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 ModelsSpatial Epidemic Models
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
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?
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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.
Dynamics:
St+1 = N − It+1,
It+1 = Yt ,1 + Yt ,2 + · · ·+ Yt ,N ;
Yt ,j ∼ Bernoulli-(1− p)It
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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.
Dynamics:
St+1 = N − It+1,
It+1 = Yt ,1 + Yt ,2 + · · ·+ Yt ,N ;
Yt ,j ∼ Bernoulli-(1− p)It
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-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.
Dynamics:
St+1 = St − It+1,
Rt+1 = Rt + It ,It+1 = Yt ,1 + Yt ,2 + · · ·+ Yt ,St ;
Yt ,j ∼ Bernoulli-(1− p)It
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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.
Dynamics:
St+1 = St − It+1,
Rt+1 = Rt + It ,It+1 = Yt ,1 + Yt ,2 + · · ·+ Yt ,St ;
Yt ,j ∼ Bernoulli-(1− p)It
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
SIR Model:Example
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Reed-Frost and Random Graphs
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
Critical Behavior: SIS EpidemicsI Population size: N →∞I # Infected in Generation t : IN
tI Initial Condition: 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.
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-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
Stochastic Logistic (SIS) ModelReed-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
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)
Regina Dolgoarshinnykh and Steve Lalley Spatial Epidemics: Critical Behavior
Mean Field ModelsSpatial Epidemic Models
Stochastic Logistic (SIS) ModelReed-Frost (SIR) ModelBranching EnvelopesCritical Behavior
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.
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
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
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.
Note 1: The total mass ‖Xt‖ is a Feller diffusion.Note 2: Watanabe is the spatial analogue of Feller’s theorem.
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.
Note 1: The total mass ‖Xt‖ is a Feller diffusion.Note 2: Watanabe is the spatial analogue of Feller’s theorem.
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 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)
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 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)
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
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