I'm modelling the spread of an infection on a population of size N, which remains constant.
Let H(t) be the number of healthy people at time t, a simple model for this, is given by the equation
$ \frac{\partial H}{\partial t} = -kH(N-H) $
Which is a non-linear ODE, which can be easily solved by separation of variables, and solves to
$ H(t) = N \frac{e^{cN-kNt}}{e^{cN-kNt}-1} $.
I'm interested in the fixed points of the ODE and their stability.
$ \frac{\partial H}{\partial t} = 0 \iff H(t) = 0 \ or \ H(t) = N $
Which means there are only two fixed points, which are either when the population is fully healthy (H(t)=N) or the population is fully infected (H(t)=0).
Is this procedure to find fixed points correct?
How can I work out which of these fixed points is the stable one and which one is the unstable one?