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Dynamics of Infectious Diseases
Stochastic dynamics,spatial models, and metapopulations
March 3, 2010
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Alternative representations of dynamics
• Continuous time, deterministic dynamics- ODEs, PDEs
• Continuous time, stochastic dynamics- continuous time Monte Carlo / stochastic simulation
• Discrete time, deterministic dynamics- maps
• Discrete time, stochastic dynamics- discrete time Monte Carlo
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Continuous time, stochastic dynamics
• continuous time Monte Carlo (“Gillespie Algorithm”)
• at every time t, each reaction has a rate or propensity
‣ ai δt = probability that reaction i will occur within time interval (t, t+δt)
S I R
infection recovery
arecovery = γYainfection = βXY/N
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Gillespie algorithms & variants
• first reaction method (exact)- pick random, exponentially-distributed time for each
reaction at rate ai
- find first reaction + execute
• direct method (exact)- pick random, exponentially-distributed time for some
reaction at total rate a0 = Σai
- pick random reaction to execute based on relative probabilities, and execute
• tau-leaping (approximate)- choose “small” δt
- choose increase δMi ≈ Poisson(ai δt)
- update state variables by amounts implicated by δMi
S I R
infection recovery
ainfection = βXY/Narecovery = γY
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Stochastic dynamics
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Stochastic dynamics
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Stochastic dynamics
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Epidemic extinction & critical community size
S I R
infection recovery
birth
death death death
importation
CCS = smallest population capable of sustaining infection without significant probability of extinction
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Epidemic extinction & critical community size
S I R
infection recovery
birth
death death death
importation
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Branching processes
Networks, Crowds, and Markets:
Reasoning About a Highly Connected World
By David Easley and Jon Kleinberg
from
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Branching processes
Networks, Crowds, and Markets: Reasoning About a Highly Connected World
By David Easley and Jon Kleinberg
• let Xn be a random variable equal to the number of infectives at level n
• let Ynj be a random variable with Ynj=1 if j is infected, 0 otherwise
• then Xn = Yn1 + Yn2 + ... + Ynm
• E[Xn] = E[Yn1] + ... + E[Ynm]• E[Xn] = pn + ... + pn = knpn
• E[Xn] = R0n [R0 = pk]
• epidemic spreads only for R0>1
p
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Branching processes
Networks, Crowds, and Markets: Reasoning About a Highly Connected World
By David Easley and Jon Kleinberg
• E[Xn] = R0n ; what is the probability qn that outbreak is still
proceeding at level n of the tree?
• let f(x) = 1 - (1-px)k
• qn = f(qn-1), i.e.,
• qn = 1 − (1 − pqn−1)k
• K&R claim that Pext = 1/R0
sequence q1, q2, ..., qn converges to nonzero probability q* for R0 > 1
slope f’(0) = R0
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Fraction of population infected (deterministic SIR)
S(t) = S(0)e−R(t)R0
• solve this equation (numerically) for R(∞) = total proportion of population infected
S(∞) = 1−R(∞) = S(0)e−R(∞)R0
R0 = 2
• outbreak: any sudden onset of infectious disease• epidemic: outbreak involving non-zero fraction of population (in limit N→∞), or which is limited by the population size
initial slope = R0
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Analytical methods
• Fokker-Planck
- PDE for the probability distribution P(x,t) for a noisy ODE of the form
• Master equation
- set of coupled ODEs for the probability of finding the population in every possible state (e.g., (S,I,R))
• Moment equations
- ODEs for the time evolution of the moments of a probability distribution
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Spatial modelsS I R
infection recovery
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Metapopulations
S I R S I R
S I R
• separate “subpopulations”, with limited interaction among them
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Stochastic vs. deterministic dynamics in metapopulations
S I R S I R
S I R
i
jdeterministic model:
Xi = # susceptibles in population iYi = # infectives in population iλi = force on infection in pop i
dXi/dt = νi − λiXi − µiXi
dYi/dt = λiXi − γiYi − µiYi
λi = βi
�
j
ρijYj
Nj
stochastic model:
Gillespie with ratesak
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S I R S I R
Deterministic dynamics in metapopulations
1 2
dXi/dt = νi − λiXi − µiXi
dYi/dt = λiXi − γiYi − µiYi
λi = βi
�
j
ρijYj
Nj
dI2/dt = β2ρ21I1 + β2I2 − γ2I2
=⇒
I2(t) =� t
0β2ρ21I1(s) exp([β2 − γ2])s)ds
Let i={1,2}, S1=S2=1, ρii = 1, ρij << 1, with infection initially in population 1:
ρ12
ρ21
Infection is instantaneously “present” in population 2, and growing exponential with rate
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S I R S I R
Stochastic dynamics in metapopulations
1 2Let i={1,2}, S1=S2=1, ρii = 1, ρij << 1, with infection initially in population 1:
ρ12
ρ21
Some chance infection will not spread, and if it does, it will be delayed
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Interactions within metapopulations
S I R S I R
S I R
• choice of interaction term ρij dictates details of dynamical behavior
• sessile hosts (e.g., plants): transmission must be wind- or vector-borne
- spatial kernel: ρij ~ f(dist(ri, rj))
• permanently migrating hosts (e.g., animals):
dXi/dt = νi − λiXi − µiXi +�
j
mijXj −�
j
mjiXi
dYi/dt = λiXi − γiYi − µiYi +�
j
mijYj −�
j
mjiYi
mij = rate at which hosts migrate to subpopulation i from j
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Interactions within metapopulations
• commuting hosts (e.g., humans):
• full dynamical equations:
- SIR + migrations
S I R S I R
i
jlij, rij
λi = βi((1− ρ)Ii + ρIj)ρ = 2q(1− q)q = lij/(rji + lij) = lji/(rij + lji)
Commuter approximation (commuter movements rapid)
q = proportion of time that individuals spend away from home
force of infection
coupling
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Gravity models
mj→k,t = transient force of infection exerted by infecteds in location j on susceptibles in location k (at time t)
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An aside on estimating parameters
Loss function
TSIR = time series SIR
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Coupling and synchrony
• weak coupling: uncorrelated dynamics among subpopulations
• strong coupling: synchronous dynamics among subpopulations
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Extinction & rescue effects
• role of coupling strength between subpopulations in determining extinction characteristics
• strong coupling: metapopulation randomly mixed
- time to extinction
• weak coupling: n independent subpopulations
T =k
e−�N= ke�N
T =k
ne−�N/n+
k
(n− 1)e−�N/n+ ... +
k
e−�N/n
< k(1 + log(n))e�N/n
time to extinction for n infected subpopulations
[faster than for strongly coupled, randomly mixed metapop]
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pop > CCS < CCS
Hypotheses:
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> CCS< CCS < CCS
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“We use wavelet phase analysis to study the spatial synchrony in timing of epidemics through the spatial phase coherence function...”
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Coupling and synchrony
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Coupling and synchrony
workflows = rates of people moving to and from their workplaces
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Coupling and synchrony
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Persistence & spatial heterogeneityTSIR =
time series SIR
+ gravity model
Edges are problematic in metapopulation
models; overestimation of
measles fadeouts in coastal areas
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Persistence & spatial heterogeneity
reparameterization to include train transport
highlights role of “social space” in addition to “geographic space”
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Time, travel, and infection
advent of faster & more densely packed steamship travel between India and Fiji allowed for the importation of measles
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