Modeling Cholera Dynamics Under Food Insecurity and Environmental Contamination: A Multi-Patch Approach

The paper correctly derives R0 = (βI + βW(1+f*))/ (γ+µ) via the next-generation matrix and identifies the endemic equilibrium branch s* = 1/R0, w* = i*, f* = 1 − a/r (equations (4)–(5) and (8) are consistent) . However, the global stability proof of the DFE (Lemma 3.3) uses V(s,i)=s−1−ln s + i and asserts dV/dt ≤ −µ(s−1)^2/s + (γ+µ)(R0−1)i ≤ 0 by bounding terms with f ≤ f* and effectively dropping the positive βW(1+f)w contribution; this step is not valid and the inequality chain in (7) does not provide an upper bound on dV/dt for the full system including w, so the LaSalle argument as written is flawed . By contrast, the candidate solution supplies a correct Lyapunov function that includes w with the precise coefficient c=βW(1+f*)/ξ to cancel the βW(1+f*)w term, proves GAS of the DFE for R0≤1 for the limit system, and then transfers GAS to the original via asymptotically autonomous dynamics and Thieme’s convergence. For R0>1, both the paper’s Appendix A and the candidate invoke persistence theory; the candidate’s persistence argument is sound and complete at the level of a standard proof sketch. The paper’s endemic stability and forward bifurcation discussion is largely qualitative or algebraically incomplete, while the candidate provides the endemic equilibrium explicitly and the correct local stability mechanism.