We add the immunity loss to the SIR model and obtain the following autonomous system.
$$
\begin{align}
s' &= -is+\alpha r \\
i' &= i s - \gamma i\\
r' &= \gamma i-\alpha r
\end{align}
\tag1
$$
with $$(s+i+r)\big|\_{t=0}=1,\ s(0)\ge0,\ i(0)\ge0,\ r(0)\ge0,$$
where prime denotes derivative w.r.t. time, $s,i,r$ represent the proportion of “susceptible”, “infected” and “recovered” individuals, $\beta$ is the kinetic constant of infectiousness, $\gamma$ that of recovery, and $\alpha$ the speed of the immunity loss. Suppose all the coefficients are positive. It is easy to supply the candidates for the long term ($t\to\infty$) asymptotic steady solution, which is to simply set the derivatives on the left hand side of the differential equations to zero, and obtain two solutions.
$$s\_\infty=1,\ i\_\infty=0,\ r\_\infty=0;\tag2$$
or
$$s\_\infty=\min(\gamma,1),\ i\_\infty=\frac{(1-\gamma)\_+}{1+\frac\gamma\alpha},\ r\_\infty=\frac{(1-\gamma)\_+}{1+\frac\alpha\gamma}.\tag3$$
I conjecture that Solution (2) is achieved either when the initial condition is exactly that, and that Solution (3) is achieved under all other conditions.


The function
${\scr L}(s,i)=s-s\_\infty\ln s+w(i-i\_\infty \ln i)$ for some positive $w$ seems to fail to play the role of a Lyapunov function. What is an appropriate Lyapunov function for this autonomous ODE?



Seeking a Lyapunov function for a SIR model with immunity loss
We examine the local stability of this system. Since the first two equations of System $(1)$ form the largest set of independent equations, the Jacobian of this system is
$$J(s,i) := \begin{bmatrix}
-i-\alpha & -s-\alpha \\
i & s-\gamma
\end{bmatrix}.
$$


At Fixed Point $(3)$, the eigenvalues are 
$$x=-\frac{1+\alpha}{2(1+\frac\gamma\alpha)}(1\pm\sqrt{1-4\delta}),\quad \delta:=\frac{1-\gamma}\alpha\Big(\frac{\alpha+\gamma}{1+\alpha}\Big)^2.$$
$\gamma<1\iff\delta>0\implies \mathbf{Re}(1\pm\sqrt{1-4\delta})>0$. So the system is locally stable there.


At Fixed Point $(2)$, the eigenvalues are
$$x\_1:=-\alpha,\ x\_2:=1-\gamma.$$
For $\gamma<1$, the fixed point is locally unstable.


Therefore, as Piyush Grover suggests in his comment above, the system is not globally stable for the domain $\big\{(s,i)\,\big|\,s+i\le1,s\ge0, i\ge0, (s,i)\ne(1,0)\big\}$ when $\gamma<1$.


