Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Spatial Epidemics: Critical Behavior Steve Lalley March 2012 with the collaboration of Regina Dolgoarshinnykh Xinghua Zheng Edwin Perkins Yuan Shao Reed-Frost (SIR) Model Reed-Frost & Erdös-Renyi Branching Envelopes Critical Behavior: Scaling Limits Spatial Epidemic Models Spatial SIR Models Superprocess Limits Spatial Epidemic Models: Critical Scaling Branching Random Walk: Local Behavior Spatial Extent of SuperBM (d = 1) Measure-Valued Spatial Epidemics Hybrid Percolation Reed-Frost (SIR) Model I Population Size N < ∞ I Individuals susceptible (S), infected (I), or recovered (R). I Recovered individuals immune from further infection. I Infecteds recover in time 1. I Infecteds infect susceptibles with probability p. Reed-Frost (SIR) Model I Population Size N < ∞ I Individuals susceptible (S), infected (I), or recovered (R). I Recovered individuals immune from further infection. I Infecteds recover in time 1. I Infecteds infect susceptibles with probability p. Critical Case: p = 1/N SIR Model:Example Reed-Frost and Erdös-Renyi Random Graphs Reed-Frost model is equivalent to the Erdös-Renyi random graph model: Individuals ←→ Vertices Infections ←→ Edges Epidemic ←→ Connected Components Reed-Frost and Random Graphs Reed-Frost and Random Graphs Reed-Frost and Random Graphs Reed-Frost and Random Graphs Reed-Frost and Random Graphs Reed-Frost and Random Graphs Reed-Frost and Random Graphs Reed-Frost and Random Graphs Branching Envelope of an Epidemic I Each epidemic has a branching envelope (GW process) I Offspring distribution: Binomial-(N, p)≈ Poisson-1 I Epidemic is dominated by its branching envelope I When It St , infected set grows ≈ branching envelope 500 400 300 200 100 200 400 600 800 1000 Critical GW Processes Theorem:(Feller;Jirina) Let YnN be a sequence of Galton-Watson processes, all with offspring distribution Poisson-1, such that Y0N = N. Then D N YNt /N −→ Yt where Yt is the Feller diffusion started at Y0 = 1: p dYt = Yt dBt . Critical GW Processes Theorem:(Feller;Jirina) Let YnN be a sequence of Galton-Watson processes, all with offspring distribution Poisson-1, such that Y0N = N. Then D N YNt /N −→ Yt where Yt is the Feller diffusion started at Y0 = 1: p dYt = Yt dBt . Corollary: (Kolmogorov) If τN is the lifetime of Y N then D τN /N −→ Exponential − 1 Critical Behavior: Reed-Frost Epidemics I Population size: N → ∞ # Infected in Generation n:= I N (n) P N # Recovered in Generation n: = R N (n)= n−1 j=0 I (j) I Initial Condition: I N (0) ∼ bN α I I Theorem: As population size N → ∞, N α I (N t)/N α I(t) D −→ R(t) R N (N α t)/N 2α The limit process satisfies I(0) = b and dR(t) = I(t) dt p dI(t) = + I(t) dBt p dI(t) = + I(t) dBt − I(t)R(t) dt Dolgoarshinnykh & Lalley 2005 if α < 1/3 if α = 1/3 Critical Behavior: Reed-Frost Epidemics I Population size: N → ∞ I # Infected in Generation n:= I N (n) I # Recovered in Generation n: = R N (n)= I Initial Condition: I N (0) ∼ bN α Pn−1 j=0 I N (j) Corollary: (Martin-Lof) If α = 1/3 then as population size N → ∞, R N (∞)/N 2/3 =⇒ τ (b) where τ (b) = first passage time of B(t) + t 2 /2 to b. Critical Behavior: Reed-Frost Epidemics I Population size: N → ∞ I # Infected in Generation n:= I N (n) I # Recovered in Generation n: = R N (n)= I Initial Condition: I N (0) ∼ bN α Pn−1 j=0 I N (j) Corollary: (Martin-Lof) If α = 1/3 then as population size N → ∞, R N (∞)/N 2/3 =⇒ τ (b) where τ (b) = first passage time of B(t) + t 2 /2 to b. Note: R N (∞) is the number of vertices in the connected components of bN 1/3 randomly chosen vertices of the Erdös-Renyi graph. D. Aldous proved an equivalent result for the size of the maximal connected component. Critical Behavior: Heuristics I Critical Epidemic with I0 = m should last ≈ m generations. I Number Rt recovered should be ≈ m2 . I Offspring in branching envelope :: attempted infections. I Misfires: Infections of immunes not allowed. I Critical Threshold: # misfires/generations ≈ O(1) Critical SIR Epidemic: E(#misfires in generation t + 1) ≈ It Rt /N so there will be observable deviation from branching envelope when It ≈ N 1/3 and Rt ≈ N 2/3 Proof Strategy I Lemma: Assume that Ln and L are likelihood ratios under Pn and P, and define Qn and Q by dQn = Ln dPn , dQ = L dP. Assume that Xn and X are random variables whose distributions under Pn and P satisfy (Xn , Ln ) =⇒ (X , L). Then the Qn −distribution of Xn converges weakly to the Q−distribution of X . Proof Strategy II PN = law of Y N QN = law of I N (Galton-Watson) (Reed-Frost) Proof Strategy II PN = law of Y N QN = law of I N (Galton-Watson) (Reed-Frost) ∞ Y dQN ≈ (1 − Rn /N)Yn+1 exp{Yn Rn /N} dPN n=0 ) (∞ ∞ X 1X 2 2 Yn+1 Rn /N ≈ exp (Yn+1 − Yn )Rn /N − 2 n=0 ≈ Girsanov LR n=0 Critical Behavior: Reed-Frost Epidemics I Population size: N → ∞ # Infected in Generation n:= I N (n) P N # Recovered in Generation n: = R N (n)= n−1 j=0 I (j) I Initial Condition: I N (0) ∼ bN α I I Theorem: As population size N → ∞, N α I (N t)/N α I(t) D −→ R(t) R N (N α t)/N 2α The limit process satisfies I(0) = b and dR(t) = I(t) dt p dI(t) = + I(t) dBt p dI(t) = + I(t) dBt − I(t)R(t) dt Dolgoarshinnykh & Lalley 2005 if α < 1/3 if α = 1/3 Spatial SIR Epidemic: I Villages Vx at Sites x ∈ Zd I Village Size:=N I Nearest Neighbor Disease Propagation SIR Rules Locally: I I I I Infected individuals infect susceptibles at same or neighboring site with probability pN Infecteds recover in time 1. Recovered individuals immune from further infection. Spatial SIR Epidemic: I Villages Vx at Sites x ∈ Zd I Village Size:=N I Nearest Neighbor Disease Propagation SIR Rules Locally: I I I I Infected individuals infect susceptibles at same or neighboring site with probability pN Infecteds recover in time 1. Recovered individuals immune from further infection. Critical Case: Infection probability pN = 1/((2d + 1)N). Spatial SIR Epidemic: I Villages Vx at Sites x ∈ Zd I Village Size:=N I Nearest Neighbor Disease Propagation SIR Rules Locally: I I I I Infected individuals infect susceptibles at same or neighboring site with probability pN Infecteds recover in time 1. Recovered individuals immune from further infection. Critical Case: Infection probability pN = 1/((2d + 1)N). Problem: What are the temporal and spatial extents of the epidemic under various initial conditions? Percolation Representation Spatial SIR epidemic is equivalent to critical bond percolation on the graph GN := KN × Zd with nearest neighbor connections: I Vertex set [N] × Zd I