spatial epidemics: critical behavior
TRANSCRIPT
Mean Field ModelsSpatial Epidemic Models
Spatial Epidemics: Critical Behavior
Regina Dolgoarshinnykh1 and Steve Lalley2
1Columbia University
2University of Chicago
October 21, 2006
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Mean Field ModelsReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Spatial Epidemic ModelsSpatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Epidemics: Basic QuestionsI How long do they last?I How far do they spread?I How many people are infected?I How do epidemics behave at criticality?
Critical: Each infection causes ≈ 1 new infection
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Stochastic Logistic (SIS) ModelI Population Size N <∞I Individuals susceptible (S) or infected (I).I Infecteds recover in time 1, then immediately susceptible.I Infecteds infect susceptibles with probability p.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
SIS Model:Example
SIS⇐⇒ Oriented Percolation on KN × Z+
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
SIS Model:Example
SIS⇐⇒ Oriented Percolation on KN × Z+
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost (SIR) ModelI 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.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Recovered individuals are immune from future infection.Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
SIR Model:Example
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random GraphsReed-Frost model is equivalent to the Erdös-Renyi randomgraph model:
Individuals←→ VerticesInfections←→ EdgesEpidemic←→ Connected Components
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Branching Envelope of an EpidemicI Each epidemic has a branching envelope (GW process)I Offspring distribution: Binomial-(N, p)
I Epidemic is dominated by its branching envelopeI When It � St , infected set grows ≈ branching envelope
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Branching envelope of SIS Epidemic: ConstructionI Branching process contains red particles and blue particlesI Red particles represent infected individualsI Each blue has ξ ∼Binomial-(N, p) blue offspringI Each red has ξ ∼Binomial-(N, p) red offspringI Red offspring choose labels randomly in [N]
I Multiple labels: all but one turn blue
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Example: SIS Epidemic and its Branching Envelope
200 400 600 800 1000
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200 400 600 800 1000
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N = 80000I0 = 200
p = 1/80000
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Critical Behavior: SIS EpidemicsI Population size: N →∞I # Infected in Generation t : IN
tI Initial Condition: IN
0 ∼ bNα.
Theorem: INNαt/Nα =⇒ Yt where
Y0 = b;
dYt =√
Yt dBt if α < 1/2
dYt =√
Yt dBt − Y 2t dt if α = 1/2
Note: When α = 1/2 the initial condition b =∞ is permitted, as∞ is an entrance boundary for the limit diffusion.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Critical Behavior: SIS EpidemicsI Population size: N →∞I # Infected in Generation t : IN
tI Initial Condition: IN
0 ∼ bNα.
Theorem: INNαt/Nα =⇒ Yt where
Y0 = b;
dYt =√
Yt dBt if α < 1/2
dYt =√
Yt dBt − Y 2t dt if α = 1/2
Note: When α = 1/2 the initial condition b =∞ is permitted, as∞ is an entrance boundary for the limit diffusion.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Critical Behavior: SIS EpidemicsI Population size: N →∞I # Infected in Generation t : IN
tI Initial Condition: IN
0 ∼ bNα.
