Download FMD

Infectious diseases in structured populations
The aim of this practical is to investigate the effect that restricted geographic movement can
have on the dynamics of infectious disease. To motivate the work, we will use data from the
2001 Foot and Mouth Disease (FMD) outbreak in the UK. This was a major epidemic that
lead to the slaughter of over half a million cattle and nearly 3.5 million sheep. The disease,
caused by a virus (FMV), is extremely contagious and although it primarily infects cows and
pigs, it is also infects sheep, goats and even rats. Although it is rarely fatal to animals, and
rarely passes to humans, meat and other goods from exposed animals cannot be sold due to
the risk of contamination. A major controversy in the 2001 outbreak was whether vaccination
should have been used as a preventative measure. Instead, the primary control measure was
slaughtering animals at farms adjacent to those that showed an outbreak. You can watch the
spread of the disease at
http://footandmouth.csl.gov.uk/secure/images/fmdspread.gif
Both at the time and subsequently mathematical models of the transmission of FMD have
been used as a way of exploring the utility of various control measures [1–5] and even
informing government policy. Some of these models ignore key aspects of population
structure, while others attempt to model its impact specifically. Within this practical you are
going to

Learn about simple models for infectious disease

Estimate epidemiological parameters from empirical data on the spread of FMD in
2001

Explore how population structure influences the dynamics of the epidemic

Look at geographic structuring to the FMD outbreak
We will be using data on the number of animals slaughtered each month in each county over
2001. These data are available from
http://footandmouth.csl.gov.uk/secure/fmdstatistics/fmdstatistics.cfm
However, I have also created a folder on the web with the relevant data in more usable forms.
http://www.stats.ox.ac.uk/~mcvean/DTC/BIO/FMD/
You will need MS EXCEL (or similar program) to do this practical. The file FMD.xls
contains both the raw data and some numerical examples used throughout the
practical. First, however, we need to introduce some basic ideas in the modelling of
disease epidemics.
The SIR model
The SIR model [6] attempts to describe the passage of an epidemic through a population, in
which infected individuals recover and gain permanent immunity from subsequent infection.
For the case of FMD infected animals did not recover, rather they were slaughtered.
However, this has the same effect; the removal of infected animals from the population.
Within the model we consider the population to be made up of three classes. S is the
proportion of animals ‘susceptible’ to infection. I is the proportion of animals infected with
the disease and that are therefore infectious. R is the proportion of the population that are
‘recovered’ (in our case ‘removed’ may be more appropriate). We wish to model the change
in population composition over time resulting from the processes of infection and
clearance/removal.
Perhaps the simplest possible model is that of the SIR model. Graphically, we can represent
the model as
S
b
I
k
R
Susceptible individuals become infected at rate b per unit time through contact with infected
animals. Infected animals are removed at rate k from the population. The dynamics of the
system can be described by the coupled differential equations
dS
 bS (t ) I (t )
dt
dI
 bS (t ) I (t )  kI (t )
dt
dR
 kI (t )
dt
It is assumed that at the start of the epidemic all bar a very few infected individuals are
susceptible. Note that in this model birth and natural death of individuals is not included –
i.e. we are only dealing with situations where the epidemic is typically much shorter in
duration than the natural lifespan of the animal.
Although we cannot obtain neat analytical solutions to the above systems of differential
equations, we can use simple numerical techniques, such as Euler’s method, to explore their
behaviour. In Euler’s method continuous time is approximated by a series of discrete time
steps. This leads to the approximation:
S (t  t )  S (t )  t
dS
dt
t
For a suitably small time step, this generally provides a good first pass solution to the
problem. In our case, we can imagine setting t to a single day. The trajectory of the
epidemic is then described by iterating the process across time steps, both for S and for I and
R.
1) Using EXCEL (or equivalent), follow the fate of an epidemic over the course of a
single year with time steps of a single day where the starting conditions are S(0) = 0.999,
I(0) = 0.001, R(0) = 0, for the parameters k = 0.1 and b = 0.2. Plot the change in
proportions (S, I and R) over the course of a year. What proportion of the initial
population remains susceptible at the end of the epidemic? Why is this not 0%? If you
get stuck on this, look at the worked example in the EXCEL spreadsheet on the web-site.
The parameters b and k clearly control the disease spread. You can think of b as the
infectiousness of the disease and k as the rate of clearance/removal. More specifically, 1/k is
the expected number of days (in our setting) that an animal is infectious before it is removed.
For FMD this is probably a matter of a few days, though there is some heterogeneity. An
extremely important quantity is the ratio b/k, usually referred to as R0, which can be
interpreted as the expected number of new cases a single new infection creates in a
completely naïve population before clearance.
2) Try different combinations of the parameters k and b, but keep the ratio b/k constant
at 2 (e.g. k = 0.05 and b = 0.1, or k = 0.2 and b = 0.4). How does the duration of the
epidemic alter? What about the proportion of the population that remains susceptible
(and therefore uninfected) by the end of the epidemic? Now try altering R0: see how the
epidemic changes for values from 0.5 to 10 (keep k fixed at 0.1).
We will now try to estimate some parameters from the FMD epidemic. First, we will
combine the information from all counties and pretend that the UK is a single, well-mixed
population (which allows us to use the above equations). We don’t actually know the total
number of cows in each population prior to the epidemic (the only records are on numbers
slaughtered). Nevertheless, we can compare predictions of the number of new cases each day
(the daily loss of the susceptible class due to infection) to the number of cattle slaughtered.
