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![Page 1: MCMC for Stochastic Epidemic Models Philip D. O’Neill School of Mathematical Sciences University of Nottingham.](https://reader034.fdocuments.us/reader034/viewer/2022051417/56649d1f5503460f949f2b28/html5/thumbnails/1.jpg)
MCMC for Stochastic Epidemic Models
Philip D. O’Neill
School of Mathematical Sciences
University of Nottingham
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This includes joint work with…
Tom Britton (Stockholm) Niels Becker (ANU, Canberra) Gareth Roberts (Lancaster) Peter Marks (NHS, Derbyshire PCT)
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Contents 1. MCMC: overview and basics 2. Example: Vaccine efficacy 3. Data augmentation 4. Example: SIR epidemic model 5. Model choice 6. Example: Norovirus outbreak 7. Other topics
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1. Markov chain Monte Carlo (MCMC) Overview and basics
The key problem is to explore a density function π known up to proportionality.
The output of an MCMC algorithm is a sequence of samples from the correctly normalised π.
These samples can be used to estimate summaries of π, e.g. its mean, variance.
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How MCMC works Key idea is to construct a discrete time
Markov chain X1, X2, X3, … on state space S whose stationary distribution is π.
If P(dy,dx) is the transitional kernel of the chain this means that
),()()( dxyPdydxS
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How MCMC works (2)
Subject to some technical conditions,
Distribution of Xn → π as n → Thus to obtain samples from π we simulate
the chain and sample from it after a “long time”.
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Example: π (x) x e-2x
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Example: π (x) x e-2x
XN = 1.2662 X1, X2, …
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Example: π (x) x e-2x
XN = 1.2662
XN+1 = 1.7840
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Example: π (x) x e-2x
XN = 1.2662
XN+1 = 1.7840
XN+2 = 0. 6629
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Example: π (x) x e-2x
Suppose Markov chain output is
..., XN = 1.2662, XN+1 = 1.7840, XN+2 = 0.6629, …. ,XN+M = 1.0312
(i.e. discard initial N values, burn-in)
032.1M
1)(
MN
Njj
XXE
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How to build the Markov chain
Surprisingly, there are many ways to construct a Markov chain with stationary distribution π.
Perhaps the simplest is the Metropolis-Hastings algorithm.
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Metropolis-Hastings algorithm Set an initial value X1.
If the chain is currently at Xn = x, randomly propose a new position Xn+1 = y according to a proposal density q(y | x).
Accept the proposed jump with probability
If not accepted, Xn+1 = x.
.)|()(
)|()(,1min
xyqx
yxqy
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Why the M-H algorithm works
Let P(dx,dy) denote the transition kernel of the chain.
Then P(dx,dy) is approximately the probability that the chain jumps from a region dx to a region dy.
We can calculate P(dx,dy) as follows:
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Why M-H works (2)
1
)|()(
)|()()|(),(
dxdyqdx
dydxqdydxdyqdydxP
)|()|()(
)(dxdyqdydxq
dx
dy
)|()()|()(),()( dxdyqdxdydxqdydydxPdx
),()(),()( Thus dxdyPdydydxPdx
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Why M-H works (3)
),()(),()( dxdyPdydydxPdxSdySdy
),()(),()( dxdyPdydydxPdx
),()()( dxdyPdydxSdy
This last equation shows that π is a stationary distribution for the Markov chain.
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Comments on M-H algorithm (1)
The choice of proposal q(y|x) is fairly arbitrary.
Popular choices include
q(y|x) = q(y) (Independence sampler)
q(y|x) ~ N(x, 2) (Gaussian proposal)
q(y|x) = q(|y-x|) (Symmetric proposal)
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Comments on M-H (2)
In practice, MCMC is almost always used for multi-dimensional problems. Given a target density π(x1, x2, … ,xn) it is possible to update each component separately, or even in blocks, using different M-H schemes.
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Comments on M-H (3)
A popular multi-dimensional scheme is the Gibbs sampler, in which the proposal for a component xi is its full conditional density
π (xi | (x1,…,xi-1, xi+1, … ,xn) )
The M-H acceptance probability is equal to one in this case.
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General comments on MCMC
How to check convergence? There is no guaranteed way.
Visual inspection of trace plots; diagnostic tools (e.g. looking at autocorrelation).
Starting values – try a range Acceptance rates – not too large/small Mixing – how fast does the chain move
around?
