Quasineutral multistability in an epidemiological-like model for defective-helper betacoronavirus infection in cell cultures

The paper identifies three planes of equilibria Π_CC^D, Π_C_D^D, Π_VD for the 6D ODE (7)–(12), derives the Jacobian spectra on each, and obtains the same threshold conditions in terms of B, β, δ as the candidate solution: Π_VD is always normally attracting; Π_C_D^D is normally attracting iff B < 1+β+δ; Π_CC^D is attracting everywhere if B < 1+β, partially attracting on a half-plane if 1+β < B < 1+β+δ (with boundary C/C_D = (1−B/(1+β+δ))/(B/(1+β)−1)), and repelling if B > 1+β+δ. These are exactly the planes, spectra, and thresholds reported in Proposition 1 and its analysis (Eqs. (14)–(21)) and Table 2 of the paper, including the characterization of ε+ and λ+ and the same splitting by B relative to 1+β and 1+β+δ . The candidate’s derivations match the paper’s formulas (e.g., η = B/(1+β), κ = 1+δ/(1+β), η/κ = B/(1+β+δ)), recover the same eigenvalue factorizations, and reach the identical stability partitions and “tristability” statements summarized in the abstract and Table 2 . Both also agree that for B > 1+β+δ the only attracting plane is Π_VD and that it is globally attracting in that regime (the paper sketches a proof and provides integral identities based on Ċ_T = −α(C_V+C_DV)) . The candidate adds a LaSalle-based argument to place ω-limit sets in {C_V=C_DV=0} and then inside the union of the three planes; while the paper does not frame this as a LaSalle theorem, it derives the key monotonicity Ċ_T = −α(C_V+C_DV) and uses it to obtain the same asymptotic conclusions (especially for B > 1+β+δ) . Overall, the logic, computations, and conclusions coincide; the approaches are essentially the same with minor presentational differences.