A discrete-time epidemic SISI model

The paper’s Theorem 2.4 asserts: (i) uniqueness when β1 k1 = b+α and α β2 k2 > b β1 k1; (ii) uniqueness when β1 k1 > b+α; and (iii) no positive solution when β1 k1 < b+α (all in the setting of the discrete-time SISI model and the fixed-point equation (2.3)) , with (2.3) explicitly given in the paper . The proof of (iii) hinges on treating g(x) as increasing and convex and deriving an inequality that forces β2 k2 > 1/α, claimed to contradict the parameter box (2.2) , which indeed includes b+β2 k2 ≤ 1 among other constraints . However, direct algebra shows the equation reduces to an upward-opening quadratic F(A)=0 with coefficients C2=(b+α)β1β2, C1=b(b+α)(β1+β2)−β1β2(b k1+α k2), C0=b^2(b+α−β1k1), and for β1 k1<b+α it is possible to have two positive solutions when αβ2k2 is large. A concrete parameter set satisfying all constraints (2.2) produces two distinct positive solutions to (2.3), contradicting (iii). Therefore (iii) is false under (2.2), while (i)–(ii) are correct by sign/discriminant analysis.