Beginner at modelling here.
To generate an equation for the spread of disease, my textbook gave me this:
x(t) denotes the number of people who have contracted the disease
y(t) denotes the number of people who have not yet been exposed
The number of interactions is jointly proportional to x(t) and y(t)
Therefore: dx/dt = kxy
Suppose a small community has a fixed population of 'n' people. If one infected person is introduced into this community, then: dx/dt = kx(n+1-x)
My question: why 'n+1'? If y=n+1-x, then wouldn't this mean the number of 'uninfected' people is greater since y(t) denotes the number of people who have not yet been exposed?
To introduce an infected person, wouldn't you change x to be x+1?
Thank you in advance!
You start with a community of $n$ people. When you introduce on infected person the community now has $n+1$ people. If $x$ are infected, the number uninfected is $n+1-x$. At the moment the infected person is introduced we have $x=1$ infected person and $n=y=n+1-x$ uninfected people.