Spatial Epidemics: Critical Behavior

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 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



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




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.




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.




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




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




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: 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



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.




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 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.




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




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.




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 Behavior

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