2. What is a Model?
A mathematical model is a description of a system using
mathematical formulas.
Models are used in a variety of fields:
Astrophysics Aeronautics Climate
Epidemics
So how do these models work?
Are they communicated satisfactorily in the media?
5. R0 – how infectious is a disease?
R0 is a mathematical quantity that tells us how “infectious” a
disease is.
It is the average number of people an infectious person will
infect, assuming that the rest of the population are uninfected.
So the larger the value of R0 the
more infectious the disease.
In Contagion, R0 = 2.
Day 1 – one person infected.
That person infects 2 new people.
2 4 8 16 32
Do this 30 times!
1 billion people!
6. R0 – how infectious is a disease?
R0 is a mathematical quantity that tells us how “infectious” a
disease is.
It is the average number of people an infectious person will
infect, assuming that the rest of the population are uninfected.
So the larger the value of R0 the
more infectious the disease.
In Contagion, R0 = 2.
Day 1 – one person infected.
That person infects 2 new people.
2 321684
Do this 30 times!
1 billion people!
13. Reality of Disease Spread
So R0 only really has any meaning at the start of an outbreak,
when everyone is susceptible.
This is why models are very powerful for informing the risk
posed by a disease.
14. R0 in the real world
Flu
HIV
Malaria
Chicken pox
Measles
Less
transmissible
More
transmissible
R0
1 2 5 10 20 100
Smallpox
15. Transmission
Rate
Recovery Rate
Susceptible Infected Recovered
A simple model
dS
dt
= -bSI
dI
dt
= bSI - gI
dR
dt
= gI
Transmission Rate
Recovery rate
Models are used to predict how many people will be infected
with a disease.
infected
recovered
susceptible
16. Controlling a disease
If R0 is greater than 1, then everyone who is infected infects
more than one person “on average”.
If R0 is less than 1, then everyone who is infected infects less
than one person “on average”.
So if R0 is less than 1, eventually the disease will die out.
No further infection.
So the aim of any
control strategy is to
force R0 below 1.
17. Vaccination
Why does vaccination work?
People are removed from the “susceptible” class, so reduces
the number of people in the population that can be infected
and hence effectively reduces R0.
If R0 is 2, then if we
vaccinate just over
half of the
population, the
epidemic will
eventually die out.The higher the value
of R0 the more people
we have to vaccinate
to control the disease.
19. Toss of the Coin
If I toss a coin ten times, how many heads will I get?
5
In fact, the probability of
getting exactly five heads in
less than a quarter (or one in
four).
One in five times you will get at
least seven heads.
So if you only do this experiment once, can you say with any
certainty, how many heads you will get?
20. Uncertainty in Disease Outbreaks
If we have an outbreak of infectious disease, can we predict with
any certainty how that disease will spread?
Day One Day Two
Day Two
Day Three
Day Three
Day Three
21. Uncertainty in Disease Outbreaks
So an outbreak of the
same disease could affect
the population in a
completely different way
“next time around”.
Day One Day TwoDay Three
Day Two
Day Four
23. What is Foot and Mouth
disease?
A viral infection of cloven hoofed animals.
Effects on livestock:
➢ Reduced milk production
➢ Reducing weight gain
➢ Death of young
National effects:
➢ Economic and political repercussions
➢ Implications for the farming industry
Susceptible animals
24. Europe, North America and Australia officially disease free (until recently).
However, there is a chance of introduction from external sources.
Endemic (always present) in much of Africa, Asia and South America (so
risk of a farm becoming infected at any time is non-zero).
Worldwide distribution of FMD
25. UK 2001 epidemic timescale
Epidemic peak occurred in
late March/early April.
FMD entered the UK in
early February.
Over 10,000 farms were affected by the epidemic (either infected or
culled as part of the control) and a total of 850,000 cattle and 4,000,000
sheep were culled.
Very long epidemic tail.
26. What's going to happen next?
To address this question we need to build models. There were three models
used throughout the epidemic.
DEFRA
The Interspread model was a simulator developed in New Zealand, originally
to predict Swine Fever.
Imperial College
This model was rapid to simulate, and relatively easy to fit to the data.
Cambridge/Warwick/Edinburgh
A spatial model – all farm locations were included. Approach was
intuitive, but model was difficult to fit to the data.
28. What should we do?
Mathematical models can only answer definite questions:
1) How do we stop the epidemic as quickly as possible?
2) How do we minimise the losses to farmers?
3) How do we minimise the political impact?
Kill all the livestock as quickly as possible
Difficult: Trade-off between short and long term losses
Almost impossible to know!
