Modelling Infectious Disease

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Modelling Infectious Disease

• And other uses for Compartment Models

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A First Model
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The Solution
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Compartments Flow
V1
V2
V3
r
r
r
r
Changes in Concentration
Excel Sheet demo
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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/p
  14.htm (read from More Background onwards)

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Results
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General Framework
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Any pattern you like
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Infectious Disease

• Susceptible pool of people
• Infected pool of people
• Recovered pool of people

S
10
?SI
dI
S
I
R
Infection Rate Contact rate Infection probability
Recovery Rate If D is the duration of
infection d 1/D
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A typical flu epidemic

• Each infected person infects a susceptible every
  2 days so ?1/2
• Infections last on average 3 days so d1/3
• London has 7.5 million people
• 10 infected people introduced

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R0

• R0 is the basic reproductive number of the
  disease
• Similar to the r and R that appear in population
  models
• R0 ?Duration ?/d
• If R0 gt 1 epidemic
• If R0 lt 1 disease dies out naturally

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Changes to Infection Rate
?1/2 d1/3
?1/1.5 d1/3
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Modifications are (almost) endless
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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

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Smallpox (Variola)

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

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Legrand et al. 2004, Epidemiol Infect, vol 132,
pp19-25
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Time to Intervention is Crucial
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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

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Minimum Vaccination Number

• Also known as Herd Immunity
• At equilibrium (stable state)R0S 1
• Vaccinate proportion q of populationR0(1-q)11
  -q1/R0qc1-(1/R0)
• This is the minimum of the pop that have to be
  vaccinated in order to stop the spread of the
  disease

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Immunisation Thresholds
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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

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Further Reading

• The bible and for a link from SIR to population
  modelsAnderson 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 spinBrown et al. 2008.
  Evolution of virulence triggering host
  inflammation allows invading pathogens to exclude
  competitors.
• Fitting models to real dataKeeling 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 diseaseTildesley 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 articleKermack McKendrick 1927.
  http//links.jstor.org/sici?sici0950-12072819270
  801291153A7723C7003AACTTMT3E2.0.CO3B2-Z