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Neil Ferguson
MRC Centre for Outbreak Analysis and Modelling
Dept. of Infectious Disease Epidemiology
Faculty of Medicine
Imperial College
What is a model and
why use one?
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Why use a model?
• Many uncertainties about emergence/spread of pathogens.
• Often limited historical data.
• Hence models necessarily simplify, make assumptions.
• So why model?
• Because without a model, judgements are made on the basis of
qualitative evidence/opinion/prejudice…
• Models at least have the benefit of
Making assumptions explicit.
Making best use of limited data.
Highlighting key factors determining policy needs.
Being quantitative (e.g. how many doses needed?)
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What do infectious diseases
have in common?
• Transmission.
• Via
Aerosol/droplets (measles, mumps,
influenza, pertussis…).
faecal-oral -water-borne/environmental
(Enteroviruses, Rotaviruses, Typhoid,
Cholera, Dysentry, tapeworms,
nematodes).
Sexual contact (HIV, gonorrhoea, syphilis,
chlamydia, HBV)
Vectors (dengue, malaria, onchocerciasis,
nosocomial infections…)
Intermediate hosts (schistosomiasis).
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One person gets infected.
That person infects others.
They infect more.
Giving a chain reaction.
Exponential growth
• The most important quantity governing an epidemic is how
many other people one person infects.
• = the Basic Reproduction Number of an epidemic – R0.
•Needs to be >1 for an epidemic to take off.
• Other quantities – e.g. Generation time=Tg – also important.
But the end result is the same…
0
1
2
3
4
5
6
7
8
1 2 3 4
t
Y
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When does exponential spread stop?
Rate
of
new
in
fecti
on
s
establish-
ment
Time
exhaustio
n o
f
suscep
tible
s
endemicity
Equilibrium,
or recurrent
epidemicsy
e(R
0-1
)/TG
t
Random effects
• Epidemic eventually begins to run out of people to infect.
• Then the number of secondary cases per case drops below R0 –
instead defined by R, the effective reproduction number = s×R0
(s = proportion still susceptible).
• Once s<1/R0 (so R<1), the epidemic goes into decline.
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Controlling infectious diseases
- what does it take (in theory)?
• To control an epidemic a policy needs to
reduce R<1 – so transmission cannot
sustain itself.
• So need to eliminate a fraction 1-1/R0 of
transmission – i.e. 50% for R0 =2, 75% for
R0 =4, 90% for R0 =10.
• This can be achieved by:
Reducing contacts
(quarantine, social distancing).
Reducing susceptibility
(vaccination, prophylaxis).
Reducing infectiousness
(e.g. treatment).
• Key issues are who is targeted, how much
effort is needed, and how fast?
persistence
100%
0 5 10 15 20
p
eradication
pc = 1-1/R0
50%
0%
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Epidemic models
• Just capture these ideas mathematically.
• A couple of minor challenges :
How do we estimate R0 (and Tg) for a particular disease and
population?
How do we estimate the effect of control measures on these
parameters?
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Deconstructing R0
• Not a fundamental biological constant.
• Determined by:
Pathogen biology (pathogenesis, lifecycle, variability).
Host factors (genetics, nutrition, age, co-morbidities).
Population structure (demography, contact patterns).
• Understanding these at a level which lets R0 to be estimated is
what a lot of quantitative infectious disease epidemiology is about.
• Need mechanistic understanding (not just curve fitting) to predict
impact of controls.
• Need DATA.
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Simple example
R0 = D ×C × p
Mean length
of time infectiousRate at which
contacts
occur
Probability of
transmission per
contact
- Highly simplified, as only applies if all contacts have an
equal risk of infection, and if contacts are not repeated.
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Data: natural history - not just SIR
• In reality, diseases develop gradually – need to allow for incubation
period (no symptoms) , variable infectiousness, morbidity/mortality.
• e.g. Smallpox:
The 2 week
incubation period is
what let smallpox to
be eradicated
‘Removed’
(immune
or dead)
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Data: transmission
• Almost never observed.
• Little quantitative data on
mechanisms.
• Some estimates of
transmission rates for small
groups (e.g. households),
derived via painstaking cohort
sudies.
• But mostly transmission
parameters have to be
estimated by matching models
to surveillance data.
