MCMC for Stochastic Epidemic Models Philip D. O’Neill School of Mathematical Sciences University of Nottingham

This includes joint work with… Tom Britton (Stockholm) Niels Becker (ANU, Canberra) Gareth Roberts (Lancaster) Peter Marks (NHS, Derbyshire PCT)

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

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.

How MCMC works Key idea is to construct a discrete time Markov chain X 1, X 2, X 3, … on state space S whose stationary distribution is π. If P(dy,dx) is the transitional kernel of the chain this means that

How MCMC works (2) Subject to some technical conditions, Distribution of X n → π as n →  Thus to obtain samples from π we simulate the chain and sample from it after a “long time”.

Example: π (x)  x e -2x

X N = X 1, X 2, …

Example: π (x)  x e -2x X N = X N+1 =

Example: π (x)  x e -2x X N = X N+1 = X N+2 =

Example: π (x)  x e -2x Suppose Markov chain output is..., X N = , X N+1 = , X N+2 = , ….,X N+M = (i.e. discard initial N values, burn-in)

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.

Metropolis-Hastings algorithm Set an initial value X 1. If the chain is currently at X n = x, randomly propose a new position X n+1 = y according to a proposal density q(y | x). Accept the proposed jump with probability If not accepted, X n+1 = x.

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:

Why M-H works (2)

Why M-H works (3) This last equation shows that π is a stationary distribution for the Markov chain.

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)

Comments on M-H (2) In practice, MCMC is almost always used for multi-dimensional problems. Given a target density π(x 1, x 2, …,x n ) it is possible to update each component separately, or even in blocks, using different M-H schemes.

Comments on M-H (3) A popular multi-dimensional scheme is the Gibbs sampler, in which the proposal for a component x i is its full conditional density π (x i | (x 1,…,x i-1, x i+1, …,x n ) )  The M-H acceptance probability is equal to one in this case.

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?

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

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

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 )

q  Disease transmission model

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

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

Efficacy Measures Furthermore it is sensible to define measures of vaccine efficacy using (A,B). VE S = 1- E[A] = 1 - f - a(1-f-c) is a protective measure VE I = 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

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 .

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

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

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

Trace plot for a

Density estimate for a

Scatterplot of a versus c

Results for VE S Posterior mean: VE S = 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.

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

4. Data augmentation Suppose we have a model with unknown parameter vector  = (  1,  2,…,  n ). Available data are y = ( y 1, y 2,…, y m ). If the likelihood π (y |  ) is intractable… …one solution is to introduce extra parameters (“missing data”) x = (x 1, x 2,…, x p ) such that π (y, x |  ) is tractable.

Data augmentation (2) The extra parameters x = (x 1, x 2,…, x p ) 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.

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!

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

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.

Epidemic curve (SARS in Canada)

Model definition Population of N individuals At time t there are: S t susceptibles I t infectives R t recovered/immune individuals Thus S t + I t + R t = N for all t. Initially (S 0, I 0,R 0 ) = (N-1,1,0).

Model definition (2) Each infectious individual remains so for a length of time T I ~ Exponential(  ). During this time, infectious contacts occur with each susceptible according to a Poisson process of rate  /N. Thus overall infection rate is  S t I t /N. Two model parameters,  and .

Data, likelihood, augmentation Suppose we observe removals at times 0 ≤ r 1 ≤ r 2 ≤ … ≤ r n ≤ . Define r = ( r 1, r 2, …, r n ). The likelihood of the data, π (r | ,  ), is practically intractable. However, given the (unknown) infection times i = ( i 1, i 2, …, i n ), π (i,r | ,  ) is tractable.

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.

MCMC algorithm – infection times It remains to update the infection times i = ( i 1, i 2, …, i n ) 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 i k by picking a new time uniformly at random in (0,  ).

Updating infection times I6I6 I6I6 Updating I 2 : Acceptance prob. π (i*,r | ,  ) / π (i,r | ,  ) I4I4 I2I2 I2*I2* I4I4

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

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

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.

Simple example Model 1 has two parameters: ,  Model 2 has one parameter:  The Markov chain moves between models and within models E.g. X n = (1, , ,  ) for model 1, ignore  Practical question – how to jump between models?

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

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

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.

Question of interest Does vomiting play a significant role in transmission? Total of 15 vomiting episodes in classrooms.

Epidemic curve in Classroom 10

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

Transmission model (2) At weekends, a susceptible remains so by avoiding infection from all infectives in the community, per-infective avoidance probability is q.

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

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

Between-model jumps Full model  sub-model: Propose qv = qc Sub-model  full model: Propose qv = qc + N(0,  2 ) Acceptance probabilities straightforward

Results Uniform(0,1) q. priors; P(M1) = 0.5 P(M1 | data) = (Full model) P(M2 | data) = (Sub model) qcqsqvq Mean S. dev

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

1. Improving algorithm performance 2. Perfect simulation 3. Some conclusions

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

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.

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)