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SIR Epidemics and Modeling

The SIR model is a fundamental framework for understanding the dynamics of infectious diseases within a population. It categorizes individuals into three groups: Susceptible (S), Infective (I), and Removed (R), indicating their role in disease transmission. As individuals transition from infective to removed due to immunity or isolation, the model highlights disease flow through basic equations. Key concepts, such as the basic reproductive ratio (R0), determine whether an infection will spread or stabilize. Examples of SIR diseases include smallpox, measles, and chickenpox.

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SIR Epidemics and Modeling

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  1. SIR Epidemics and Modeling By: Hebroon Obaid and Maggie Schramm

  2. 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

  3. Examples of SIR diseases • Smallpox • Measles • Mumps • Foot and mouth • Chicken pox

  4. 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/dτ= -βIS dI/dτ= βIS-γI dR/dτ= γI βIS γIS  I  R

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

  6. Further Info • R0 for various diseases • AIDS 2 to 5 • Smallpox 3 to 5 • Measles16 to 18 • Malaria > 100 • http://www.shodor.org/master/biomed/epidemio/sir/runsir.html • http://plus.maths.org/issue14/features/diseases/index-gifd.html • herd immunity- vaccination for an individual in a community provides some sort of immunity for the entire community

  7. The End

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