Equilibrium selection with tied maximal reproduction numbers in infection-age PDE models

Determine, for the infection-age structured competitive epidemiological model with n strains defined by S'(t) = Λ − μ_S S(t) − S(t)∑_{k=1}^n ∫_0^∞ β_k(a) x_k(t,a) da, ∂_t x_k + ∂_a x_k = − μ_k(a) x_k, and boundary condition x_k(t,0) = S(t) ∫_0^∞ β_k(a) x_k(t,a) da, which endemic equilibrium the solution converges to for a given initial condition in the case where more than one strain attains the maximal basic reproduction number R_{0,k} > 1, thereby characterizing the asymptotic selection mechanism in the PDE setting.

Background

In ODE models of multi-strain competition, the asymptotic behavior is fully characterized, and when several strains share the largest basic reproduction number, the limit equilibrium is determined by the initial condition. Prior PDE formulations with infection-age structure lacked such a complete description in the tie case.

The cited literature addressed uniqueness-of-maximum cases (competitive exclusion) or provided only partial results (e.g., for n=2). The authors point out that identifying the limit equilibrium under ties of the maximal R0 has remained unresolved in the PDE framework.