
This answer would simply lead to a decay of the number of infectious persons

no matter what the rate of change is a negative factor times the current value.

This doesn't make sense

With this answer, the rate of change which depend only on the number of susceptible persons.

It would not matter to the rate of change of the number of infectious persons

whether we started with 1 or 10 or 1 million infectious persons.

This doesn't make sense. So we have to have a solution that incorporates both quantities.

The number of susceptible persons and the number of infectious persons. So what about that ratio?

This would mean that if the number of infectious person is low, this ratio is high.

We divide by a small number and the rate would be high which again doesn't make sense.

And there is another reason why this doesn't makes sense,

the unit would be 1/days times persons/persons.

This canceled and unit of the result is 1/days but what we need is persons per day,

so this cannot be true for several reasons.

This is what is left and it makes sense. Let's look at the units.

A number divided by days and persons, times persons squared.

What we are left with is person per day, that's what we need and this product has the right behavior.

If we increase the number of infectious persons, we increase that rate.

We have more encounters. That makes sense.