-
0.20
0.40
0.60
0.80
1.00
0 1 2 3 4 5 6
-
0.20
0.40
0.60
0.80
1.00
0 1 2 3 4 5 6
-
0.20
0.40
0.60
0.80
1.00
0 1 2 3 4 5 6
-
0.20
0.40
0.60
0.80
1.00
0 1 2 3 4 5 6
Number of people infected
Pro
po
rtio
n
Household data for flu
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Data: surveillance
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
1400.00
1965 1969 1973 1977 1981 1985 1989 1993 1997 2001 2005
Influenza-like i l lness (ICD9 487) first (F) and new (N) episodes
Incidence rates per 100,000 Total
• e.g. For flu
• GP consultation rates
for E&W (RCGP).
• Affected by healthcare
seeking behaviour.
• Often not flu (e.g.
RSV).
• Only measures
disease, not infection.
• Unknown
ascertainment.
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Data: contact patterns
Defining ‘relevant’ contacts often a challenge – STIs the easiest:
Gregson et al, Lancet 2002
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Genetic/antigenic data
• Increasing volumes of
pathogen sequence
data.
• Population structure
and polymorphisms still
often not well
understood.
• Antigenic (strain) data
also often available –
and linked to genetic
data for RNA viruses,
but not for many more
complex pathogens.
• Molecular basis of
transmissibility very
poorly understood.
ThD1-0041/82China.Guangzhou/80
ThD1-0037/88ThD1-0036/88
ThD1-0336/91ThD1-0031/91ThD1-P0153/92ThD1-0123/92
ThD- K0127/94ThD1-0398/89
ThD1-0848/90ThD1-K0379/93
ThD1-0009/88ThD1-CN0323/91
ThD1-0179/87ThD1-0384/87ThD1-0875/87
ThD1-0412/86ThD1-0336/88ThD1-0746/87
ThD1-K0229/90Djibouti/98
Taiwan.765101/87ThD1-0001/89
ThD1-0178/92ThD1-0074/93ThD1-0191/93
ThD1-K0485/93ThD1-0641/90
ThD1-K0053/94ThD1-K0109/92
ThD1-S0088/92ThD1-0540/85
ThD1-0128/89ThD1-0118/83
Thailand.PUO 359/80ThD1-S0008/81ThD1-0096/81ThD1-S0081/82
ThD1-0153/81ThD1-0240/86ThD1-0023/81
ThD1-0233/80ThD1-0005/02
ThD1-0762/97ThD1-0277/97
ThD1-0002/95ThD1-0175/02
ThD1-0134/00
ThD1-0067/99ThD1-0289/97
Thailand.23-1NIID/02ThD1-0499/01
ThD1-S0102/01ThD1-0075/02
ThD1-0116/97ThD1-0081/98
ThD1-0483/01ThD1-K0013/01
ThD1-K0163/01ThD1-0876/99
ThD1-0388/98ThD1-0141/00
ThD1-K0080/01ThD1-K0851/01
ThD1-0280/97ThD1-0562/99
ThD1-0438/95ThD1-0726/99
ThD1-K0107/98ThD1-K0079/00
ThD1-K0035/00ThD1-K0051/99
Cambodia.61-1NIID/01ThD1-0119/91
ThD1-0097/94ThD1-0488/94
ThD1-0153/00ThD1-0762/99
ThD1-A0153/95ThD1-S0197/96
ThD1-0301/93ThD1-K0080/97
ThD1-K0056/96ThD1-K0060/98
ThD1-K0048/97ThD1-K0062/97
ThD1-K0052/95ThD1-K0088/95
ThD1-K0407/01ThD1-K0113/99
ThD1-0861/90ThD1-K0022/93
Japan.Mochizuki/43Hawaii/45
Indonesia.A88Tahiti.44-1NIID/01
Australia.HAT17/83Indonesia.17-1NIID/02
Philippines.PRS 228682/74Thailand.2543/63
ThD1-NB0038/83ThD1-0127/80ThD1-0442/80ThD1-0673/80
Myanmar.PRS 228686/76Myanmar.32514/98
Venezuela.28164/97Brazil/90
Peru.DEI 0151/91Argentina.297/00
Angola.RIO H 36589/88Colombia.INS 371869/96
Brazil.BE AR 404147/82Aruba.495-1/85
Singapore.S275/90Cote DÕIvoire:Abidjan/98
Nigeria.IBH 28328/68Cote DÕIvoire:Dakar.A-1520/85
0.005 subst itutions/site
I
II
III
Thai strains
1980-1994
Thai strains 1990-2002
Thai strains 1980-1983
98
96
100
100
100
100
98
100
98
100
93
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Data: interventions
• Trials (e.g.
efficacy/effectiveness)
.