Edges connect vertices (i, x) and (j, y ) if dist(x, y ) ≤ 1 Percolation Representation Spatial SIR epidemic is equivalent to critical bond percolation on the graph GN := KN × Zd with nearest neighbor connections: I Vertex set [N] × Zd I Edges connect vertices (i, x) and (j, y ) if dist(x, y ) ≤ 1 Problem: At critical point p = 1/(2d + 1)N, I How does connectivity probability decay? I How does size of largest connected cluster scale with N? I Joint distribution of largest, 2nd largest,· · · ? Critical Spatial SIS Epidemic: Simulation 100 75 400 50 25 300 0 200 50 100 100 Village Size: 20224 Initial State: 2048 infected at 0 Infection Probability: p = 1/20224 Branching Envelope The branching envelope of a spatial SIR epidemic is a branching random walk: In each generation, I A particle at x puts offspring at x or neighbors x + e. I #Offspring are independent Binomial−(N, pN ) or Poisson-NpN I Critical BRW : p = pN = 1/((2d + 1)N). Branching Envelope The branching envelope of a spatial SIR epidemic is a branching random walk: In each generation, I A particle at x puts offspring at x or neighbors x + e. I #Offspring are independent Binomial−(N, pN ) or Poisson-NpN I Critical BRW : p = pN = 1/((2d + 1)N). Associated Measure-Valued Processes XtM = measure that puts mass 1/M at √ x/ M for each particle at site x at time t. Watanabe’s Theorem Let XtM be the measure-valued process associated to a critical nearest neighbor branching random walk. If X0M =⇒ X0 then M XMt =⇒ Xt where Xt is the Dawson-Watanabe process (superBM). The DW process is a measure-valued diffusion. Note 1: The total mass kXt k is a Feller diffusion. Note 2: In 1D, Xt has a continuous density X (t, x). Scaling Limits: SIR Spatial Epidemics in d = 1 √ Let XtM,N be the measure that puts mass 1/M at x/ M for each infected individual at site x at time t. Theorem: Assume that the epidemic is critical and that M = N α . If X0M,N ⇒ X0 then M,N XMt =⇒ Xt where I If α < 2/5 then Xt is the Dawson-Watanabe process. I If α = 2/5 then Xt is the Dawson-Watanabe process with killing at rate Z t X (s, x) ds 0 Lalley 2008 Scaling Limits: SIR Spatial Epidemics in d = 2, 3 √ Recall: XtM,N is the measure that puts mass 1/M at x/ M for each infected individual at site x at time t. Theorem: Assume that the epidemic is critical and that M = N α . If X0M,N ⇒ X0 and X0 satisfies a smoothness condition then M,N XMt =⇒ Xt where I If α < 2/(6 − d) then Xt is the Dawson-Watanabe process. I If α = 2/(6 − d) then Xt is the Dawson-Watanabe process with killing rate L(t, x) = Sugitani local time density. Lalley-Zheng 2010 Sugitani’s Local Time Theorem: Assume that d = 2 or 3 and that the initial configuration X0 = µ of the super-BM Xt satisfies Smoothness Condition: Z tZ φt (x − y ) dµ(y ) 0 x∈Rd is jointly continuous in t, x, where φt (x) is the heat kernel (Gaussian density). Then for each t ≥ 0 the occupation measure Z t Lt := Xs ds 0 is absolutely continuous with jointly continuous density L(t, x). Critical Scaling: Heuristics (SIR Epidemics, d = 1) I Duration: ≈ M generations. I I # Infected Per Generation: ≈ M √ # Infected Per Site: ≈ M √ # Recovered Per Site: ≈ M M I # Misfires Per Site: ≈ M 2 /N I # Misfires Per Generation: ≈ M 5/2 /N I So if M ≈ N 2/5 then # Misfires