Epidemic Modelling
The following solution was sent in by
Thomas from Dalton Primary School, New York. If you repeated
Thomas's experiment with the same simulation parameters you would
get different results. Can you think why? It is because the results
depend on probabilities. To get reliable results that we can base
decisions on we need to find the average (or mean) results from
many repetitions of the same experiment with exactly the same
parameters.
Thomas's results are interesting because they
show very different outcomes according to whether the sick people
circulate in the village, or stay at home or are put in total
isolation. Lewis from Highcliff Primary School also says that
isolation is a good policy but when do you think it is advisable
and why?
Thomas's results
I modelled a large village being affected by a very lethal and
infectious disease and looked at the impact of mobility and
isolation on the length of the epidemic, the number of deaths, the
number of infections, and the number of recoveries.
| Mobility |
Duration |
Deaths |
Not infected |
Recovered |
|
|
|
|
|
| Normal |
169 |
342 |
33 |
125 |
| Static |
35 |
15 |
481 |
4 |
| Isolated |
9 |
1 |
498 |
1 |
Conclusions:
- Reduced mobility and isolation had an enormous impact on the
duration of the epidemic, the number of deaths, and the infection
rate.
- Isolation was more effective than people remaining static when
infected.
- This suggests that when there is a dangerous epidemic (High
infection rate, high death rate), effective public health policies
would be to tell people to stay at home and to isolate those who
are sick.
Ruth from Manchester High School for Girls
sent us this careful investigation of a different aspect of this
problem. She repeated each experiment several times and drew
conclusions from the mean of several runs.
Ruth's Results:
I am investigating whether the incubation period of an illness
affects how useful it is to isolate infected individuals.
| The simulation parameters were: |
|
| Grid Dimension |
25 |
| Initial Population |
150 |
| Initially infected |
25 |
| Initially immune |
0 |
| Days ill |
8 |
| Probability of death |
0.9 |
| Probability of infection |
0.9 |
| Probability of static |
0.1 |
| Gain immunity |
true |
Independent Variables:
Days before infectious
Behaviour if ill
RESULTS
Normal when ill \begin{array}{lllll} & \text{Duration} &
\text{Deaths} & \text{Never Ill} & \text{Recovered} \\
\text{Mean} &19.8& 135 &0.2& 14.8\\ \text{St. dev.}
&2.7& 3.6& 0.4& 3.5 \end{array}
If the behaviour when ill is normal, the number of days before
infectiousness makes no difference.
Isolated when ill
0 days before infectious \begin{array}{lllll} & \text{Duration}
& \text{Deaths} & \text{Never Ill} & \text{Recovered}
\\ \text{Mean} &9 &21.6& 125& 3.4\\ \text{St. dev.}
&0&1.5 &0 &1.5 \end{array}
1 days before infectious \begin{array}{lllll} & \text{Duration}
& \text{Deaths} & \text{Never Ill} & \text{Recovered}
\\ \text{Mean} & 15.2 & 81.8 & 56.8 & 11.4 \\
\text{St. dev.} & 1.6 & 5.5 & 8.4 & 3.4
\end{array}
2 days before infectious \begin{array}{lllll} & \text{Duration}
& \text{Deaths} & \text{Never Ill} & \text{Recovered}
\\ \text{Mean} & 18.6 & 113.4 & 26.8 & 9.8 \\
\text{St. dev.} & 4.4 & 7.2 & 6.4 & 4.6 \end{array}
3 days before infectious \begin{array}{lllll} & \text{Duration}
& \text{Deaths}& \text{Never Ill} &\text{Recovered}\\
\text{Mean}& 19 &126.2& 10.8 &13 \\ \text{St. dev.}
& 1.5& 6.3 &7.5 &1.7 \end{array}
4 days before infectious \begin{array}{lllll}
&\text{Duration}& \text{Deaths}& \text{Never Ill}
&\text{Recovered} \\ \text{Mean} & 19.2 &129.2 &4.4
&16.4 \\ \text{St. dev.} &1.9 &4.1& 2.2& 4.2
\end{array}
5 days before infectious \begin{array}{lllll} &\text{Duration}
&\text{Deaths} &\text{Never Ill}& \text{Recovered} \\
\text{Mean} &18.8 &133 &0.6 &16.4\\ \text{St. dev.}
&1.9 &2.8 &0.8 &2.9 \end{array}
6 days before infectious \begin{array}{lllll}
&\text{Duration}& \text{Deaths}& \text{Never Ill}
&\text{Recovered} \\ \text{Mean} &21.8& 133.2
&0.6& 16.2 \\ \text{St. dev.}& 4.7 &4.2 &0.8
&4.5 \end{array}
7 days before infectious \begin{array}{lllll} &\text{Duration}
&\text{Deaths}& \text{Never Ill}& \text{Recovered} \\
\text{Mean}& 20& 133& 0.4& 15.8\\ \text{St. dev.}
&1.7& 4.1 &0.8 &3.3 \end{array}
8 days before infectious \begin{array}{llll} &\text{Duration}
&\text{Deaths} &\text{Never Ill} &\text{Recovered} \\
\text{Mean}& 19.6 &136.8& 0.2 &13 \\ \text{St.
dev.} &0.8 &1.7 &0.4 &1.7 \end{array}
This disease is very lethal and infectious. If nothing is done, it
will kill most of the population of the town. Isolation is an
effective way of reducing the death toll and duration of the
epidemic.
If the time before infectiousness is a large proportion of the
duration of the illness, it makes very little difference to the
outcome whether or not infected people are isolated. The percentage
difference between the number of deaths when isolated and not
isolated is under 1.5% when the period before infectiousness is
over half the length of the illness (5 days or more) but is over
80% when the period before infectiousness is 0 days and is almost
40% when it is 1 day. The variation in the length of the epidemic
follows a similar pattern with less than 1% variation between
isolation and non-isolation for 7 or 8 days before infectiousness
but over 50% difference for 0 days and almost 20% for 1 day.
These results show that, while isolating infected individuals will
almost always reduce the death toll and end the epidemic sooner, it
is most effective when the incubation period of the illness is
relatively short. As the incubation period increases, the amount of
time that infected individuals are isolated for, and therefore the
amount of time they are not infecting others for, decreases, so it
is not unexpected that this is the case. The results suggest that,
as isolating infected people would be quite difficult and
expensive, it is only worth doing so if the incubation period of
the infection, when they are infectious but show no symptoms, is
quite short compared to the period when they do show symptoms so
would be isolated.