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Stochastic spatio-temporal modelling
methods in epidemiology and ecology
Gavin J Gibson
Heriot-Watt University
NERC-EMS Workshop on Inference for Stochastic
Population Models in Epidemiology and Ecology
14-17 September 2004
•Contact distribution models
•Lattice-based models
•Implementation and simulation
•Inference and application
Contact distribution models for
population spread (individual-based, pure
birth process)
Assumptions:
•Each individual in the population produces
offspring according to Poisson process with
rate l.
•Offspring displaced from parent by vector r
chosen randomly from a probability
distribution f(r) the contact distribution.
Time between births (for individual) are
i.i.d. Exp(l) random.
offspring
Parent
r
Often we choose f to be radially symmetric.
Therefore simulation involves choosing a radius,
r, from a distribution along with an angle q
uniformly in (0, 2p). New offspring placed at x
+ (r cosq, r sinq) where x is parent’s position.
If we choose r2 to have an Exp(l) distribution
then r will be a BVN(0, I/(2l)) distribution.
The qualitative nature of spatial distributions of
the populations we generate depends on the
properties of f. If f is light-tailed, births will tend
to be close to parents. If heavy-tailed, significant
probabilities of births displaced a long way from
parents. (See pictures later.)
Adding complexity:
•Survival probabilities of offspring related to
local density
•Death/removal of individuals
•Consumption of resources
•Environmental heterogeneity
•Heterotropic dispersal (prevailing winds,
etc.)
Lattice-based epidemic models, forest-fire
models are natural developments of contact
distribution models
x
y
Infective challenge Fq(x, y)
Stochastic SI spatio-temporal model:
Pr(Ix(t + dt) = 1 | Ix(t) = 0) =

I y  t ) 1
Fq  x, y ) dt
Ix(t)  “Infectious status of individual
located at x at time t.”
q is vector of parameters
Examples of F include:
• a nearest-neighbour interaction
• Fq(x, y) = be-a|x – y|
• Fq(x, y) = b|x-y|-2a
Extensions of the model include:
•Finite infective period (spatio-temporal SIR
or SIS model)
•Addition of latent period (spatio-temporal
SEIR model)
•Pre-symptomatic period
•Addition of further transmission routes e.g.
primary infection corresponding to infection
from external sources.
Studying spatio-temporal models by
simulation is relatively straightforward.
Simulation of a simple SI process on a lattice
At time t, infective set Y = {y | Iy(t) = 1}. To
compute time and location of next infection:
1. For each susceptible, x, calculate its total
infection rate R  x )   Fq  x, y ) dt
I y  t ) 1
2. Choose time till next infection
T ~ Exp(Sx R(x))
3. Choose location x with probability
proportional to R(x).
4. Update t and Y and go back to 1.
If interactions exponentially bounded then
emerging patterns start to look essentially
like expansion of foci. Wave dynamics
result.
Heavy-tailed interaction functions produce
patterns that are ‘patchy’. Apparent
expansion from several foci.
When using models for prediction it is
important to be able to estimate
characteristics of spatial interaction
functions.
What about inference?
Suppose we observe the process through time –
how can we estimate parameters?
Given complete data {x, t(x)} we can calculate
a likelihood
L(q) = “Pr({x, t(x)} | q)”.
Example: nearest-neighbour interaction
Fq(x, y) = q if x and y are NN, 0 otherwise.
Observed over period [0, 2]
1.0
0.8
0
0.5
Numbers denote observed infection times.
In this way a likelihood can be built up.
However, we don’t observe populations
continuously in practice. In a real experiment
infection times would be censored (known to
lie within some interval) e.g. if we observe
infected set at distinct observation times t1, t2,
…, tn.
The problem is now one involving missing
data. Problem can be solved in a Bayesian
framework. Let y be the observations and x the
exact times of infection. Then investigate
p(q, x |y)  p(q)f(x, y|q)
using MCMC.
An example (1): Citrus tristeza virus.
(see GJG, Applied Statistics 1997)
Data: 2 snapshots of the epidemic at times
1 year apart (Marcus et al.)
