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?

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?)

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).

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

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.

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%

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?

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.

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.

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)

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

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.

Data: contact patterns

Defining ‘relevant’ contacts often a challenge – STIs the easiest:

Gregson et al, Lancet 2002

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

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

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!

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!

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

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

• 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

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

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.

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

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.

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%).

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

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)

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.

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

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

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

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.

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

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.