On qualitative analysis of a discrete time SIR epidemical model

The paper computes the Jacobian at E0 and E1 and applies a 2×2 discrete-time stability criterion to obtain the exact stability regions in terms of β0, β1, β2, and rmax, culminating in Theorem 3.2 (disease-free E0 stable for 1<r<3 and 0<β<β0; endemic E1 stable for either 1<r≤3 with β0<β<β2 or 3<r<rmax with β1<β<β2) . The candidate solution reproduces the same Jacobians, rewrites J(E1) in the convenient form [A,−K; C,1], and then verifies the three standard Schur/Jury inequalities (equivalent to the paper’s 2×2 conditions) to derive exactly the same thresholds and cases, including the identities 1+aS*=β/(β−aK), S*/(1+aS*)=K/β, and I*/(1+aS*)^2=N/β^2 and the necessity β>β0 from 1−τ+Δ>0. It also clarifies monotonicity of the relevant quadratic for 1<r≤3. Thus, both are correct and follow the same stability-program (Jacobian → trace/determinant → 2×2 Jury/Schur test), with the model giving slightly more algebraic detail while the paper leans on CAS assistance for solving the inequalities .