29. Did the models work?
Remember, like the coin toss, we need to run our model many times to
determine the behaviour of the epidemic.
Keeling et al. (2001) Science.
Average
prediction from
epidemic
Each time we run
the model, we get
a slightly
different answer.
So as well as
showing the
average, we need
to show the best
and worst case.
32. The Time-course of the Epidemic
Reported Cases
Slaughtered Premises
MAFF attempts to cull
all bordering infected farms
33. But then we had …The Phoenix Factor
The media finds Phoenix the calf:
34. But then we had …The Phoenix Factor
The media finds Phoenix the calf:
Ross Board, 11 with his pet calf
Phoenix, saved from slaughter after
surviving the cull of the rest of the
herd.
35. But then we had …The Phoenix Factor
The media finds Phoenix the calf:
Ross Board, 11 with his pet calf
Phoenix, saved from slaughter after
surviving the cull of the rest of the
herd.
The calf was reprieved on April 25th
after the government changed their
Policy on slaughter of contiguous
farms
36. But then we had …The Phoenix Factor
The media finds Phoenix the calf:
Ross Board, 11 with his pet calf
Phoenix, saved from slaughter after
surviving the cull of the rest of the
herd.
37. The Time-course of the Epidemic
Reported Cases
Slaughtered Premises
MAFF attempts to cull
all contiguous premises
Cattle from high biosecurity
farms are reprieved from the cull
38. Why do we want to cull healthy
animals?
Need to create a firebreak to stop infection spreading around the
country.
39.
40. When will the Epidemic end?
Feb 23rd
Cases
Models cannot predict this precisely.
Average end date
Best Case scenario
Worst Case
scenario
41. Feb 23rd
Confidence intervals were thought
to be too confusing - so were omitted
from the press release.
Jun 9th
Cases
The average end date corresponded exactly to the date of the general election.
Modelling lost all credibility.
When will the Epidemic Die Out?
42. Vaccination
Throughout the UK epidemic we were continually asked about
vaccination.
Problems with vaccination:
1) There is a significant delay (several days) between vaccination and protection.
2) Animals infected before protection are likely to spread the disease but may not
show signs of infection.
However, we considered the effect of vaccinating in a ring
around all infected farms.
43. IP
What size of ring
should we use?
Given the size of the
outbreak, it is
impossible to
vaccinate all farms in
a ring immediately,
so how should we
prioritise farms for
vaccination?
44. Model Predictions - Vaccination
Control strategy Optimal ring size (in km) Average Epidemic
Size
Furthest farms first 8.5 1752
Closest farms first 8.6 1791
Large cattle farms first 10.5 1535
Most livestock (cattle 12.5 1343
and sheep) first
Random vaccination 9.0 1688
Tildesley et al. (2006) Nature.
45. Model Predictions - Vaccination
Control strategy Optimal ring size (in km) Average Epidemic
Size
Furthest farms first 8.5 1752
Closest farms first 8.6 1791
Large cattle farms first 10.5 1535
Most livestock (cattle 12.5 1343
and sheep) first
Random vaccination 9.0 1688
Tildesley et al. (2006) Nature.
71. The Small World Effect
In 1967 the sociologist Stanley Milgram conducted an experiment
to analyse the path characteristics in social networks.
He chose 50 people at random in Kansas and Nebraska to deliver a
letter to a stock broker in Cambridge, Massachusetts.
They didn’t know who the recipient was and could only pass the
letter to people they knew.
72. The Small World Effect
The letters that reached
their target were passed
through six people on
average.
One letter arrived at its destination within 4 days.
This led to the theory of “the six degrees
of separation” or the “small world effect”
that everyone in the world is linked by 6
steps or fewer.
73. The Six Degrees of Kevin Bacon
The actor Kevin Bacon once commented that he’s worked with
everyone in Hollywood or worked with someone who’s worked
with them.
College students in the USA used that statement and Milgram’s work
to invent “The Six Degrees of Kevin Bacon”
The theory is that any actor can be linked to Kevin Bacon through
movies they’ve co-starred in in 6 steps or fewer.
74. The Six Degrees of Kevin Bacon
Bacon Number=1
B No.=2
B No.=3
75. Human Behaviour
In order to understand how a disease may spread in the human
population, we need to have a good understanding of human
behaviour.
Of course, it’s impossible to know exactly who contacts whom
and the risk of disease spread (remember the Simpsons!).
So we need a way to approximate this behaviour to provide data
for mathematical models.
One way to do this is to use contact networks.
76. The Warwick Contact Survey
In the 1990s, academics at Warwick
University kept a diary over the summer
of everyone they came into contact with.
