SIR Epidemics and Modeling

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SIR Epidemics and Modeling
By Hebroon Obaid and Maggie Schramm
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SIR Diseases

• Individuals leaving the infective class play no
  further role in the disease
• They may be immune, dead, or removed by isolation
  and therefore are at no further risk of
  contracting the disease.
• Thus the entire population can be categorized in
  one of three categories

S? susceptible, those who have not been
infected I ? infective, those who are currently
infected R ? removed, those who have a permanent
immunity
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Examples of SIR diseases

• Smallpox
• Measles
• Mumps
• Foot and mouth
• Chicken pox

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Basic Equations for the Model
Flow of the disease
S ? I ? R
The entire population (N)
N S(t) I(t) R(t)
Movement between classes
dS/dt -ßIS dI/dt ßIS-?I dR/dt ?I
ßIS ?I

S ? I
? R
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Equations

• Let variables equal the parameters.. Simplify the
  equations
• u S/N , v I/N , w R/N, t ? t
• Taking the derivative, the equations become
• du/dt -R0uv, dv/dt (R0u-1)v, dw/dt v
• where R0 ßN/ ? which is the basic reproductive
  ratio, i.e. the rate of contraction over the rate
  of recovery times the population
• If R0lt 1, then the population will eventually
  stabilize at about 0 infected people
• If R0gt1, then the population will stabilize at
  some number higher than 0, indicating a presence
  of infection

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Further Info

• R0 for various diseases
• AIDS 2 to 5
• Smallpox 3 to 5
• Measles16 to 18
• Malaria gt 100

• http//www.shodor.org/master/biomed/epidemio/sir/
  runsir.html
• http//plus.maths.org/issue14/features/diseases/i
  ndex-gifd.html
• herd immunity- vaccination for an individual in
  a community provides some sort of immunity for
  the entire community

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• The End