• Observational
studies.
• Extrapolation
(nearly) always
needed to predict
population effects.
-1
0
1
2
3
4
5
6
0 50 100 150 200 250 300
e.g. impact of antivirals on
HIV viral load
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Knowledge synthesis
Model
Natural history
Epidemiology
Demography
Contact patterns
Interventions
Evolution Fundamental
parameters
Detailed
predictions
Control policy
optimisation
Insight into
mechanism
Not all models are mathematical!
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Roles of mathematical modelling
• Quantifying risk.
• Knowledge synthesis:
Data analysis.
Extrapolating to the future.
Optimising control policies.
• Has benefit of:
making assumptions explicit.
being testable/disprovable.
• Not all knowing, can’t predict with no data!
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Model complexity
• Many possible choices: deterministic/stochastic,
compartmental/individual-based, spatial/non-spatial,
age-structured...?
• Fundamentally, complexity should be driven by need
– what does the model need to do?
• And by data
– what assumptions/level of detail can be justified?
The art of modeling is knowing what to leave out.
Ydt
dZ
YN
XY
dt
dYN
XY
dt
dX
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Good news: models can be
(much) simpler than reality and still work
UK
e.g. Measles dynamics -
0
200
400
600
800
1000
0 1 2 3 4 5 6 7 8 9 10
Time (years)
Y
Very
simple
seasonal
SIR model
SIR model with
age structure
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• Modelling pandemic emergence in Indonesia.
• Simulation of 230 million people, with detailed
representation of population.
• But ‘only’ 5 transmission parameters.
More complex
model
0
500000
1000000
1500000
2000000
2500000
0 30 60 90 120 150
Dai
ly c
ases
Day
R0=2
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Model validation
• Key parameters should be
estimated from data.
• Models should reproduce past
epidemics (goodness-of-fit).
• But rarely get comparable
‘training’ and ‘validation’
datasets – no 2 epidemics are
quite alike.
• Sensitivity analysis important
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Trends in modelling
• Traditionally, most focus on endemic diseases (childhood diseases,
parasitic infections) – because equilibrium properties of models could be
determined analytically, and long-term control (e.g. vaccination).
• HIV and later emerging epidemics – and more powerful computers – have
moved field towards modelling dynamics of (novel) epidemics.
• Foot and Mouth Disease and SARS (& HIV/BSE!) showed potential of real-
time modelling.
• For endemic diseases, more focus on seasonal and spatial dynamics.
• Much more attention to rigorous model fitting/parameter estimation.
• Integrating genetics and epidemic modelling.
• And being relevant to public/veterinary health.
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Emerging infections – why worry?
• Pandemic = global epidemic of a
new disease.
• Starts with a zoonosis mutating to
be transmissible.
• SARS – near-pandemic.
• H5N1/Nipah/VHFs/???... – the
next pandemic?
• Can profoundly affect society.
•Risk may be increasing –
encroachment on habitats, higher
human/livestock densities…
• Black Death and syphilis
• Influenza and HIV/AIDS
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Detecting emergence
• Need to detect growing
clusters of cases of new
disease.
• Need innovative
surveillance (e.g. electronic
syndromic surveillance,
web crawling).
• Need new analytical
methods to analyse cluster
data.
• And rapid field
investigation.
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Outbreak analysis & modelling:
past examples
• UK Foot and Mouth Disease livestock
epidemic (2001) – modelling guided
control policy.
0
50
100
150
200
250
300
350
400
450
18-Feb 4-Mar 18-Mar 1-Apr 15-Apr 29-Apr 13-May 27-May 10-Jun 24-Jun 8-Jul
Date
Co
nfi
rme
d d
aily
ca
se
in
cid
en
ce
A: Several Days to Slaughter
B: Slaughter on infected premises
within 24 hours
C: Slaughter on infected and
neighbouring farms within 24 and 48
hours, respectively
Data up to 29 March
Data from 30 March
A
B
C
Model predictions by Dr Neil Ferguson, Dr Christl Donnelly & Prof. Roy Anderson, Imperial College
0
20
40
60
80
100
120
22-F
eb
1-M
ar
8-M
ar
15-M
ar
22-M
ar
29-M
ar
5-A
pr
12-A
pr
19-A
pr
26-A
pr
3-M
ay
10-M
ay
17-M
ay
24-M
ay
31-M
ay
• SARS 2003 – estimates of
transmissibility (R0~3) and mortality
(~15%).