At the end of the experiment, all
participants and their contacts were built
into a network, to highlight the risk of a
disease spreading in the population.
Some interesting trends emerged…
People are clearly not randomly
connected and are observed to form into
clusters with some “very connected”
people.
77. Implications of clusters – airport network
Barabasi and Bonabeau (1999) Scientific American.
Nearest neighbours are connected A few hubs with lots of connections
78. Implications of clusters – airport network
Barabasi and Bonabeau (1999) Scientific American.
Random attack
Targeted attack
So the airline network is vulnerable to
targeted attacks.
82. The 2009 H1N1 flu pandemic
In March 2009, over half of the
population of the town of La
Gloria, Veracruz, Mexico, became
infected by an unknown
respiratory illness.
By the end of March, cases had
been reported in the USA.
By the beginning of May, 36 of
the 50 states of the USA had
reported cases of H1N1.
La Gloria,
Veracruz
83. Worldwide Cases
Yellow – suspected cases
Red – confirmed cases
Black – confirmed deaths
1st May 2009
1st June 2009
84. Worldwide Cases
Yellow – suspected cases
Red – confirmed cases
Black – confirmed deaths
1st August 2009
21st December 2009
85. The Role of Models
Model was used
to predict the
daily number of
cases of swine flu
in the UK.
Baguelin et al. (2010) Vaccine.
86. H1N1 Pandemic in China
First case of H1N1 reported on 11th
May (passenger flying from USA to
Chengdu, China).
Passenger was isolated – all other travellers on the same plane
were located and quarantined where possible.
As epidemic progressed, same policy was introduced for each
new reported case.
The number of cases in China was lower (proportionally) than
in other countries, owing to this draconian control measure.
87. 0 20 40 60 80 100 120 140 160
10
0
10
1
10
2
10
3
10
4
10
5
10
6
Time
Cumulativenumberofreportedcases
Model output
Repoted data
Model Results
Initially the model fits the data well.
Model deviates around day 60.
Coincides with the school summer holidays so we expect
that the disease won’t spread as easily.
88. Model Results
When we build in the effect of school closure into our model, we
get a much better result:
It’s clear that
school closure is an
effective control
measure.
Models show that
vaccination of
school children is
the best strategy
for disease control.
89. • Internet-based live data collection.
• Recruit members of the public to record their symptoms each
week.
UK - www.flusurvey.org.uk
93. The host level
V. low mortality rate.
Ducks shed virus, but mainly asymptomatic.
Free grazing ducks “commute” to rice paddies –
lots of contacts with wild birds!
V. high mortality rate >90% in some breeds.
Death occurs in a few days.
Kept in huge numbers.
94. Infection in humans and birds
Long Range (Between Country!) scale transmission
95. Modelling Bird Flu
Van Boeckel et al 2011. Agriculture Ecosystems and Environment
Tagging Wild Birds
Testing for infection in
poultry and humans
Developing a model to determine risk of disease spread to
humans and methods for disease control.
97. Lessons Learned
• Mathematical models can be used to predict
potential for disease spread.
• Control policies can be established to
minimise number of infected cases.
98. Lessons Learned
• Mathematical models can be used to predict
potential for disease spread.
• Control policies can be established to
minimise number of infected cases.
• However, results from mathematical models
should only form part of the decision making
process.
101. Acknowledgements
Ellen Brooks Pollock (Cambridge)
Colleen Webb (Colorado State)
Matt Keeling (Warwick)
Gwilym Enstone (Warwick)
Gary Smith (U. Penn)
Matt Ferrari (Penn State)
Uno Wennergren (Linkopings)
Ken Eames (LSHTM)
Marleen Werkman
Thomas van Boeckel
Peter Dawson
Benjamin Hu
My Group
Editor's Notes
So the prediction of 1 billion people becoming infected so quickly is a massive exaggeration. Kate Winslet also says throughout the movie that R0 keeps increasing, to 4, 8 etc. She would have no way of knowing this (though she would know if it had increased).
So the prediction of 1 billion people becoming infected so quickly is a massive exaggeration. Kate Winslet also says throughout the movie that R0 keeps increasing, to 4, 8 etc. She would have no way of knowing this (though she would know if it had increased).
So the prediction of 1 billion people becoming infected so quickly is a massive exaggeration. Kate Winslet also says throughout the movie that R0 keeps increasing, to 4, 8 etc. She would have no way of knowing this (though she would know if it had increased).
Bad for the airline network, good for controlling disease, if we can target the most highly connected hubs.
