mass /N α to the point ( t, x/N α/ ) for each infectedindividual at site x at time [ tN α ] . Assume that X N converges weakly to a limitmeasure X as the village size N → ∞ . Then as N → ∞ , (16) X N [ N α t ] D −→ X t , where X t is a measure-valued process with initial value X whose law depends onthe value of α and the type of epidemic (SIS or SIR) as follows:(a) SIS: If α < then X t is a Dawson-Watanabe process with variance σ .(b) SIS: If α = then X t is a Dawson-Watanabe process with variance σ andkilling rate (17) θ ( x, t ) = X ( x, t ) / . (c) SIR: If α < then X t is a Dawson-Watanabe process with variance σ .(d) SIR: If α = then X t is a Dawson-Watanabe process with variance σ andkilling rate (18) θ ( x, t ) = X ( x, t ) Z t X ( x, s ) ds. The Dawson-Watanabe process with variance σ and (continuous, adapted)killing rate θ ( t, x, ω ) is characterized [2] by a martingale problem similar to (15)above: for each test function φ ∈ C c ( R ),(19) h X t , φ i − h X , φ i − σ Z t h X s , ∆ φ i ds + Z t h X s , θϕ i ds is a martingale. The law of this process is mutually absolutely continuous relativeto that of the Dawson-Watanabe process with no killing, and there is an explicitformula for the Radon-Nikodym derivative – see [2]. Arguments similar to those given above for the mean-field
SIS and
SIR epidemicscan be used to guess the critical thresholds for the spatial
SIS-
SIR- attrition of the red population (since blue particles created by red parentsare potential red offspring that are not realized!). S. P. Lalley
Consider first the
SIS- N α parti-cles, distributed (say) uniformly among the N α/ sites nearest the origin. Then byFeller’s limit theorem (recall that the total population size in a branching randomwalk is a Galton-Watson process), the branching envelope can be expected to sur-vive for O P ( N α ) generations, and at any time prior to extinction the populationwill have O P ( N α ) members. These will be distributed among the sites at distance O P ( N α/ ) from the origin, and therefore in dimension d = 1 there should be about O P ( N α/ ) particles per site. Consequently, for the SIS − O P ( N α − ), and so the total attritionrate per generation should be O P ( N α/ − ). If α = 2 /
3, then the total attritionrate per generation will be O P (1), just enough so that the total attrition throughthe duration of the branching random walk envelope will be on the same order ofmagnitude as the population size N α .For the SIR − SIS − O P ( N α ) generations, and upto the time of extinction the population should have O P ( N α ) individuals, about O P ( N α/ ) per site. Therefore, through N α generations, about N α × N α/ numbers j will be retired, and so the attrition rate per site per generation should be O P ( N α/ × N α/ ), making the total attrition rate per generation O P ( N α/ ). Hence, if α = 2 / O P (1), just enough so that thetotal attrition through the duration of the branching random walk envelope will beon the same order of magnitude as the population size. d ≥ In higher dimensions, the critical behavior of the
SIS - d and SIR - d epidemics appearsto be considerably different. We expect that there will be no analogous thresholdeffect, in particular, we expect that the epidemic will behave in the same manner asthe branching envelope up to the point where the infected set is a positive fraction ofthe total population. This is because in dimensions d ≥
2, the particles of a criticalbranching random walk quickly spread out, so that (after a short initial period)there are only O P (1) (in dimensions d ≥
3) or O P (log N ) (in dimension d = 2)particles per site. (With N particles initially, a critical branching random walktypically lives O ( N ) generations, and particles are distributed among the sites atdistance O ( √ N ) from the origin; in dimensions d ≥
2, there are O ( N d/ ) such sites,enough to accomodate the O ( N ) particles of the branching random walk withoutcrowding.) Consequently, the rate at which “multiple” infections are attempted(that is, attempts by more than one contagious individual to simultaneously infectthe same susceptible) is only of order O P (1 /N ) (or, in dimension d = 2, order O P (log N/N )).The interesting questions regarding the evolution of critical epidemics in dimen-sions d ≥ o ( N ) generations) in which the particles spread out from their initial sites. Thesewill be discussed in the forthcoming University of Chicago Ph. D. dissertation ofXinghua Zheng. There is an obvious gap in the heuristic argument of Section 3.3 above: Even if thetotal number of infected individuals is, as expected, on the order N α , and even if ritical Scaling of Stochastic Epidemic Models these are concentrated in the sites at distance on the order N α/ from the origin, itis by no means obvious that these will distribute themselves uniformly (or at least locally uniformly) among these sites. The key step in filling this gap in the argumentis to show that the particles of the branching envelope distribute themselves moreor less uniformly on scales smaller than N α/ .Consider, as in Section 3.1, a sequence of branching random walks, indexed by M = 1 , , . . . , with offspring distribution Binomial-( N, p M ) as above, and p M givenby (13). Let Y Mt ( x ) be the number of particles at site x at time [ t ], and let X M ( t, x )be the continuous function of t ≥ x ∈ R obtained by linear interpolation fromthe values(20) X M ( t, x ) = Y Mt ( √ M x ) √ M for M t ∈ Z + and √ M x ∈ Z . Theorem 6 ([5]) . Assume that d = 1 . Assume also that the initial particle config-uration is such that all particles are located in an interval [ − κ √ M , κ √ M ] and suchthat the initial particle density satisfies (21) X M (0 , x ) = ⇒ X (0 , x ) as M → ∞ for some continuous function X (0 , x ) with support [ − κ, κ ] . Then as M → ∞ , (22) X M ( t, x ) = ⇒ X ( t, x ) , where X ( t, x ) is the density function of the Dawson-Watanabe process with killingrate a and initial value X (0 , x ) . The convergence is relative to the topology of uni-form convergence on compacts in the space C ( R + × R ) of continuous functions. Since the measure-valued processes associated with the densities X M ( t, x ) areknown to converge to the Dawson-Watanabe process, by Watanabe’s theorem, toprove Theorem 6 it suffices to establish tightness. This is done by a somewhattechnical application of the Kolmogorov-Chentsov tightness criterion, based on acareful estimation of moments. See [5] for details.It is also possible to show that convergence of the particle density processes holdsin Theorem 5. References [1]
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