Centre for the Mathematical Modelling of Infectious Diseases. London School of
... Overview. Introduction to infectious disease modelling. Research outline.
Helen Johnson, John Edmunds and Richard White Centre for the Mathematical Modelling of Infectious Diseases London School of Hygiene & Tropical Medicine
Overview Introduction to infectious disease modelling Research outline Preliminary work Ideas for the future
Modelling transmission and control of infectious diseases Very non-linear
Potential to invade a population crudely measured by R0 - the number of secondary infections caused by one infection in a totally susceptible population Don’t need to cure all infections to eliminate
disease Just get R0 < 1
100%
Endemic Prevalence
Long history: Ross Macdonald modelling Malaria in ~1910
Hyperendemic malaria >>> Measles
80% 60% HIV
40%
Flu
20% 0% 0 2 4 6 8 10 12 14 16 18 20
Basic reproduction number
For sexually transmitted infections R0 = c β D
C - number of sexual partnership per unit time
β – transmission probability per partnership D – Duration of infectiousness
Historically used deterministic compartmental models using ODEs Very quick, fitted using ML ..
Complex (e.g. individual-based) models are
increasingly used for decision-making in the control of infectious diseases Suitability depends on how well fitted to empirical
data and how well models can be analysed Fitting of complex infectious disease models is
often poor since most formal methods (incl. distance-based and likelihood-based measures) require models run many times Urgent need to develop methods to robustly
calibrate and analyse complex models for infectious disease control
Aims and objectives AIM: To develop and evaluate methods to calibrate and analyse complex individual-based stochastic models, and apply them to explore the impact of anti-retroviral therapy (ART) on HIV/AIDS in Africa. OBJECTIVES: Develop novel methods for Bayesian Emulation of stochastic models Compare the accuracy and efficiency of existing MCMC, ABC and novel Bayesian Emulation methods for the calibration and analysis of individual-based stochastic models Develop and evaluate a novel hybrid model calibration strategy combining the strengths of both ABC and Emulation methods Apply developed and evaluated methods to predict the impact of ART on HIV/AIDS in Africa Work with collaborators to apply developed and evaluated methods to other health questions
Who London School of Hygiene and Tropical Medicine (White,
Johnson, Edmunds, Hayes)
Infectious disease modelling
Statistics group in the Department of Mathematical Sciences,
Durham University (Goldstein, Vernon) Linear Bayes Emulation
Department of Probability and Statistics, Sheffield University
(Oakley)
Fully Bayesian Emulation
Cambridge University (Wood, McKinley) Approximate Bayesian Computation MRC/UVRI Uganda Research Unit on AIDS (Nsubuga, Levin) HIV/AIDS data
1: Develop novel methods for Bayesian Emulation of stochastic models Use both the fully Bayesian and the Bayes Linear approaches Fully Bayesian: assume form of the output distribution and
emulate it directly Bayes Linear: emulate means and variances only Both emulators will then be used in an adapted version of the History Matching process that results in the iterative discarding of large, unacceptable parts of the input parameter space. As the Bayes Linear approach involves purely means, variances and covariances => should be able to handle models with more inputs Initially emulate univariate, then, multivariate outputs using existing multivariate emulation techniques
2: Compare the accuracy/efficiency of existing MCMC, ABC and novel Bayesian Emulation methods Model Mukwano – an individual-based stochastic model of the
transmission of sexually transmitted infections (Santhakumaran et al, STI, 2010) 5 to 50+ inputs ; 1 to 20 outputs Experimental design is based on that used by Rutter et al, JASA, 2009 Iterative design using an increasingly complex model In each iteration, evaluate accuracy and efficiency of MCMC, ABC &
Emulation methods in estimating parameters A priori, we expect increasing model complexity will prevent MCMC, then ABC methods, and perhaps even Emulation from calibrating the models in ‘reasonable‘ times
Objectives 3, 4 & 5 3.
Develop and evaluate a novel hybrid model calibration strategy combining the strengths of both ABC and Emulation methods Use Bayes Linear Emulation & history matching to evaluate large areas of parameter space Use ABC generate probabilistic statements about goodness of fit
4.
Apply developed & evaluated methods to predict impact of antiretroviral therapy (ART) on HIV/AIDS in Africa
5.
Work with collaborators to apply developed and evaluated calibration methods to other public health questions that require complex models, including HPV and cervical cancer in UK and Sweden
Preliminary work: methods First steps to emulating a stochastic function: build an emulator formed on the basis of averaged output values from the complex stochastic model Tentative steps towards model fitting: using emulator runs to exclude parameter space
Preliminary work: emulating a stochastic function Initial investigation to assess emulator performance at
matching mean stochastic model output. Sampled 88 sets of 11 behavioural input parameters
using space-filling design (LHS) -> Mukwano Trained emulator with the mean output from 1000
complex model runs Used emulator to predict HIV prevalence at
a further 18 points
Preliminary work: emulating a stochastic function Behavioural input parameters (11): Time between sex(1) New partner acquisition rate (p.a.) for males/females in long-term/casual partnerships (4) Mean partnership duration for long-term/casual partnerships (2) Proportion of males/females who are non-monogamous (2) Tendency of males/females who are non-monogamous to take additional partners (2) Output: HIV prevalence in males (2000)
Preliminary work: results 8% Male HIV prevalence (2000)
Emulator
Complex model
7% 6% 5% 4% 3%
`
2% 1% 0% 1
2
3
4
5
6
7 8 9 10 11 12 13 14 15 16 17 18 19 Parameter set number
Comparison of complex model and emulator output for 19 parameter sets in wave 1.
