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- 1. Preparing for Armageddon How mathematical models can save the planet from infectious disease outbreaks
- 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?
- 3. Hollywood Does History …
- 4. Hollywood Does Science… Global ice age occurs in around 48 hours! 200 ft rise in water levels!
- 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!
- 7. Reality of Disease Spread
- 8. Reality of Disease Spread
- 9. Reality of Disease Spread
- 10. Reality of Disease Spread
- 11. Reality of Disease Spread
- 12. Reality of Disease Spread
- 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.
- 18. Playing the odds
- 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
- 22. The UK 2001 Foot and Mouth Disease Epidemic
- 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.
- 27. The DataCPH County, Parish Holding Location of Farm House Number and type of livestock Link Probable source of infection Report and Slaughter dates
- 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.
- 30. Comparison between Model and Data Infected Farms
- 31. The Role Of Politics
- 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. When will the Epidemic end? Feb 23rd Cases Models cannot predict this precisely. Average end date Best Case scenario Worst Case scenario
- 40. 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?
- 41. 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.
- 42. 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?
- 43. 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.
- 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. Human Epidemics
- 46. 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.
- 47. 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.
- 48. 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.
- 49. The Six Degrees of Kevin Bacon Bacon Number=1 B No.=2 B No.=3
- 50. 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.
- 51. 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.
- 52. Implications of clusters – airport network Barabasi and Bonabeau (1999) Scientific American. Nearest neighbours are connected A few hubs with lots of connections
- 53. Implications of clusters – airport network Barabasi and Bonabeau (1999) Scientific American. Random attack Targeted attack So the airline network is vulnerable to targeted attacks.
- 54. Implications of clusters – disease spread Farms Markets
- 55. Implications of clusters – disease spread Farms Markets
- 56. Implications of clusters – disease spread Farms Markets
- 57. 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
- 58. Worldwide Cases Yellow – suspected cases Red – confirmed cases Black – confirmed deaths 1st May 2009 1st June 2009
- 59. Worldwide Cases Yellow – suspected cases Red – confirmed cases Black – confirmed deaths 1st August 2009 21st December 2009
- 60. 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.
- 61. 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.
- 62. 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.
- 63. 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.
- 64. • Internet-based live data collection. • Recruit members of the public to record their symptoms each week. UK - www.flusurvey.org.uk
- 65. Preparing for Armageddon…
- 66. Thailand and Vietnam Three waves of infection in 2004 and 2005. Sporadic outbreaks in South East Asia ever since.
- 67. Avian Influenza in South East Asia
- 68. 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.
- 69. Infection in humans and birds Long Range (Between Country!) scale transmission
- 70. 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.
- 71. Lessons Learned • Mathematical models can be used to predict potential for disease spread.
- 72. Lessons Learned • Mathematical models can be used to predict potential for disease spread. • Control policies can be established to minimise number of infected cases.
- 73. 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.
- 74. And don’t forget, it’s all about playing the odds!
- 75. 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

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