Theorem: INNαt/Nα =⇒ Yt where
Y0 = b;
dYt =√
Yt dBt if α < 1/2
dYt =√
Yt dBt − Y 2t dt if α = 1/2
Note: When α = 1/2 the initial condition b =∞ is permitted, as∞ is an entrance boundary for the limit diffusion.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Example: Entrance Boundary
100 200 300 400 500 600 700
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100 200 300 400 500 600 700
500
1000
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N = 80000I0 = 10000
p = 1/80000
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Reed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Critical Behavior: Reed-Frost (SIR) EpidemicsI Population size: N →∞I # Infected in Generation t := IN
tI # Recovered in Generation t : = RN
tI Initial Condition: IN
0 ∼ bNα
Theorem: (N−αIN
tN−2αRN
t
)D−→
(I(t)R(t)
)The limit process satisfies I(0) = b and
dR(t) = I(t) dt
dI(t) = +√
I(t) dBt if α < 1/3
dI(t) = +√
I(t) dBt − I(t)R(t) dt if α = 1/3
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial SIS and SIR EpidemicsI Villages Vx at Sites x ∈ ZI Village Size:=NI Nearest Neighbor Disease PropagationI SIR or SIS Rules Locally:
I Infected individual will infect susceptible at same orneighboring site with probability (3N)−1
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial SIS and SIR EpidemicsI Villages Vx at Sites x ∈ ZI Village Size:=NI Nearest Neighbor Disease PropagationI SIR or SIS Rules Locally:
I Infected individual will infect susceptible at same orneighboring site with probability (3N)−1
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial SIS and SIR EpidemicsI Villages Vx at Sites x ∈ ZI Village Size:=NI Nearest Neighbor Disease PropagationI SIR or SIS Rules Locally:
I Infected individual will infect susceptible at same orneighboring site with probability (3N)−1
Random Graph Formulation:I SIR Epidemic⇐⇒ Percolation on Z×KN
I SIS Epidemic⇐⇒ Oriented Percolation on Z2 ×KN .
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial SIS and SIR EpidemicsI Villages Vx at Sites x ∈ ZI Village Size:=NI Nearest Neighbor Disease PropagationI SIR or SIS Rules Locally:
I Infected individual will infect susceptible at same orneighboring site with probability (3N)−1
Associated Measure-Valued Processes
X Mt = X M,N
t : measure that puts mass 1/Mat x/
√M for each particle at
site x at time t .
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Critical Spatial SIS Epidemic: Simulation
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0
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Village Size: 20224Initial State: 2048 infected at 0
Infection Probability: p = 1/20224
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Branching Envelope of a Spatial Epidemic
Nearest Neighbor Branching Random Walk:
I Particle at x puts offspring at x − 1, x , x + 1I #Offspring are independent Binomial−(N, p)
I Critical BRW : p = pN = 1/3N.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Branching Envelope of a Spatial Epidemic
Nearest Neighbor Branching Random Walk:
I Particle at x puts offspring at x − 1, x , x + 1I #Offspring are independent Binomial−(N, p)
I Critical BRW : p = pN = 1/3N.
Associated Measure-Valued Processes
X Mt : measure that puts mass 1/M
at x/√
M for each particle atsite x at time t .
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Watanabe’s Theorem ILet X M
t be the measure-valued process associated to a criticalnearest neighbor branching random walk. If
X M0 =⇒ X0
thenX M
Mt =⇒ Xt
where Xt is the Dawson-Watanabe process (superBM). TheDW process is a measure-valued diffusion.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Watanabe’s Theorem IILet X M
t be the measure-valued process associated to a criticalnearest neighbor branching random walk with particles killed atrate a/M. If
X M0 =⇒ X0
thenX M
Mt =⇒ Xt
where Xt is the Dawson-Watanabe process with killing rate a.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Dawson-Watanabe Process in 1DAbsolute Continuity: With probability 1, Xt has a continuousdensity X (t , x) relative to Lebesgue, and X (t , x) is jointlycontinuous in t , x . (Konno-Shiga)
Note: In d ≥ 2 the measure Xt is singular.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Dawson-Watanabe Process in 1DAbsolute Continuity: With probability 1, Xt has a continuousdensity X (t , x) relative to Lebesgue, and X (t , x) is jointlycontinuous in t , x . (Konno-Shiga)
Note: In d ≥ 2 the measure Xt is singular.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Scaling Limits: SIS Spatial EpidemicsTheorem: Let X N
t = X M,Nt be the measure-valued process
associated with critical SIS spatial epidemic with village size Nand scaling M = Nα. If X N
0 ⇒ X0 then
X NMt =⇒ Xt
whereI If α < 2/3 then Xt is the Dawson-Watanabe process.I If α = 2/3 then Xt is the Dawson-Watanabe process with
location-dependent killing rate X (t , x)2.
