I've experimented with a few different SI and SIR models in an attempt to find some closed orbits. So far, I've had no luck. (Note that $\beta$ is the infection probability/rate, $b$ is constant birthrate, $d$ is constant death rate, $v$ is probability/rate of recovery from infection, and $S$ are healthy but susceptible people, $I$ are infected people).
$\dot S=-\beta I(1-I)+bS-dS+vI\\\dot I = \beta I(1-I) + bI - dI - vI$
This is symmetric over $S=I$, so swapping the two would yield the same system - I am not sure if this matters or not in terms of finding periodic solutions.
I also tried mixing it up, giving $S$ all the births (ie, nobody is born infected).
$\dot S=-\beta I(1-I)+b-dS+vI\\\dot I = \beta I(1-I) - dI - vI$
This did not yield any limit cycles either, at least from what I gathered.
I also tried expanding to an SIR model (where suspectiple people get infected, infected people become immune ($R$)).
$\dot S=-\beta IS \\ \dot I = \beta IS - vI \\ \dot R = vI$
Clearly, for a fixed point to exist $I=0$. But then all the other components are zero, which implies that there is not a limit cycle (since $S,I,R \gt 0$).
I've also tried adding birth/death rate terms to the SIR model, but still cannot achieve a limit cycle.
I am wondering if the problem is that I am not looking hard enough to find a model with a limit cycle, or alternatively, if the limitation that $S+I=1$ or $S+I+R=1$ somehow makes limit cycles difficult to achieve.
My guess is that $S+I=1$ defines a subspace that is a line over two dimensions, and since periodic orbits are not possible on a one-dimensional surface, so it's not going to work. However, if $b$ and $d$ are not equal, $S+I$ is an exponential function of time, so in this case I don't see why periodic orbits are so difficult to find.
You will not find any system of this kind that would possess a limit cycle. If you are looking for an explanation, you can read this book.
The simplest model of this form that does have a limit cycle includes a nonlinear transmission rate of the form $$ -\beta S^pI^q\\ $$ where $p$ and $q$ are parameters. In particular, for large $p$ Hopf bifurcation can occur and limit cycle can appear. Details, if necessary, can be found in the same book.