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26
Pandemic modelling
2-4 months to peak at
source, 1-3 months to
spread to West.
Travel restrictions
would only buy a few
weeks at most.
1/3 of UK population
would become ill, 0.5-
1 million new sick
people per day at
peak.
1st wave over ~3
months after 1st UK
case.
Thailand GB
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Modelling and preparedness:
assessing control options
Treatment & prophylaxisSchool closureVaccination
Containment at source (i.e. Stopping spread when
there are only a few tens of
cases)
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Opposite scenario: eradication
- why is polio holding on in India?
• New analyses by Nick Grassly at
Imperial (published in Science, Lancet)
showed that the key problem was poor
vaccine efficacy in some parts of India.
• Trivalent oral vaccine only giving ~9%
protection in Uttar Pradesh – less than
half that achieved in the rest of India.
• So children were getting 15 doses and
still getting Polio.
• Poor efficacy linked to environmental
factors (competing infections with cross-
immunity).
• Now switching to new high potency
monovalent vaccine.
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Impact of new vaccine on Type 1 Polio
Uttar Pradesh, India
Sep 06 Oct 06 Nov 06
Dec 06 Jan 07 Feb 07
Mar 07 Apr 07
* data as on 28th June 2007
May 07
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Inferring the effectiveness of public
health measures from observational data
• Data on public health
measures often very limited.
• e.g. no data for masks.
• Can we use historical data
to reduce the uncertainty?
• We asked if public health
interventions provide a
plausible quantitative
explanation of the variation
between US cities?
0
50
100
150
200
250
300
0 90 180 270
Weekly
excess
mo
rtali
ty/1
00k
Days since Sept 7 1918
St Louis
Philadelphia
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Correlations
• Both peak and total
mortality weakly correlated
with timing of pandemic wave
and previous year’s mortality.
• Peak mortality correlated
with ‘early’ interventions.
• Peak mortality strongly
correlated with presence of 2
autumn peaks, total mortality
weakly so.
Results in agreement with 2
other analyses.
R² = 0.19
0
100
200
300
400
500
600
700
800
900
500 1000 1500 2000
To
tal
mo
rtali
ty
1917 mortality
a
R² = 0.24
0 2 4 6 8 10
First week wheremortality > 20/100,000
b
R² = 0.69
0100200300400500600700800900
1000
0 200 400
To
tal
mo
rtali
ty
Mortality to 12 daysafter intervention start
c
R² = 0.71
0
50
100
150
200
250
300
0 200 400
Peak w
eekly
mo
rtali
tyMortality to 12 days
after intervention starts
d
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Results of 1918 analysis
• Public health measures explain 1918 pattern well.
• Transmission cut by >50% in some cities.
• But measures often started too late, always lifted too early.
• Evidence of spontaneous behaviour change.
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Estimating the impact of
school closure
• New analyses of seasonal flu
surveillance data allows effect of school
closure to be estimated.
• Have looked for evidence of changes
in transmission in different age groups in
and out of school terms in sentinel
surveillance data.
• Fitted stochastic model with schools
and households to the surveillance data:
Schools account for 16.5% of
transmission overall.
Overall, school closure in a pandemic
might reduce attack rates by ~5% (from
32% to 27%) overall – but reduces
attack rates in children by a quarter.
Paris-1985
05
01
50
Aix-Marseille-1985
01
50
30
0 Lille-1985
05
01
50
Paris-1989
02
00
40
0 Aix-Marseille-1989
02
00
40
0 Lille-1989
01
00
20
0
Paris-1997
04
08
0
Aix-Marseille-1997
05
01
50
Lille-1997
04
01
00
Paris-2001
04
01
00
Aix-Marseille-2001
01
00
20
0
Lille-2001
01
50
35
0
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Infectious disease modelling
-future challenges
General:
More mobile, more populous world –
diseases spread faster so need faster/better
responses.
Prioritising/targeting – emerging infections
vs the rest, insufficient resources overall.
Modelling has to deliver health benefits.
Technical:
Better natural history / transmission
models.
Quantifying and validating proxy
measures of ‘infectious contact’ patterns.
Inference methods
Data on transmission/interventions.
Maintaining simplicity.