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Contents 1. MCMC: overview and basics 2. Example: Vaccine efficacy 3. Data augmentation 4. Example: SIR epidemic model 5. Model choice 6. Example: Norovirus outbreak 7. Other topics
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2. Example: Vaccine Efficacy Outbreak of Variola Minor, Brazil 1956 Data on cases in households (size 1 to 12) 338 households: 126 had no cases 1542 individuals:
809 vaccinated, 85 cases
733 unvaccinated, 425 cases
Objective: estimate vaccine efficacy
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Disease transmission model
Population divided into separate households. Divide transmission into community-acquired
and within-household.
q = P( individual avoids outside infection ) = P ( one individual fails to infect another
in the same household )
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q
Disease transmission model
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Vaccine response model
For a vaccinated individual, three responses can occur: complete protection; vaccine failure; or partial protection and infectivity reduction.
c = P(complete protection) f = P(vaccine failure) a = proportionate susceptibility reduction b = proportionate infectivity reduction
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Vaccine response : (A,B)
A convenient way of summarising the random response is to suppose that an individual’s susceptibility and infectivity reduction is given by a bivariate random variable (A,B). Thus
P[ (A,B) = (0, -)] = c
P[ (A,B) = (1,1) ] = f
P[ (A,B) = (a,b) ] = 1-c-f
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Efficacy Measures
Furthermore it is sensible to define measures of vaccine efficacy using (A,B).
VES = 1- E[A] = 1 - f - a(1-f-c) is a protective measure VEI = 1 - E[AB] / E[A] = 1 - [f + ab(1-f-c)] / [f + a(1-f-c)] is a measure of infectivity reduction Note both are functions of basic model
parameters
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Bayesian inference
Object of inference is the posterior density
( | n ) = ( a,b,c,f,q, | n )
where n is the data set. By Bayes’ Theorem (| n ) (n | ) ( ), where (| n ) is the likelihood, and () is the prior density for .
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MCMC details
There are six parameters: a,b,c,f,q, Each parameter has range [0,1] Update each parameter separately using a
Metropolis-Hastings step with Gaussian proposal centered on the current value
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MCMC pseudocode Initialise parameters (e.g. a = 0.5, b=0.5,…) User input burn-in (B), sample size (S),
thinning gap (T)
LOOP: counter from –B to (S x T) Update a, update b, …, update IF (counter > 0) AND (counter/T is integer) THEN store current values END LOOP
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Updating details for a
Propose ã~ N(a, 2) Accept with probability
Note that the (symmetric) proposal cancels out The other parameters are updated similarly
1)(),,,,,|(
)~(),,,,,~|(
aqfcba
aqfcba
n
n
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Trace plot for a
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Density estimate for a
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Scatterplot of a versus c
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Results for VES
Posterior mean: VES = 1 – E(A) = 0.84
Posterior Standard Deviation = 0.03
These results are easily obtained using the raw
Markov chain output for the model parameters.
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Contents 1. MCMC: overview and basics 2. Example: Vaccine efficacy 3. Data augmentation 4. Example: SIR epidemic model 5. Model choice 6. Example: Norovirus outbreak 7. Other topics
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4. Data augmentation Suppose we have a model with unknown
parameter vector = (1,2,…,n).
Available data are y = ( y1, y2,…, ym ). If the likelihood π (y | ) is intractable… …one solution is to introduce extra
parameters (“missing data”)
x = (x1, x2,…, xp)
such that π (y , x | ) is tractable.
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Data augmentation (2)
The extra parameters x = (x1, x2,…, xp) are simply treated as unknown model parameters as before.
To obtain samples from π ( y | ), take samples from π (y , x | ) and ignore x.
Such a scheme is often easy using MCMC.
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Data augmentation (3)
Can also add parameters to improve the mixing of the Markov chain (auxiliary variables).
Choosing how to augment data is not always obvious!
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Contents 1. MCMC: overview and basics 2. Example: Vaccine efficacy 3. Data augmentation 4. Example: SIR epidemic model 5. Model choice 6. Example: Norovirus outbreak 7. Other topics
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4. SIR Epidemic Model
Suppose we observe daily numbers of cases during an epidemic outbreak in some fixed population.
Objective is to say something about infection rates and infectious period duration of the disease.
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Epidemic curve (SARS in Canada)
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Model definition
Population of N individuals At time t there are:
St susceptibles
It infectives
Rt recovered/immune individuals
Thus St + It + Rt = N for all t.
Initially (S0, I0 ,R0 ) = (N-1,1,0).
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Model definition (2)
Each infectious individual remains so for a length of time TI ~ Exponential().
During this time, infectious contacts occur with each susceptible according to a Poisson process of rate /N.
Thus overall infection rate is St It/N.
Two model parameters, and .
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Data, likelihood, augmentation Suppose we observe removals at times
0 ≤ r1 ≤ r2 ≤ … ≤ rn ≤ .
Define r = ( r1, r2 , …, rn ). The likelihood of the data, π (r | , ), is
practically intractable. However, given the (unknown) infection
times i = ( i1, i2 , …, in ), π (i ,r | , ) is tractable.
