Bayesian Experimental Design with Stochastic
Epidemic Models
Gavin J Gibson & Alex Cook
School of Mathematical & Computer Sciences and Maxwell Institute for
Mathematical Sciences, Heriot-Watt University, Edinburgh, UK
Chris Gilligan
Department of Plant Sciences, Cambridge University, UK
Acknowledgements: BBSRC
•Bayesian inference for stochastic epidemic models
•The design problem
•The need for approximation
•Application to the design of microcosm experiments
•Future work
Epidemics: Observable and unobservable processes
Consider SEIR model for an epidemic in a closed population that
mixes homogeneously.
History of individual i.
time
S
susceptible
E
I
exposed
?
Diagnostic tests:
R
infective
?
?
•Limited frequency of tests
•Certain states indistinguishable (e.g. S & E)
• False negative and positive results
removed
?
Stochastic model:
S E: If j is in state S at time t, then
Pr(j is exposed in (t, t+dt)) = bI(t)dt
E I:
TEj ~ EE
(random time in E)
I R:
TI j ~ II
(random time in I)
Parameters:  = (b, E, I)
How can we estimate the parameters from the observations?
Likelihood and Bayesian inference in a nutshell…
Given observations y (from model M with unknown
parameters, ) likelihood principle says all evidence about 
contained in likelihood
L(|y) = Pr(y|) “the probability of the observations given ”.
Bayesian approach. Represent prior beliefs re  as density
(). Update in light of y using Bayes’ Theorem
(|y)  ()L(|y).
When  multivariate, inference on individual components made
using marginal posterior density.
Bayesian inference for epidemic models
•Let y represent observed data (incomplete), z represent
complete data (times and nature of all events).
•Problem is that Pr(z|) usually tractable, but
Pr( y |  )   Pr( z |  )dz
Integrate over all z
consistent with y
Solution: Consider ‘hidden’ aspects, x, of the data as additional
unknown parameters. Investigate the joint posterior density (,
x |y). Make inference on  by marginalisation.
Now, by Bayes’ Theorem
(, x |y)  ()(x, y|)
Likelihood for
augmented data
(tractable)
Using Markov chain methods we can generate samples {(i, xi)}
from (, x|y), where
(, x |y)  ()(x, y|)
Construct Markov chain with this stationary distribution which
iterates by proposing & accepting/rejecting changes to the
current state (i, xi) to obtain (i+1, xi+1).
Updates to (some components of)  can often be carried out by
Gibb’s steps.
Updates to x, usually requires M-H and RJ type approaches.
Algorithms of this type have been developed and applied to the
analysis of data (see e.g. GJG (1997), GJG & ER (1998, 2001),
GS & GJG (2004a, 2004b), O’Neill & Roberts (1999), O’Neill
& Becker (2001), Forrester et al. (2006))
Example. Epidemics of R solani in radish (with Plant Sciences, Cambridge)
Population size 50 (with and without added T. viride).
No Trichoderma
Trichoderma added
A simple model for this is the SEI with primary, secondary
infection + quenching (Gibson et al., PNAS, 2004)
For any susceptible individual at time t, the probability of
becoming exposed in the time interval [t, t+dt) is given by
R(t) = (rp + rsI(t))exp{-at}.
Exposed individuals become infectious at fixed time m after
exposure.
Using vague uniform priors over finite windows we obtain
the following posteriors for the model parameters, suggesting
that Trichoderma acts on primary infection process.
PRIMARY (rp)
SECONDARY (rs)
QUENCHING (a)
LATENT PERIOD (m)
Statisticians (like me) have tended to focus on analyses of
particular data sets, often historical to illustrate the value of the
Bayesian approach. Emphasis placed on tackling the complexity
arising from partial observation and the need to respect the
likelihood principle.
Experimental design requires us to address the uncertainty in
future realisations of a process and the range of ways in which it
could be observed.
To address these two sources of complexity simultaneously in
the context of epidemic models (stochastic, spatial, nonlinear) is
a major challenge.
Experimental design with stochastic epidemic models:
Propose to use Bayesian framework (Muller et al., 1999) for
identifying optimal designs.
 ~ () represents current belief regarding .
z ~ (z | ) – ‘complete’ future realisation of process.
d(z ) – censored/filtered version of z arising from design d
chosen from some suitable space of designs.
U(d(z )) utility function quantifying information in d(z ), and or
cost of the experiment.
Aim: Select d to maximise the expectation of U(d(z ))
Identifying optimal designs
‘Trick’ is to treat the design, d, as an additional variable and
assign joint distribution to (, z, d) so that the joint density is,
f(, z, d)  ()(z|)U(d(z))
Integrating w.r.t. z, we obtain f(d|)  E(U(d)|), the
expectation of U conditional on .
Integrating with respect to  we find that f(d)  E(U(d)).
Hence, identifying the optimal design for future experiment
given current belief re  is equivalent to identifying the mode of
f(d). In theory we could investigate f using MCMC methods.
Looks simple – where do the difficulties arise?
If you’re a Bayesian then you should base your utility on the
posterior density you obtain in an experiment.
One sensible(?) measure would be Kullback-Leibler
divergence between prior and posterior. Let y = d(z)
Then U(d(z)) =
   | y  
   | y log     d
May be nasty!
However, the posterior (|y) is usually difficult to obtain.
Therefore we will have to use approximations to it.
Go back to the R solani system.
Example. Epidemics of R solani in radish (with Plant Sciences, Cambridge)
Let’s look at problem of selecting a sparse set of m observation
times for microcosm experiments. We focus on a simpler
model in which the latent period is assumed to be zero.
e.g. m = 3
I(t)
t1
t3
t2
For this experiment y = (I(t1), I(t2), I(t3)),  = (rp, rs, a).
   | y  
d
    