Preliminary work: fitting methods 2 wave fitting procedure Wave 1: 88 sets of 11 behavioural input parameters -> Mukwano.
Point estimates for male HIV prevalence in 2000 calculated from mean of 1000 runs.
Trained emulator with ensemble of input parameter sets and
Mukwano output
Made predictions for 1000 further input parameter sets Determined fitting sets according to UNAIDS HIV
prevalence data for adults (15-49) in Ghana in 2000. (2.2% -2.7%)
Preliminary work: fitting methods Wave 2: Restricted input parameter ranges very crudely,
sampling between the maximum and minimum values that had yielded fits from the emulator in wave 1 Generated point estimates for male HIV prevalence in 2010 for
these new input parameter sets using averaged output from Mukwano Trained a second emulator with new ensemble and made
predictions for 1000 further input parameter sets Again identified fits to the UNAIDS HIV prevalence data
for Ghana
Preliminary work: wave 1 fit results 35
emulator predictions cover parameter space well.
Aver age time between sex (days)
1000 parameter sets for
30 25
20 15
All inputs
10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of males available for concurrent partnerships
Preliminary work: wave 1 fit results 35
for emulator predictions cover parameter space well. However, fits are only
found for a muchreduced range of time between sex, roughly corresponding to once every 2 to 10 days
Aver age time between sex (days)
1000 parameter sets
30 25
20
All inputs
15
Fits 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of males available for concurrent partnerships
35
35
30
30
25 20
All inputs
15
Fits 10 5 0
Average time between sex (days)
Aver age time between sex (days)
Preliminary work: wave 1 fit results 25 20
All inputs
15
Fits 10 5
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of males available for concurrent partnerships
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Proportion of females available for concurrent partnerships
Preliminary work: wave 1 fit results Proportion available for concurrent sexual partnerships, by gender
1000 parameter sets for
emulator predictions cover parameter space well.
1.0 0.9 0.8
Females
0.7 0.6 0.5
All inputs
0.4 0.3 0.2 0.1 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Males
Preliminary work: wave 1 fit results Proportion available for concurrent sexual partnerships, by gender
1000 parameter sets for
emulator predictions cover parameter space well.
1.0 0.9 0.8
Fits are found for
widely varying proportions of availability for concurrency in both males and females.
Females
0.7 0.6 0.5
All inputs
0.4
Fits
0.3 0.2 0.1 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Males
Preliminary work: wave 2 fit results 1000 parameter sets for Average time between sex (days)
emulator predictions cover reduced parameter space well.
35 30
25 20
All inputs
15 10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of males available for concurrent partnerships
Preliminary work: wave 2 fit results 1000 parameter sets for
Start to see some more
detail in the distribution of fitting parameters
Average time between sex (days)
emulator predictions cover reduced parameter space well.
35 30
25 20
All inputs 15
Fits
10 5 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of males available for concurrent partnerships
Preliminary work: wave 2 fit results Parameter ranges for the
Proportion available for concurrent sexual partnerships, by gender
proportion of males/females available for concurrency are not diminished by much.
1.0 0.9 0.8
Females
0.7 0.6 0.5
All inputs
0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Males
Preliminary work: wave 2 fit results Parameter ranges for the
Proportion available for concurrent sexual partnerships, by gender
proportion of males/females available for concurrency are not diminished by much.
1.0 0.9
However, the reduced definition
It is not possible to fit the
observed HIV prevalence if a high proportion of males and a low proportion of females are available for concurrent sexual partnerships
0.7
Females
of other input parameter ranges (e.g. time between sex) restricts the ‘allowed’ availability for concurrency
0.8
0.6 0.5
All inputs
0.4
Fits
0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Males
Preliminary work: limitations Many!! No attempt to emulate variance Method for reduction of parameter space is very crude. Need to restrict
parameter ranges on the basis of probability distribution of fits .This would allow better resolution of local maxima in fitting probability Need to conduct further waves with ever more stringent fitting criteria
Only fitting to one output Too few complex model runs -> large stochasticity Too few training runs for emulator
Further work Develop novel methods for Bayesian Emulation of stochastic
models
Compare the accuracy and efficiency of existing MCMC, ABC
and novel Bayesian Emulation methods for the calibration and analysis of individual-based stochastic models
Develop and evaluate a novel hybrid model calibration strategy
combining the strengths of both ABC and Emulation methods
Apply developed and evaluated methods to predict the impact of
ART on HIV/AIDS in Africa
Work with collaborators to apply developed and evaluated
methods to other health questions
Future Large amounts of health spending is directed based on
the use of complex stochastic models (e.g. HIV, Malaria, Gates Foundation) -> urgent need to develop methods for their calibration and analysis . This work aims to begin addressing this need
Preliminary results are promising
Acknowledgements Andy Cox, Toby Ealden, Katie O’Brien, LSHTM Ian Vernon & Michael Goldstein, Durham Jeremy Oakley, Sheffield
Lower bound Time between sex
Upper bound
Mode
1/365
1/12
1/106.8
cm0
0
0.2
cf0
0
0.1
cm1
0
2.5
cf1
0
1
d0
5
10
d1
0
1
θm1
0
1
0.5
θf1
0
1
0.15
P1 (m)
0
1
0.7
P1 (f)
0
1
0.5
