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Modelling Infectious Disease

Modelling Infectious Disease. … And other uses for Compartment Models. Plumbing. Tracking the concentration of dissolved particles through pipes. A simple conceptual model. Amount of solutes at the start = x(t=0)=x(0)=18 Concentration of solutes at any time = x / V

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Modelling Infectious Disease

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  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 • 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) Volume rate rate

  4. V r r A simple mathematical model

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

  6. Varying the rate of flow

  7. Compartments & Flow V1 V2 V3 r r r r Changes in Concentration

  8. Evaluate the Model • Choose some parameters • V1 = 80 • V2 = 100 • V3 = 120 • r = 20 • Define the initial conditions • x1(0) = 10 • x2(0) = 0 • x3(0) = 0 • http://math.fullerton.edu/mathews/N310/projects2/p14.htm (read from “More Background” onwards)

  9. Results

  10. General Framework

  11. Land Air Sea Any pattern you like…

  12. From plumbing to infectious diseases

  13. I R Infectious Disease • Susceptible pool of people • Infected pool of people • Recovered pool of people S

  14. bSI vI S I R 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. R0as a useful statistic • R0 is the basic reproductive number of the disease • Similar to the r and R that appear in population models • R0 = N*b*Duration = N(b/v) • If R0 > 1 epidemic • If R0 < 1 disease dies out naturally

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

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

  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 virusgenus Orthopox • Eradicated 1979 • Remains a biological threat • Huge vaccine stocks are held by many Governments

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

  22. Time to Invervention is crucial

  23. 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 Endemic Infections

  24. Minimum Vaccination Number • Also known as Herd Immunity • At equilibrium (stable state)R0S = 1 • Vaccinate proportion qof populationR0(1-q)=11-q=1/R0qc=1-(1/R0) • This is the minimum % of the pop that have to be vaccinated in order to stop the spread of the disease

  25. Immunisation Thresholds

  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 • 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-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/display_person.xml?netid=grenfell&display=All

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