Per Generation ≈ 1. Critical Scaling: Heuristics (SIR Epidemics, d = 1) I Duration: ≈ M generations. I I # Infected Per Generation: ≈ M √ # Infected Per Site: ≈ M √ # Recovered Per Site: ≈ M M I # Misfires Per Site: ≈ M 2 /N I # Misfires Per Generation: ≈ M 5/2 /N I So if M ≈ N 2/5 then # Misfires Per Generation ≈ 1. But how do we know that the infected individuals in generation n don’t “clump”? Critical Scaling: Heuristics (SIR Epidemics, d = 3) I # Infected Per Generation ≈ Duration ≈ M I # Sites Reachable ≈ M 3/2 . I # Infected Per Infected Site: ≈ O(1) I I √ # Recovered Per Site: ≈ M 2 /M 3/2 = M √ # Misfires Per Generation: ≈ M × M/N So if M ≈ N 2/3 then # Misfires Per Generation ≈ 1. Proof Strategy I Lemma: Assume that Ln and L are likelihood ratios under Pn and P, and define Qn and Q by dQn = Ln dPn , dQ = L dP. Assume that Xn and X are random variables whose distributions under Pn and P satisfy (Xn , Ln ) =⇒ (X , L). Then the Qn −distribution of Xn converges weakly to the Q−distribution of X . Proof Strategy II Theorem: (Dawson) The law Q of the Dawson-Watanabe process with location-dependent killing rate θ(x, t) is mutually a.c. relative to the law P of the Dawson-Watanabe process with no killing (superBM), and the likelihood ratio is Z Z 1 2 hXt , θ(t, ·) i dt dQ/dP = exp − θ(t, x) dM(t, x) − 2 where M is the orthogonal martingale measure attached to the superBM Xt . Proof Strategy III P M = Law of Mth branching random walk. Q M,N = Law of corresponding spatial epidemic. Y Y dQ M,N M,N = 1 + R (t, x) dP M timest sitesx where R M,N (t, x) is a function of the number of misfires at site x at time t. So the problem is to show that under P M , as M → ∞, XX R M,N (t, x) t x converges to the exponent in Dawson’s likelihood ratio. Local Behavior for Branching Random Walk: d = 1 Ynk (·) = branching random walk on Z with Poisson-1 offspring distribution and initial state Y0k , with scaling as in Watanabe’s theorem. Local Behavior for Branching Random Walk: d = 1 Ynk (·) = branching random walk on Z with Poisson-1 offspring distribution and initial state Y0k , with scaling as in Watanabe’s theorem. √ Theorem: If Y0k ([ kx]) → Y0 (x) where Y0 (x) is continuous with compact support then √ Yktk ([ k x]) √ =⇒ X (t, x) k where X (t, x) is the Dawson-Watanabe density process. Lalley 2008 Local Time for Branching Random Walk: d = 2, 3 Ynk (·) = Unk (·) = branching random walk on Zd with Poisson-1 offspring distribution and initial state Y0k scaling as in Watanabe’s theorem. Pt k i=0 Yi (·) Lalley-Zheng 2010 Local Time for Branching Random Walk: d = 2, 3 Ynk (·) = Unk (·) = branching random walk on Zd with Poisson-1 offspring distribution and initial state Y0k scaling as in Watanabe’s theorem. Pt k i=0 Yi (·) √ Theorem: If Y0k ([ kx]) → Y0 (x) where Y0 satisfies hypotheses of Sugitani then in d = 2, 3, √ Uktk ([ k x]) =⇒ L(t, x) k 2−d/2 where L(t, x) is Sugitani local time. Lalley-Zheng 2010 Spatial Extent of Super-BM in d = 1 I Xt = Dawson-Watanabe process, S R := t≥0 support(Xt ) I uD (x) := − log P(R ⊂ D | X0 = δx ) I Spatial Extent of Super-BM in d = 1 I Xt = Dawson-Watanabe process, S R := t≥0 support(Xt ) I uD (x) := − log P(R ⊂ D | X0 = δx ) I Theorem (Dynkin): For any finite interval D, uD (x) is