Kermack-McKendrick Model (SIR model)
Overview
{% S(t) %} is the number of individuals susceptible to the disease
{% I(t) %} is the number of individuals who are infected
{% R(t) %} is the number of individuals removed from the population
Kermack-McKendrick Model (SIR model)
The basic model originally proposed by Kermack and McKendrick is
{% I'(t) = \beta S(t) I(t) - \alpha I(t) %}
That is, the rate of change of the infected individuals is the sum of two factors.
- {% -\beta S(t) I(t) %} - new infected individuals come from the unifected population, {% S(t) %}.
Infections come from contact with the infected population, {% I(t) %}
- The number of infected individuals are removed from the population at a rate of {% \alpha I(t) %}
- note, the individuals that are removed from the infected population are not added back to the
unifected population. That is, those individuals have become immune.
The rate of change of {% S(t) %} is just the negative of the first factor of the rate of change of {% I(t) %}.
{% S'(t) = -\beta S(t) I(t) %}
The rate of change of R is just proportional to {% I(t) %}
{% R'(t) = \alpha I(t) %}
The following constraint holds in general:
{% N = S+I+R %}
That is, the total population is constant and is the sum of the three terms. As a general rule, the R term can
be ignored, as it does not enter into the equations for S or I, I being the variable we are primarily interested
in. When R is ignored, the SIR model is effectively an
SI model
The original SIR model given above can be numerically solved using the
euler methodology.
The quantity {% R_0 = \beta S/\alpha %} is the reproduction number which specifies the number of secondary infections
caused by an infected individual. For {% R_0 %} < 1, the disease dies out.
