Uniqueness of positive bounded entire solutions for the limiting epidemic equations

Under Assumptions~--, let $R_0 > \mathbb{E}\left[\frac{1}{\gamma_*}\right]$ and suppose that $\overline{\mathfrak F}(0)>0$. For functions $x,y:\mathbb{R}\to\mathbb{R}$, the limiting equations are

$$y(t)=\int_{-\infty}^{t}\overline{\lambda}(t-s)x(s)y(s)\,ds,$$

and

$$\int_{-\infty}^{t}\mathbb{E}\left[\exp\left(-\int_s^t\gamma(r-s)y(r)\,dr\right)\right]x(s)y(s)\,ds=1.$$

A solution is positive and bounded when both component functions are positive and bounded on $\mathbb{R}$.

Uniqueness conjecture. The limiting system above has a unique positive and bounded solution on $\mathbb{R}$.

This uniqueness statement is presented as an equivalent formulation of the convergence conjecture: uniqueness of the limiting entire solution would force all subsequential time-translation limits to equal the constant endemic equilibrium. The constant pair $(1/R_0,\overline{\mathfrak F}_*)$ is known to solve the system, but uniqueness remains open.