Download Spatial Epidemics: Critical Behavior

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
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Population Size N < ∞
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Individuals susceptible (S), infected (I), or recovered (R).
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Recovered individuals immune from further infection.
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Infecteds recover in time 1.
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Infecteds infect susceptibles with probability p.
Reed-Frost (SIR) Model
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Population Size N < ∞
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Individuals susceptible (S), infected (I), or recovered (R).
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Recovered individuals immune from further infection.
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Infecteds recover in time 1.
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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
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Each epidemic has a branching envelope (GW process)
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Offspring distribution: Binomial-(N, p)≈ Poisson-1
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Epidemic is dominated by its branching envelope
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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
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Population size: N → ∞
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# Infected in Generation n:= I N (n)
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# 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
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Population size: N → ∞
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# 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
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Critical Epidemic with I0 = m should last ≈ m generations.
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Number Rt recovered should be ≈ m2 .
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Offspring in branching envelope :: attempted infections.
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Misfires: Infections of immunes not allowed.
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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:
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Villages Vx at Sites x ∈ Zd
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Village Size:=N
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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:
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Villages Vx at Sites x ∈ Zd
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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
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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:
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Vertex set [N] × Zd
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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:
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Vertex set [N] × Zd
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Edges connect vertices (i, x) and (j, y ) if dist(x, y ) ≤ 1
Problem: At critical point p = 1/(2d + 1)N,
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How does connectivity probability decay?
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How does size of largest connected cluster scale with N?
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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,
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A particle at x puts offspring at x or neighbors x + e.
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#Offspring are independent Binomial−(N, pN ) or
Poisson-NpN
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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,
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A particle at x puts offspring at x or neighbors x + e.
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#Offspring are independent Binomial−(N, pN ) or
Poisson-NpN
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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
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If α < 2/5 then Xt is the Dawson-Watanabe process.
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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
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If α < 2/(6 − d) then Xt is the Dawson-Watanabe process.
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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)
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Duration: ≈ M generations.
I
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# Infected Per Generation: ≈ M
√
# Infected Per Site: ≈ M
√
# Recovered Per Site: ≈ M M
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# Misfires Per Site: ≈ M 2 /N
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# Misfires Per Generation: ≈ M 5/2 /N
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So if M ≈ N 2/5 then # Misfires Per Generation ≈ 1.
Critical Scaling: Heuristics (SIR Epidemics, d = 1)
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Duration: ≈ M generations.
I
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# Infected Per Generation: ≈ M
√
# Infected Per Site: ≈ M
√
# Recovered Per Site: ≈ M M
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# Misfires Per Site: ≈ M 2 /N
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# Misfires Per Generation: ≈ M 5/2 /N
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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)
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# Infected Per Generation ≈ Duration ≈ M
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# Sites Reachable ≈ M 3/2 .
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# 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
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Xt = Dawson-Watanabe process,
S
R := t≥0 support(Xt )
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uD (x) := − log P(R ⊂ D | X0 = δx )
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Spatial Extent of Super-BM in d = 1
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Xt = Dawson-Watanabe process,
S
R := t≥0 support(Xt )
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uD (x) := − log P(R ⊂ D | X0 = δx )
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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
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Xt = Dawson-Watanabe process,
S
R := t≥0 support(Xt )
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uD (x) := − log P(R ⊂ D | X0 = δx )
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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
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If θ < θc (d) then X dies out a.s.
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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
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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