A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation

The paper proves AFE stability via the Jacobian at (1,0,0) with eigenvalues −µ (double) and βκ−(µ+γ)=(µ+γ)(R0−1), so AFE is LAS iff R0<1, matching the model’s step 1. It derives, for ν>0, the cubic f(a)=x3 a^3+x2 a^2+x1 a+x0 for endemic equilibria with the same coefficients used by the model, and notes f(0)=Rµ^2(1−R0) and f(1)>0, implying existence for R0>1 and nonexistence when R0<1 and Rφ<1; this aligns with the model’s step 2. The paper states necessary conditions for two and three positive equilibria and a sufficient condition for uniqueness beyond the upper saddle-node (R*0), which the model reproduces with slightly more detail (e.g., the Vieta/sign argument forcing Rφ>1 for multiple positive roots). Overall, the reasoning tracks the paper’s theorems and remarks and reaches the same conclusions with minor elaborations. Key matches: AFE Jacobian/eigenvalues and stability threshold (Theorem 3.1) ; cubic coefficients and endpoint facts (f(0), f(1)) for ν>0 ; region-based conditions for 2, 3, and 1 positive equilibria (Theorems 3.2–3.4) ; model and parameter definitions, including Rφ, Rµ, g(ŝ) .