Note: X Nt is the measure that puts mass 1/M at x/
√M for each
infected individual at site x .
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Scaling Limits: SIR Spatial EpidemicsTheorem: Let X N
t = X M,Nt be the measure-valued process
associated with critical SIR spatial epidemic with village size Nand scaling M = Nα. If X N
0 ⇒ X0 then
X NMt =⇒ Xt
whereI 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 rate
X (t , x)
∫ t
0X (s, x) ds
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Dawson-Watanabe with killingDefinition: The Dawson-Watanabe process Xt with killing rateθ(x , t) is a solution to a martingale problem: For φ ∈ C∞
c (R),
〈Xt , φ〉 −12
∫ t
0〈Xs,∆φ〉ds −
∫ t
0〈θ(·, s)Xs, φ〉ds
is a martingale.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Critical Scaling: Heuristics (SIS Epidemics)I # Infected Per Generation: ≈ MI Duration: ≈ M generations.I # Infected Per Site: ≈
√M
I # Collisions Per Site: ≈ M/NI # Collisions Per Generation: ≈ M3/2/N
So if M ≈ N2/3 then # Collisions Per Generation ≈ 1.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Critical Scaling: Heuristics (SIR Epidemics)I # Infected Per Generation: ≈ MI Duration: ≈ M generations.I # Infected Per Site: ≈
√M
I # Recovered Per Site: ≈ M√
MI # Collisions Per Site: ≈ M2/NI # Collisions Per Generation: ≈ M5/2/N
So if M ≈ N2/5 then # Collisions Per Generation ≈ 1.
But how do we know that the infected individuals in generationn don’t “clump”?
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Critical Scaling: Heuristics (SIR Epidemics)I # Infected Per Generation: ≈ MI Duration: ≈ M generations.I # Infected Per Site: ≈
√M
I # Recovered Per Site: ≈ M√
MI # Collisions Per Site: ≈ M2/NI # Collisions Per Generation: ≈ M5/2/N
So if M ≈ N2/5 then # Collisions Per Generation ≈ 1.
But how do we know that the infected individuals in generationn don’t “clump”?
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Watanabe’s Theorem RevisitedTheorem: Let X M(t , x) be the (rescaled) density processassociated to a critical nearest neighbor branching randomwalk with particles killed at rate a/M. If
X M0 =⇒ X0 in C(R)
thenX M(Mt , x) =⇒ X (t , x) in C(R)
where X (t , x) is the the DW density process with killing rate a.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial Extent of Dawson-Watanabe ProcessI Xt = Dawson-Watanabe processI R(X ) := ∪t≥0support(Xt)
I uD(x) := − log P(R(X ) ⊂ D |X0 = δx)
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial Extent of Dawson-Watanabe ProcessI Xt = Dawson-Watanabe processI R(X ) := ∪t≥0support(Xt)
I uD(x) := − log P(R(X ) ⊂ D |X0 = δx)
Theorem (Dynkin): For any finite interval D, uD(x) is themaximal nonnegative solution in D of the differential equation
u′′ = u2
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial Extent of Dawson-Watanabe ProcessI Xt = Dawson-Watanabe processI R(X ) := ∪t≥0support(Xt)
I uD(x) := − log P(R(X ) ⊂ D |X0 = δx)
Solution: Weierstrass P− Function
uD(x) = PL(x/√
6) =1
6x2 +∑ω∈L∗
{1
6(x − ω)2 −1ω2
}
where the period lattice L is generated by Ceπi/3 for C > 0depending on D = [0, a] as follows:
C =√
6a
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Spatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM
Spatial Extent of Dawson-Watanabe ProcessI Xt = Dawson-Watanabe processI R(X ) := ∪t≥0support(Xt)
I uD(x) := − log P(R(X ) ⊂ D |X0 = δx)
General Initial Conditions: For any finite Borel measure µ withsupport ⊂ D,
− log P(R(X ) ⊂ D |X0 = µ) =
∫uD(x) µ(dx)
=
∫PL(x/
√6) µ(dx)
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior