A refractory density approach to a multi-scale SEIRS epidemic model

The paper formulates the macroscopic refractory-density PDE system (∂t+∂t*)ρ = −ρ H(U,t*), (∂t+∂t*)U = −U/τ + k(t)I + D(t*), ν(t)=∫0^∞ ρH + I0 δ(t), I(t)=∫_{t−τI}^t ν (Eqs. 9a–9d), with boundary ρ(t,0)=ν(t) and U(t,0)=V_T, plus initial data with an atom at t* = ∞; it also states mass conservation via the boundary/source identity and gives both escape- and white-noise hazards (Eqs. 6 and 8) . At the mesoscopic level, it introduces ρ=S ν_N(t−t*), a survivor S solving (∂t+∂t*)S=−SH(U), and a compensator H̄ in ν for the residual mass, with ν_N(t)=ν(t)+√(ν(t))/N ξ(t) (Eqs. 11a–11d) and emphasizes that ρ is an abstract measure; it asserts convergence as N→∞ but supplies no proof beyond references and simulations . The candidate solution reproduces the macroscopic PDEs and boundary conditions, derives the inflow trace ρ(t,0)=ν(t), proves mass conservation, and outlines a rigorous fixed-point/characteristic argument yielding well-posedness and convergence (under stated Lipschitz/boundedness conditions). Small mismatches remain: (i) the paper’s mesoscopic U-equation includes an unexplained additive u(t) term, which the candidate omits (likely a typographical slip), and (ii) the compensator H̄ is modeled differently (paper: a weighted average; model: a limit at t*→∞), though both reduce to 0 in the escape-noise case and serve the same purpose. Net, the paper’s arguments are correct but incomplete on well-posedness and convergence, whereas the model’s solution fills these gaps with a coherent proof sketch.