Epidemic Modelling

4489 /www/nrich/html/content/id/4489/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://motivate.maths.org/content/DiseaseDynamics/Activities">Warm-up</a></li> <li><a href="http://nrich.maths.org/6086">Try this next</a></li> <li><a href="http://nrich.maths.org/6633">Think higher</a></li> <li><a href="http://plus.maths.org/content/do-you-know-whats-good-you-0#epidemiology">Read: mathematics</a></li> <li><a href="http://www.flusurvey.org.uk/">Read: science</a></li> <li><a href="https://motivate.maths.org/content/MultiMediaResources">Explore further</a></li> </ul> <div> </div> Mathematics is used in medical research, engineering and finance to model the real world because it is much safer and cheaper to try out theoretical models than it is to experiment with living subjects, to build and test expensive prototypes, or to invest real money in untried schemes. <a href="http://motivate.maths.org/content/DiseaseDynamics">Find out</a> about involvement of schools in on-going research into epidemic modelling. Using this probability environment you can be a researcher and use mathematical modelling to investigate the spread of different sorts of diseases. Others have contributed their findings but you can join in this ongoing research and let us know what you find out. You will be able to model some of the characteristics of your chosen disease. (Click on Configuration Data). <a href="/content/id/4489/epidemic2.swf">Full Screen Version</a> <mdo:flash height="400" id="/content/id/4489/epidemic2.swf" width="440"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/4489/epidemic2.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="440"></param> </mdo:flash> Your experiment takes place in a community modelled by a square grid. You will see individuals scattered in the community, each occupying one square. The sick individuals are red, the individuals with immunity are dark green and the other healthy individuals are a lighter green. You can set the conditions, watch the epidemic run its course several times, and record the data produced by the computer. You can then observe the effects of changing the conditions of the model. You can model different diseases by choosing different probabilities of catching the disease and of dying from it and also choosing the time between contracting the disease and becoming infectious to others. You can test the effects of the sort of policy decisions that a Public Health Official makes with regard to vaccination and whether the sick people should stay at home or be put in isolation. At the end of the given number of days the sick individual either becomes healthy or dies and is removed from the village. Now decide on what you want to investigate, carry out repeated trials, and send us a report of your findings. As there is so much scope for different investigations we'll publish all the interesting reports received. This model will work for diseases that are spread by contact (such as flu, the common cold, measles, meningitis, etc.) but not for sexually transmitted diseases and not for vector borne diseases such as those spread by insects like for example malaria. <hr></hr> To learn more see <a href="http://motivate.maths.org/conferences/conf109/c109_projects.shtml#top">t</a>he <a href="http://motivate.maths.org/content/DiseaseDynamics">Disease Dynamics Schools Pack</a></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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? <h4>Thomas's results</h4> <div>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.</div> <table border="0"> <tbody> <tr> <td style="text-align: center;">Mobility</td> <td style="text-align: center;">Duration</td> <td style="text-align: center;">Deaths</td> <td style="text-align: center;">Not infected</td> <td style="text-align: center;">Recovered</td> </tr> <tr> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">Normal</td> <td style="text-align: center;">169</td> <td style="text-align: center;">342</td> <td style="text-align: center;">33</td> <td style="text-align: center;">125</td> </tr> <tr> <td style="text-align: center;">Static</td> <td style="text-align: center;">35</td> <td style="text-align: center;">15</td> <td style="text-align: center;">481</td> <td style="text-align: center;">4</td> </tr> <tr> <td style="text-align: center;">Isolated</td> <td style="text-align: center;">9</td> <td style="text-align: center;">1</td> <td style="text-align: center;">498</td> <td style="text-align: center;">1</td> </tr> </tbody> </table> <h4>Conclusions:</h4> <ol> <li>Reduced mobility and isolation had an enormous impact on the duration of the epidemic, the number of deaths, and the infection rate.</li> <li>Isolation was more effective than people remaining static when infected.</li> <li>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.</li> </ol> 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. <h4>Ruth's Results:</h4> <div>I am investigating whether the incubation period of an illness affects how useful it is to isolate infected individuals.</div> <table border="0"> <tbody> <tr> <td>The simulation parameters were:</td> <td></td> </tr> <tr> <td>Grid Dimension</td> <td>25</td> </tr> <tr> <td>Initial Population</td> <td>150</td> </tr> <tr> <td>Initially infected</td> <td>25</td> </tr> <tr> <td>Initially immune</td> <td>0</td> </tr> <tr> <td>Days ill</td> <td>8</td> </tr> <tr> <td>Probability of death</td> <td>0.9</td> </tr> <tr> <td>Probability of infection</td> <td>0.9</td> </tr> <tr> <td>Probability of static</td> <td>0.1</td> </tr> <tr> <td>Gain immunity</td> <td>true</td> </tr> </tbody> </table> <div>Independent Variables:</div> <div>Days before infectious</div> <div>Behaviour if ill</div> <h4>RESULTS</h4> 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. <h4>Isolated when ill</h4> 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. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This is an entirely open problem to which there are no 'correct' answers. Indeed the answers are not even known. You are doing genuine research. You can choose the characteristics of the disease you want to model and study. The environment is designed so that the researcher can use a mathematical model to explore the possibilities of the spread of the disease before it actually hits a real population. Doctors can estimate the probability of death and the duration of the infectious period for different diseases. Based on this knowledge the public health officials may decide on advising the public whether sick individuals should stay at home to limit the risk of spreading the infection, or even advise that all infected individuals are put into isolation to curb the spread of the disease. These projects can be carried out with only a basic knowledge of probability and statistics or with a higher level of knowledge. The more mathematics you use to analyse your results the more potential value there might be in your findings. <ul> <li>Decide how many individuals there are in your population.</li> <li>Decide how many individuals are infected with the disease at the start.</li> <li>Decide on how many individuals (if any) will be vaccinated in advance.</li> <li>Decide on the probability $p$ of catching the disease on contact with it.</li> <li>Decide on the probability $q$ of dying and $1-q$ of recovering from the disease.</li> <li>Decide how many days the disease lasts.</li> <li>Decide on the incubation period during which time the sick person is in circulation and passes on the disease to others.</li> <li>Choose one of these three options: After the incubation period, and while the illness lasts, the sick individual can either (1) circulate in the population as normal or (2) stay at home (modelled by remaining static) or (3) be put in isolation and have no contact with others.</li> <li>Decide whether recovered individuals can be re-infected or not. If not they will develop an immunity in which case they become dark green on the display.</li> <li>Model a day as a trial during which every healthy individual either stays in the same place or moves one square in one of 8 directions.</li> <li>Note that paths touching the edge of the board continue along the 'reflected' direction back into the village. To avoid collisions both individuals stop in their tracks.</li> <li>If a healthy individual, who is not immune, lands on a square that touches the space of a sick individual, either corner to corner or edge to edge, then the healthy individual is immediately infected.</li> </ul> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There are many variables in this model. You will need to choose your variables and carry out the trial a sufficient number of times to suggest a theory about the likely outcome when the disease has 'run its course'. Then change only one variable and repeat the experiment and see whether the outcome changes and if so how it changes. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> There are many different projects possible. We invite accounts of your experiments and will publish reports of interesting results found. These projects can be carried out with only a basic knowledge of probability and statistics or with a higher level of knowledge. The more mathematics you use to analyse your results the more potential value there might be in your findings. </mdoxml> 2 3 0 0 0 1 1 Epidemic Modelling Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths. Information and Communications Technology Interactivities Using, Applying and Reasoning about Mathematics Real world Using, Applying and Reasoning about Mathematics Mathematical modelling Using, Applying and Reasoning about Mathematics Making and proving conjectures Probability Experimental probability Advanced Probability and Statistics Probability distributions, expectation and variance Applications biology Admin Computer-based Applications Maths Supporting SET Applications STEM - General Stage 5 Statistics Mapping Document Sm - Sampling and hypothesis testing Stage 5 Statistics Mapping Document SM - Statistical Modelling