SIR Epidemics and Modeling By: Hebroon Obaid and Maggie Schramm

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

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

Basic Equations for the Model S  I  R N = S(t) + I(t) + R(t) Flow of the disease: The entire population (N): Movement between classes: dS/dτ = -βIS dI/dτ = βIS-γI dR/dτ = γI βIS γI S  I  R

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

Further Info R 0 for various diseases R 0 for various diseases AIDS 2 to 5 Smallpox 3 to 5 Measles16 to 18 Malaria > 100    herd immunity- vaccination for an individual in a community provides some sort of immunity for the entire community

The End