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How can we model the spreading of a contagious disease?
The straight forward approach may be to model individual persons as either infectious or susceptible
If one infectious and one susceptible person meet, there is a certain chance
that the susceptible person becomes infectious too and then the first infectious person
as well as the newly infected person further spread the disease.
This seems to cry out loud for a spatial model in which
we take care of the location of these persons that would be complex.
Luckily, there is another way. This is a much simpler way of thinking about such diseases.
The SIR Model. We do not look at the spatial distribution at all.
We simply say that our population is made up of susceptible, infectious, and recovered persons
and we only look at the total number of these persons.
Susceptible persons can get infected and then become infectious.
The rate of that process is controlled by the number of infectious people.
The more infectious people there are, the larger the probability is
that a susceptible person meets an infectious person.
Hopefully, people are going to recover from the disease,
so that is the process that takes us from infectious to recovered.
And then the standard version of that model, we assume
that the recovered persons are immune and cannot be infected again.
So they form a third compartment.
This is a much simpler way of dealing with our problem.
We forget about the spatial distribution altogether and simply construct a mechanical model if you will
in which the population consists of three compartments,
susceptible persons, infectious persons, recovered persons
depending on the number of infectious persons, susceptible persons become infected
and hence infectious and after a while, infectious persons recover and we assume that
recovered persons are immune against the disease and cannot be infected again.
This is the SIR Model. Susceptible, Infectious, Recovered.