Aims: Understand spatial aspects of
transmission.
Models: simple SI spatio-temporal with
interactions
•Fq(x, y) = be-a|x – y|
• Fq(x, y) = b|x-y|-2a
A simplification: Suppose we did the
following experiment. Given the locations of
the 1st 131 infections, record the locations
the next 45 infections (without measuring
times or orderings). Call this set of locations
X and let W denote the set of all possible
orderings of those locations.
For this experiment the likelihood
L(q|X) =
 Pr  w | a , b )   Pr  w | a )
wW
wW
Therefore we can forget about b, since sets
of infected sites and orderings thereof are
independent of it.
Estimate a in Bayesian framework by
investigating
p(w, a| X)  p(a)Pr(w|a)
by MCMC. (See GJG, 97)
•Consider a discrete parameter space for a.
•Updates to a can be done by a Gibb’s step.
• Updates to w can be done by Metropolis
methods by proposing swaps to adjacent
pairs in the ordering.
Extensions: Single patterns can be analysed
if we propose a model for the diseased set at
an earlier time. For example, we might
assume that the epidemic arises from a single
infection randomly placed in the population.
An example (2):
R. Solani (fungal pathogen) in radish (host)
Host plants infected through:
•primary infection (inoculum in soil);
•secondary infection from previously
infected plants;
•Infectivity/susceptibility varies as plants
develop.
Experiments aim to quantify the dynamics of
spread and how they depend on a range of
factors (inoculum density, presence of biocontrol, etc.)
Microcosm experiments
Small experimental populations, highly
controlled conditions
Lattice of seeds in
sandy matrix
Primary inoculum
(randomly placed)
Observe over time
Symptomatic
seedling
Spatio-temporal model for symptom progress
The model uses a percolation approach.
•The population is represented as being located
at the vertices of a square lattice L
•At time t = 0, a subset X0 L is inoculated with
the fungus. Any x X0 develops symptoms at
time T ~ Exp(a) (if not already symptomatic)
•Secondary infection is nearest-neighbour
x
y
If x develops symptoms at time t, then
neighbour y develops symptoms at t + Txy where
Txy ~ Exp(f(b, t)) (if not already symptomatic).
Txy are independent over y (c.f. bond
percolation)
Txy ~ Exp(b0 exp( b1 (log (t  4) / b2 ))2 )
(cf. Filipe et al. B. Math. Biol. (2004))
Application of model to microcosm data
All cases 18 x 23 grid of plants, tmax – 21days,
roughly daily sampling:
High inoculum: 45 randomly chosen sites
Low inoculum: 15 randomly chosen sites
•Patterns not always connected so that purely nn transmission gives vanishing likelihood! Add
small infection rate (10-7) for ‘spurious’ primary
infection.
•Some sites fail to germinate
Histogram estimates of posterior densities for 4
parameters, based on 105 iterations.
Missing O, Primary inoc. +
Symptomatic day 9 X
Parameter estimation using MCMC
(Gibson et al. (submitted))
Main features:
•Propose (independent) priors for a, b0, b1,
b2;
•Investigate joint posterior
p(q, x| y)  p(q)L(q |x, y)
Daily recordings
Precise
infection times
MCMC methods uses mixture of Gibb’s steps
and Metropolis steps to investigate this
posterior density.
High inoculum
Low inoculum
High inoculum
Low inoculum
High inoculum
Low inoculum
High inoculum
Low inoculum
Conclusions?
•Little evidence of differences between the
treatments
•Evidence of within-treatment differences
between replicates
•Clear evidence of a non-stationarity in
secondary infection rates
But ….. all this depends on how appropriate
the model is. How can we assess the fit of
spatio-temporal models and select between
competing models in any given scenario?
Summing up:
•Growing body of methodology for fitting
spatio-temporal stochastic models to data.
•Maximise insights in studies where spatial
information is recorded
•Essential for assessing control strategies for
spatio-temporal processes
•More advanced applications, increasing
complexity (see e.g. Lara Jamieson’s talk)
•Many challenges! Inferences only as good
as the model is appropriate.