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Dynamics of Infectious Diseases

Stochastic dynamics,spatial models, and metapopulations

March 3, 2010

Saturday, March 6, 2010

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


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


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


Stochastic dynamics


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


Epidemic extinction & critical community size

S I R

infection recovery

birth

death death death

importation


Branching processes

Networks, Crowds, and Markets:

Reasoning About a Highly Connected World

By David Easley and Jon Kleinberg

from


http://www.arts.cornell.edu/econ/deasley/


http://www.cs.cornell.edu/home/kleinber/


Branching processes

Networks, Crowds, and Markets: Reasoning About a Highly Connected World


• 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






Branching processes

Networks, Crowds, and Markets: Reasoning About a Highly Connected World


• 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






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


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


Spatial modelsS I R

infection recovery


Metapopulations

S I R S I R

S I R

• separate “subpopulations”, with limited interaction among them


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


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


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


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


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


Gravity models

mj→k,t = transient force of infection exerted by infecteds in location j on susceptibles in location k (at time t)


An aside on estimating parameters

Loss function

TSIR = time series SIR


Coupling and synchrony

• weak coupling: uncorrelated dynamics among subpopulations

• strong coupling: synchronous dynamics among subpopulations


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]


pop > CCS < CCS

Hypotheses:


> CCS< CCS < CCS


“We use wavelet phase analysis to study the spatial synchrony in timing of epidemics through the spatial phase coherence function...”



workflows = rates of people moving to and from their workplaces


Persistence & spatial heterogeneityTSIR =

time series SIR

+ gravity model

Edges are problematic in metapopulation

models; overestimation of

measles fadeouts in coastal areas


Persistence & spatial heterogeneity

reparameterization to include train transport

highlights role of “social space” in addition to “geographic space”


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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