3) For FMD, we can use a value of k = 0.125 (meaning that it takes eight days on average
to spot and remove an infectious animal). Using the data on the number of cows
slaughtered each month (summed across all counties) and assuming that the infection
started out on the 26th of Jan at a single farm (i.e. I(26)=0.001) estimate as R0, for FMD
(just use your eye to find the ‘best’ value). Do the same for sheep. How do the estimates
of R0 compare? How well do your predictions fit the data?
Cumbria, North Yorkshire and Northumberland are geographically close within the UK.
However, they had very different outbreaks. We’ll now look at their specific epidemics.
4) Plot the profile of the outbreaks of FMD in cattle in each of Cumbria, North
Yorkshire and Northumberland and estimate (again by eye) R0, for each county (again
assuming a starting frequency of 0.001 on Jan 26th and a value of k = 0.125). How well
do the models predict the epidemic structures? Repeat for outbreaks of FMD in sheep
in each population.
There are two obvious discrepancies between the predictions of the model and the observed
numbers. First, for Cumbria, both the sheep and cattle epidemics last longer than the models
typically predict. Second, Northumberland has not one, but two separate epidemics. Both of
these features suggest that spatial dynamics in the disease might be important. For example,
the fact the nearby farms tend to be infected (see the movie) suggests an ‘advancing wave’ of
infection rather than a single population. Likewise, the recurrence of the disease in
Northumberland suggests that repeated migration of infected animals has the potential to lead
to secondary epidemics at the local scale.
To explore the effect of population structure, we are going to examine the dynamics
of epidemics in a pair of populations that are linked by migration. Although this is clearly a
simplification of the UK structure, it will nevertheless allow us to see how much geographical
structure can influence infectious disease dynamics. To do this, we need to specify S1(t),
S2(t), etc., where the subscript indicates the proportion in populations 1 or 2. If we let m be
the migration parameter, we can approximate the change in proportions per day as
S1 (t  t )  S1 (t )  bS1 (t ) I 1 (t )  mS1 (t )  S 2 (t )t
I 1 (t  t )  I 1 (t )  bS1 (t ) I 1 (t )  kI1 (t )  mI 1 (t )  I 2 (t )t
R1 (t  t )  R1 (t )  kI1 (t )  mR1 (t )  R2 (t )t
Similar expressions apply to the proportions in the second population (note that the migration
terms will need to be reversed for population 2).
5) Using EXCEL and the equations above, describe the dynamics of an epidemic in two
populations linked by migration. The parameters to use are k1 = k2 = 0.125, b1 = b2 = 0.2.
Start the epidemic with a frequency of infected individuals of 0.001 in the first
population and zero in the second population. Use a migration parameter of m = 0.001.
Describe the behaviour of the epidemic. Is the total proportion of the population left
susceptible after the epidemic the same in the two populations? If not, why not? Now
try values of m = 0.01 and m = 0.0001.
These examples show how epidemics can spread in structured populations (and also how the
dynamics in linked populations are not identical). However, what is perhaps more interesting
is how populations with different epidemiological parameters behave when linked by
migration.
6) Describe the dynamics of an epidemic in two populations linked by migration where
k1 = k2 = 0.125, b1 = 0.2, b2 = 0.0625. Start the epidemic with a frequency of infected
individuals of 0.001 in the first population and zero in the second population. Use a
migration parameter of m = 0.01. For population 1, compare this to the epidemic you
get for m = 0 in terms of A) duration, B), total number of cases and C) the shape of the
epidemic curve(s). In particular, look at whether the presence of a second population
prolongs or shortens the epidemic measured in the first population. How do the model
predictions compare to the epidemics in the Cumbria and N. Yorkshire?
In reality, each county within the UK is a structured population, so even the dynamics of
infection within a single county should be modelled as structured populations.
7) Using the migration model developed above, we might approximate the dynamics of
infection within a single county as the sum of the epidemics in a series of linked subpopulations with different epidemiological parameters. By way of example, consider
two linked populations with the parameters k1 = k2 = 0.125, b1 = 0.2, b2 = 0.15. Start the
epidemic with a frequency of infected individuals of 0.001 in the first population and
zero in the second population. Use a migration parameter of m = 0.001. Look at the
sum of the number of cases (per day) in each sub-population. Does this look more like
the Cumbria epidemic? Why might different regions within a county have different
rates of infection and clearance?
If you have time, you may wish to explore a more complex model, such as a ‘stepping-stone’
model in which a series of linearly-arranged populations exchange migrants with their
immediate neighbours.
Write-up
To provide a record of the practical I would like you to keep track of the analysis you have
done, recording the answers to specific questions, including plots of any graphs and
discussion of any aspects of the models (their limitations, choices you had to make in
implementation, etc.) that are relevant to their use in interpreting the FMD epidemic. The
write-up is to be handed in by the end of the week.
Reference List
1. Ferguson NM, Donnelly CA, Anderson RM (2001) Transmission intensity and impact
of control policies on the foot and mouth epidemic in Great Britain. Nature
413: 542-548.
2. Ferguson NM, Donnelly CA, Anderson RM (2001) The foot-and-mouth epidemic in
Great Britain: pattern of spread and impact of interventions. Science 292: 11551160.
3. Tildesley MJ, Savill NJ, Shaw DJ, Deardon R, Brooks SP et al. (2006) Optimal reactive
vaccination strategies for a foot-and-mouth outbreak in the UK. Nature 440:
83-86.
4. Keeling MJ, Woolhouse ME, May RM, Davies G, Grenfell BT (2003) Modelling
vaccination strategies against foot-and-mouth disease. Nature 421: 136-142.
5. Keeling MJ, Woolhouse ME, Shaw DJ, Matthews L, Chase-Topping M et al. (2001)
Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a
heterogeneous landscape. Science 294: 813-817.
6. Bailey NTJ (1975) The Mathematical Theory of Infectious Diseases and its
Applications. London: Charles Griffin and Company.