We wish to approximate    | y  log 
First approximation: Approximate the prior with a discrete
uniform prior by drawing a random sample from .
Second approximation: Approximate the likelihood L()
using moment-closure methods (e.g. Krishnarajah et al., 2005)
Basic idea: L() = P(I(t1) = y1|)×P(I(t2) = y2 | I(t1) = y1, )
× P(I(tk) = yk | I(tk-1) = yk-1, ) …….
Approximate each term individually using moment-closure.
First suppose a = 0. Now, for t > tk, let n0 = N – I(tk),
C = rp + rsI(tk), J(t) = I(t) – I(tk).
Effective primary
infection rate
Infections after tk
Remaining
susceptibles
Idea of MC is to approximate the evolution of moments of
population variables by a system of differential equations.
For our simple model:
Pr(J(t+dt) – J(t) = 1) = (C(n0-J(t)) + rsJ(t)(n0-J(t))dt
= {Cn0 + ( rsn0 - C)J(t) – rsJ2(t)}dt

d E(J(t)) = Cn + (r n - C)E(J(t)) – r E(J2(t))
0
s 0
s
dt
This equation involves the 2nd moment of J(t).
Pr(J2(t+dt) – J2(t) = 2J(t) + 1) = (C(n0-J(t)) + rsJ(t)(n0-J(t))dt
= {Cn0 + ( rsn0 - C)J(t) – rsJ2(t)}dt
E(J(t+dt)-J(t)|J(t)) =
(2J(t) + 1){Cn0 + (rsn0 - C)E(J(t)) – rsE(J2(t)}dt
d

E(J2(t)) = Cn0 + (rsn0 + 2Cn0 – C)E(J(t))
dt
- (2n0rs – rs -2C)E(J2(t)) – 2CE(J3(t))
Unfortunately, 3rd moment appears on r.h.s.! In general,
equation for kth moment includes terms in (k+1)th moment.
Solution: Close the system of equations by assuming that
J(t) has a particular distribution. Here we assume that
J(t) ~ BetaBin(n0, a(t), b(t)).
a(t) and b(t) are determined by the first two moments.
E(J3(t)) is then a function of a(t) and b(t), and hence the first
two moments. Substituting this function into the differential
equation for E(J2(t)) allows the system to be closed and
solved using Euler’s method.
Previous work (Krishnarajah et al., BMB 2005) indicates
that the BetaBin distribution provides a good approximation
to the distribution of J(t).
Comparison of moment-closure with exact probability function.
No Trichoderma
t =1
t=5
t = 15
Trichoderma
Estimating L(rp, rs, a)
I(t3)
I(t)
I(t2)
L = p1p2p3
I(t1)
t1
Integrate ODE to get
distribution for I(t1),
given I(0) = 0.
t2
Integrate ODE to get
distribution for I(t2),
given I(t1).
t3
Integrate ODE to get
distribution for I(t3),
I(t2).
Full algorithm for experimental design
Recall we wish to draw from joint density
f(, z, d)  z|)U(d(z))
where d represents a set of m distinct sampling times
arranged between t = 0 and tmax.
 = (rp, rs, a) and is a priori uniform over a finite set of points
(random sample from continuous ).
z represents complete process and comprises the infection
times for all individuals in the population.
Outline of steps for updating (i, zi, di)
1. Propose new (′, z′) by drawing ′ from the prior and
simulating realisation z′ from the model.
Accept with probability min{1, U(di(z′))/U(di(zi))},
otherwise reject.
2. Propose changes to sampling times e.g. using M-H
methods. Proposed d′ can be di with perturbation applied
to one of sampling times.
t1
t2
tj
tm
di
d′
Accept with prob. min{1, U(d′(zi+1))/U(di(zi+1))}
Note:
•The algorithm identifies optimal design for a single-replicate
experiment. Optimal designs for multi-replicate experiments
look broadly similar.
•The utility can be ‘sharpened’ by adapting the algorithm to
propose k independent (, z) combinations. Now f(d) 
[E(U(d(z))]k.
We illustrate some optimal designs for the unquenched SI
model. (See Cook et al., in prep.)
Now a more practically relevant situation – designing
experiments for the R solani system (without
Trichoderma).
rp
rs
Sample from joint posterior gives new prior, ′.
a
We consider 2 situations:
Progressive design: How should you repeat experiment to
maximise expected information change w.r.t. ′? (Uses ′ to
propose new z’s and ′ as prior in utility calculation.)
Confirmatory (pedagogic) design: How should you repeat
experiment to maximise expected information change w.r.t. ?
(Uses ′ to propose new z’s and  as prior in utility calculation.)
Designs constrained to be subset of sampling times in original
experiment.
pedagogic
rp
progressive
rs
a
Results: Designs and their application to analysis of original
data. Posteriors shown are for pedagogic designs.
Further work:
•Designs for imperfect diagnostic tests
•Adaptive designs
•Extension to spatio-temporal modelling – designs include
location of host or inoculum.
•Non-Markovian systems.
Efficient approximation of intractable likelihoods is key to all
of these problems.