the maximal nonnegative solution in D of the differential equation u 00 = u 2 Spatial Extent of Super-BM in d = 1 I Xt = Dawson-Watanabe process, S R := t≥0 support(Xt ) I uD (x) := − log P(R ⊂ D | X0 = δx ) I Solution: Weierstrass P− Function X √ 1 1 1 + − uD (x) = PL (x/ 6) = 6x 2 6(x − ω)2 6ω 2 ∗ ω∈L where the period lattice L is generated by Ceπi/3 for C > 0 depending on D = [0, a] as follows: √ C = 6a Spatial Extent of Super-BM in d = 1 I I I Xt = Dawson-Watanabe process, S R := t≥0 support(Xt ) uD (x) := − log P(R ⊂ D | X0 = δx ) Solution: Weierstrass P− Function X √ 1 1 1 + − uD (x) = PL (x/ 6) = 6x 2 6(x − ω)2 6ω 2 ∗ ω∈L where the period lattice L is generated by Ceπi/3 for C > 0 depending on D = [0, a] as follows: √ C = 6a Consequence: For any finite Borel measure µ with support contained in the interior of D, Z √ − log P(R ⊂ D | X0 = µ) = PL (x/ 6) µ(dx) Measure-Valued Spatial Epidemics The scaling limits of (near-)critical spatial epidemics at the critical threshold in dimensions d = 1, 2, 3 are Dawson-Watanabe processes with killing. These processes X (t, x) are solutions of the SPDE √ 1 ∂X = ∆X + θX − LX + X W 0 (t, x) ∂t 2 where W 0 (t, x) = space-time white noise L(t, x) = local time density θ= infection rate parameter. Measure-Valued Spatial Epidemics The scaling limits of (near-)critical spatial epidemics at the critical threshold in dimensions d = 1, 2, 3 are Dawson-Watanabe processes with killing. These processes X (t, x) are solutions of the SPDE √ 1 ∂X = ∆X + θX − LX + X W 0 (t, x) ∂t 2 where W 0 (t, x) = space-time white noise L(t, x) = local time density θ= infection rate parameter. Question 1: Can the epidemic survive forever? Question 2: Can the epidemic survive locally? Survival Threshold Theorem 1: In dimension d = 1 the epidemic dies out with probability 1 (that is, Xt = 0 eventually) for any value of θ. In dimensions d = 2, 3 there exist critical points 0 < θc (d) < ∞ such that I If θ < θc (d) then X dies out a.s. I If θ > θc (d) then X survives w.p.p. Theorem 2: For any infection rate θ, in each dimension d = 1, 2, 3 the epidemic almost surely dies out locally, that is, for any compact set K Xt (K ) = 0 Lalley-Perkins-Zheng 2011 eventually. Critical Hybrid Percolation I Vertex set [N] × Zd I Edges connect vertices (i, x) and (j, y ) if dist(x, y ) ≤ 1 I Percolation: Remove edges with probability (1 − p). Critical Hybrid Percolation I Vertex set [N] × Zd I Edges connect vertices (i, x) and (j, y ) if dist(x, y ) ≤ 1 I Percolation: Remove edges with probability (1 − p). Problem: At critical point pd = 1/(2d + 1)N, I How does connectivity probability decay? I How do sizes of largest connected clusters scale with N? Critical Hybrid Percolation I Vertex set [N] × Zd I Edges connect vertices (i, x) and (j, y ) if dist(x, y ) ≤ 1 I Percolation: Remove edges with probability (1 − p). Problem: At critical point pd = 1/(2d + 1)N, I How does connectivity probability decay? I How do sizes of largest connected clusters scale with N? Conjecture (Theorem?): (L.-Shao) Let (C1 , C2 , . . . ) be the largest, 2nd largest, etc., cluster sizes and (D1 , D2 , . . . ) be the corresponding cluster diameters. Then in d = 1, as N → ∞, D N −3/5 (C1 , C2 , . . . ) −→ D N −1/5 (D1 , D2 , . . . ) −→ and