We add the immunity loss to the SIR model and obtain the following autonomous system. $$ \begin{align} s' &= -\beta is+\alpha r \\ i' &= \beta i s - \gamma i\\ r' &= \gamma i-\alpha r \end{align} \tag1 $$ 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 $$s_\infty=\min\Big(\frac\gamma\beta,1\Big),\ i_\infty=\frac{\big(1-\frac\gamma\beta\big)_+}{1+\frac\gamma\alpha},\ r_\infty=\frac{\big(1-\frac\gamma\beta\big)_+}{1+\frac\alpha\gamma}.\tag3$$ 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.
How does one prove or disprove this conjecture?
I tried using a Lyapunov function such as ${\scr L}(s,i)=s-s_\infty\ln s+i-i_\infty \ln i$. But I have not succeeded in carrying out the derivation.
Prologue: It has an answer on Mathoverflow.net.