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MCMC algorithm
Specifically,
It follows that if π() ~ Gamma distribution then π( | …) ~ Gamma distribution also.
Same is true for . So can update and using a Gibbs step.
0
tt
1
r
2
ii )dt γIIS(expIIS ),|,( t j j j βγβγβn
j
n
j
ri
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MCMC algorithm – infection times
It remains to update the infection times i = ( i1, i2 , …, in )
Various ways of doing this. A simple way is to use a M-H scheme to
randomly move the times. For example, propose a new ik by picking a
new time uniformly at random in (0,).
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Updating infection times
I6
I6
Updating I2 :
Acceptance prob. π (i*,r | ,) / π (i,r | ,)
I4I2
I2*I4
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Extensions
Epidemic not known to be finished by Non-exponential infectious periods Multi-group models (e.g. age-stratified
data) More sophisticated updates of infection
times Inclusion of latent periods
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Contents 1. MCMC: overview and basics 2. Example: Vaccine efficacy 3. Data augmentation 4. Example: SIR epidemic model 5. Model choice 6. Example: Norovirus outbreak 7. Other topics
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5. Model Choice
Bayesian model choice problems can also be implemented using MCMC.
So-called “transdimensional MCMC” (alias “Reversible Jump MCMC”) is used.
The basic idea is to construct the Markov chain on the union of the different sample spaces and (essentially) use M-H.
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Simple example
Model 1 has two parameters: , Model 2 has one parameter: The Markov chain moves between models
and within models E.g. Xn = (1, , , ) for model 1, ignore Practical question – how to jump between
models?
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Contents 1. MCMC: overview and basics 2. Example: Vaccine efficacy 3. Data augmentation 4. Example: SIR epidemic model 5. Model choice 6. Example: Norovirus outbreak 7. Other topics
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6. Example: Norovirus outbreak
Outbreak of gastroenteritis in summer 2001 at school in Derbyshire, England.
A single strain of Norovirus virus found to be the causative agent.
Believed to be person-to-person spread
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Outbreak data
15 classrooms, each child based in one. Absence records plus questionnaires. 492 children of whom 186 were cases. Data include age, period of illness, times
of vomiting episodes in classrooms.
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Question of interest
Does vomiting play a significant role in transmission?
Total of 15 vomiting episodes in classrooms.
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Epidemic curve in Classroom 10
0
2
4
6
8
10
12
1 3 5 7 9 11 13 15 17
Day of outbreak
Num
ber
ill f
or f
irst
day
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Stochastic transmission model
Assumption: A susceptible on weekday t remains so on day
t+1 if they avoid infection from each infective child;
per-infective daily avoidance probabilities are classmate : qc schoolmate: qs in class, vomiters : qv
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Transmission model (2)
At weekends, a susceptible remains so by avoiding infection from all infectives in the community,
per-infective avoidance probability is q.
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Two models
M1 : Full model: qc, qv, qs, q Vomiters treated separately
M2 : Sub-model: qc, qv=qc, qs, q Vomiters classed as normal infectives
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MCMC algorithm
Construct Markov chain on state space
S = { ( qc, qs, qv, q, M) }
where M = 1 or 2 is the current model
Model-switching step to update M Random walk updates for the q’s
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Between-model jumps
Full model sub-model: Propose qv = qc
Sub-model full model: Propose qv = qc + N(0, 2 )
Acceptance probabilities straightforward
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Results
Uniform(0,1) q. priors; P(M1) = 0.5
P(M1 | data) = 0.0001 (Full model) P(M2 | data) = 0.999 (Sub model)
qc qs qv q
Mean 0.997 0.998 0.936 0.999
S. dev 0.0014 0.00018 0.018 0.000066
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Contents 1. MCMC: overview and basics 2. Example: Vaccine efficacy 3. Data augmentation 4. Example: SIR epidemic model 5. Model choice 6. Example: Norovirus outbreak 7. Other topics
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7. Other topics
1. Improving algorithm performance 2. Perfect simulation 3. Some conclusions
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Improving algorithm performance
Choose parameters to reduce correlation Trade-off between ease of computation
and mixing behaviour of chain Choice of M-H proposal distributions Choice of blocking schemes
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Perfect simulation
Detecting convergence can be a real problem in practice.
Perfect simulation is a method for constructing a chain that is known to have converged by a certain time.
Unfortunately it is far less applicable than MCMC.
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Some conclusions
MCMC methods are hugely powerful The methods enable analysis of very
complicated models Sample-based methods easily permit
exploration of both parameters and functions of parameters
Implementation is often relatively easy Software available (e.g. BUGS)