Spread of SARS-CoV-2 in a SIS model with vaccination and breakthrough infection

The paper specifies the 4-ODE SIS model with three susceptible classes and explores it numerically, reporting a plateau (endemic) prevalence and monotone parameter sensitivities, but it provides no mathematical proofs and even calibrates parameters via R0 = β/γ from a single-class SIS model rather than the 3-class system it actually studies . The model solution gives a rigorous framework: it derives the correct DFE, an R0 consistent with the 3-class chain, an endemic-equilibrium equation G(i)=0, Hirsch-based convergence arguments, and comparative statics. However, its proof of uniqueness of the endemic equilibrium is incomplete: showing G′(i*) < 0 proves the root is simple, not unique; global convergence relies on (unproven) uniqueness. Hence, the paper is incomplete (no proofs; questionable R0 calibration), and the model solution is also incomplete (gap in uniqueness/global convergence).