Spatial Epidemics: Critical Behaviorgalton.uchicago.edu/~lalley/Talks/Epidemics.pdfRegina...

Mean Field ModelsSpatial Epidemic Models

Spatial Epidemics: Critical Behavior

Regina Dolgoarshinnykh1 and Steve Lalley2

1Columbia University

2University of Chicago

October 9, 2006

Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior


Mean Field ModelsStochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior

Spatial Epidemic ModelsSpatial SIS and SIR ModelsBranching Random Walks and Superprocess LimitsSpatial Epidemic Models: Critical ScalingSpatial Extent of SuperBM



Stochastic Logistic (SIS) ModelReed-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?




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.

Dynamics:

St+1 = N − It+1,

It+1 = Yt ,1 + Yt ,2 + · · ·+ Yt ,N ;

Yt ,j ∼ Bernoulli-(1− p)It




SIS Model:Example

SIS⇐⇒ Oriented Percolation on KN × Z+




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.

Dynamics:

St+1 = St − It+1,

Rt+1 = Rt + It ,It+1 = Yt ,1 + Yt ,2 + · · ·+ Yt ,St ;

Yt ,j ∼ Bernoulli-(1− p)It




Recovered individuals are immune from future infection.Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior



SIR Model:Example




Reed-Frost and Random GraphsReed-Frost model is equivalent to the Erdös-Renyi randomgraph model:

Individuals←→ VerticesInfections←→ EdgesEpidemic←→ Connected Components




Reed-Frost and Random Graphs




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




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




Example: SIS Epidemic and its Branching Envelope

200 400 600 800 1000

100

200

300

400

500

200 400 600 800 1000

100

200

300

400

500

N = 80000I0 = 200

p = 1/80000




Feller’s TheoremI Z N

t : Galton-Watson processesI Offspring Distribution F : mean 1, variance σ2 <∞.I Initial Conditions: Z N

0 = aN.

=⇒ Z NNt/N D−→ Yt

I Feller process:

Y0 = a

dYt = σ√

Yt dBt




Feller’s TheoremI Z N

t : Galton-Watson processesI Offspring Distribution F : mean 1− b/N, variance σ2 <∞.I Initial Conditions: Z N

0 = aN.

=⇒ Z NNt/N D−→ Yt

I Feller process with drift:

Y0 = a

dYt = σ√

Yt dBt−bYt dt




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.




Example: Entrance Boundary

100 200 300 400 500 600 700

500

1000

1500

2000

100 200 300 400 500 600 700

500

1000

1500

2000

N = 80000I0 = 10000

p = 1/80000




Critical Behavior: SIS EpidemicsI Population size: N →∞I # Infected in Generation t : IN


0 ∼ bNα.

Corollary: If α = 1/2 then∑t≥0

INt /N =⇒ τ(b)

where τ(b) = first passage time to zero of Ornstein-Uhlenbeckprocess started at b.

Proof: Time change.




Critical Heuristics: SIS EpidemicsI Critical Epidemic with I0 = m should last ≈ m generations.

I Offspring in branching envelope :: attempted infections.I Collisions: Duplicate infections not allowed.I Critical Threshold: # collisions/generations ≈ O(1)

Critical SIS Epidemic:

E(#collisions in generation t + 1) ≈ I2t /N

so observable deviation from branching envelope when

It ≈√

N




Critical Heuristics: SIS EpidemicsI Critical Epidemic with I0 = m should last ≈ m generations.I Offspring in branching envelope :: attempted infections.

I Collisions: Duplicate infections not allowed.I Critical Threshold: # collisions/generations ≈ O(1)




It ≈√

N




Critical Heuristics: SIS EpidemicsI Critical Epidemic with I0 = m should last ≈ m generations.I Offspring in branching envelope :: attempted infections.I Collisions: Duplicate infections not allowed.

I Critical Threshold: # collisions/generations ≈ O(1)




It ≈√

N




Critical Heuristics: SIS EpidemicsI Critical Epidemic with I0 = m should last ≈ m generations.I Offspring in branching envelope :: attempted infections.I Collisions: Duplicate infections not allowed.I Critical Threshold: # collisions/generations ≈ O(1)




It ≈√

N




Critical Behavior: Reed-Frost (SIR) EpidemicsI Population size: N →∞I # Infected in Generation t := IN

tI # Recovered in Generation t : = RN


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




Critical Behavior: Reed-Frost (SIR) EpidemicsI Population size: N →∞I # Infected in Generation t := IN

tI # Recovered in Generation t : = RN


0 ∼ bNα

Corollary: If α = 1/3 then

RN∞/N2/3 =⇒ τ(b)

where τ(b) = first passage time of B(t) + t2/2 to b.

(Martin-Lof; Aldous)




Critical Heuristics: SIR EpidemicsI Critical Epidemic with I0 = m should last ≈ m generations.I Offspring in branching envelope :: attempted infections.I Collisions: Infections of immunes not allowed.I Critical Threshold: # collisions/generations ≈ O(1)

Critical SIR Epidemic:

E(#collisions in generation t + 1) ≈ It(N − St)/N

so observable deviation from branching envelope whenIt ≈ N1/3.



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 .






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 .




Critical Spatial SIS Epidemic: Simulation

50

100

100

200

300

400

0

25

50

75

100

50

100

Village Size: 20224Initial State: 2048 infected at 0

Infection Probability: p = 1/20224




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.




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 .




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.

Note 1: The total mass ‖Xt‖ is a Feller diffusion.Note 2: Watanabe is the spatial analogue of Feller’s theorem.




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.




Dawson-Watanabe Process in 1DSuperposition Principle: Let Xµ

t be a Dawson-Watanabeprocess with killing rate a and initial state

Xµ0 = µ.

If Xµ and X ν are independent Dawson-Watanabe processeswith initial conditons µ and ν then

Xµt ∪ X ν

tD= Xµ+ν

t

Absolute 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)




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 .




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




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.




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”?




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.




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)





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





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





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)


Spatial Epidemics: Critical Behaviorgalton.uchicago.edu/~lalley/Talks/Epidemics.pdfRegina...

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