1. Modelling Infectious Disease … And other uses for Compartment Models

2. Plumbing Tracking the concentration of dissolved particles through pipes

3. A simple conceptual model rate Volume Amount of solutes at the start = x(t=0)=x(0)=18 Concentration of solutes at any time = x/V Water coming in removes an amount of x at a constant rate Need a model to calculate x(t)

4. A simple mathematical model rr V

5. The Solution X(0) = 18 r = 10 V = 100

6. Varying the rate of flow

7. Compartments & Flow rrrr V1V1 V2V2 V3V3 Changes in Concentration

8. Evaluate the Model Choose some parameters V 1 = 80 V 2 = 100 V 3 = 120 r = 20 Define the initial conditions x 1 (0) = 10 x 2 (0) = 0 x 3 (0) = 0 http://math.fullerton.edu/mathews/N310/projects2/p14.ht m (read from “More Background” onwards)

9. Results

10. General Framework

11. Any pattern you like… Land Sea Air

12. From plumbing to infectious diseases

13. Infectious Disease Susceptible pool of people Infected pool of people Recovered pool of people S IR

14. SIR bSIvI Infection Rate: Contact rate Infection probability Recovery Rate If D is the duration of infection: v = 1/D

15. A “typical” flu epidemic Each infected person infects a susceptible every 2 days so bN=1/2 (N = S+I+R) Infections last on average 3 days so v=1/3 London has 7.5 million people 10 infected people introduced See accompanying notes on parameter meanings

16. R 0 as a useful statistic R 0 is the basic reproductive number of the disease Similar to the r and R that appear in population models R 0 = N*b*Duration = N(b/v) If R 0 > 1 epidemic If R 0 < 1 disease dies out naturally

17. Changes to Infection Rate b=0.5/N v=1/3 b=2/N v=1/3

18. Modifications are (almost) endless Susceptible Exposed Infected Recovered SEIR Susceptible CarrierInfected Recovered Carrier Type Diseases: TB, Typhoid

19. Typhoid Mary 1869-1938 Healthy carrier of typhoid Infected 47 people in the US Quarantined twice under the mental health act We still do this!! – e.g. TB

20. Smallpox (Variola) Enveloped DNA virus genus Orthopox Eradicated 1979 Remains a biological threat – Huge vaccine stocks are held by many Governments

21. Legrand et al. 2004, Epidemiol Infect, vol 132, pp19-25 Uninfected contacts (located) Vaccinated successfully Exposed contacts (missed) Susceptible Infectious Removed Exposed contacts (located) Quarantine

22. Time to Invervention is crucial

23. Endemic Infections These are persistent infections in the population that tick along at a relatively stable level, never going extinct. This happens when the number of Infectious people remains constant

24. Minimum Vaccination Number Also known as Herd Immunity At equilibrium (stable state) R 0 S = 1 Vaccinate proportion q of population R 0 (1-q)=1 1-q=1/R 0 q c =1-(1/R 0 ) This is the minimum % of the pop that have to be vaccinated in order to stop the spread of the disease

25. Immunisation Thresholds DiseaseR0R0 Threshold q c =1-(1/R 0 ) Measles1593% Smallpox786% Mumps580%

26. Conclusions Compartment models are versatile – Flow of liquids between tanks – Diffusion of nutrients across sediment boundaries – Spread of disease through populations Endless elaborations can be made – Spatial structure – Population structure

27. Further Reading The bible and for a link from SIR to population models: Anderson & May. 1979. Population biology of infectious diseases: Part 1. Nature 280, 361-367. May & Anderson. 1979. Population biology of infectious diseases: Part 2. Nature 280, 455-461. For an evolutionary spin: Brown et al. 2008. Evolution of virulence: triggering host inflammation allows invading pathogens to exclude competitors. Fitting models to real data: Keeling & Grenfell, 2001. Understanding the persistence of measles: reconciling theory, simulation and observation. Proc Roy Soc B 269, 335-343. Indeed, anything by Bryan Grenfell is worth reading: http://www.cidd.psu.edu/people/bio_grenfell.html http://www.cidd.psu.edu/people/bio_grenfell.html Foot-and-mouth disease: Tildesley et al. 2006. Optimal reactive vaccination strategies for a foot-and-mouth outbreak in the UK. Nature 440, 83-86. (and refs therein, esp the first 2) The original article:Kermack & McKendrick 1927. http://links.jstor.org/sici?sici=0950- 1207%2819270801%29115%3A772%3C700%3AACTTMT%3E2.0.CO%3B2-Zhttp://links.jstor.org/sici?sici=0950- 1207%2819270801%29115%3A772%3C700%3AACTTMT%3E2.0.CO%3B2-Z

28. Tasks for next tutorial Why do some infectious diseases such as measles epidemics cycle? – Intrinsic (properties of the infective process itself) – Extrinsic (environmental) See Bryan Grenfell’s research on measles as a starter http://www.princeton.edu/eeb/people/displa y_person.xml?netid=grenfell&display=All http://www.princeton.edu/eeb/people/displa y_person.xml?netid=grenfell&display=All