Effects of density dependence, partial cross immunity and non-linear incidences on coinfection dynamics

The paper’s Proposition 7.1 claims global asymptotic stability of the disease-free equilibrium G2 = (S**,0,0,0,0) under S** < min{(µ′1−µ0)/α1,(µ′2−µ0)/α2,(µ′3−µ0)/α̂3}, where α̂3=β1+β2+α3. The authors choose a Lyapunov function v=S−S** ln S+I1+I2+I12+R and compute v′ exactly as v′=−(b/K)(N−S**)^2−(α1S**−(µ′1−µ0))I1−(α2S**−(µ′2−µ0))I2−(α̂3S**−(µ′3−µ0))I12−(µ′4−µ0)R, which is strictly negative off G2 under the stated hypothesis, consistent with their model (1) and assumptions (2) and the definition S**=K(b−µ0)/b (all explicitly stated in the paper) . However, instead of completing the argument via LaSalle or a strict Lyapunov argument on a positively invariant, bounded set (boundedness is earlier established), the paper ends by invoking “Lemma 2 in [6]” and even misprints “N=S converges to S*,” which is dimensionally and notationally inconsistent with G2’s S**; it also omits an explicit justification for R(t)→0 (though this follows from the last equation once N→S**) . In contrast, the candidate solution proves forward invariance, uniform boundedness of trajectories (via N), uses the entropy-type Lyapunov function V=S−S**−S**ln(S/S**)+I1+I2+I12+R, derives the same v′-structure with the key identity (b(1−N/K)−µ0)(N−S**)=−(b/K)(N−S**)^2, shows that V′=0 only at G2, and then correctly applies LaSalle’s invariance principle (precompactness ensured by the N-bound) to conclude global convergence, also noting the technical domain S(0)>0 for the log term. Net: the paper’s statement is right and its v′ formula is correct, but the proof as written is incomplete and contains a notational error at the final step; the model solution is complete and correct. Supporting details: system (1) and parameter definitions , the bound S**=K(b−µ0)/b and boundedness results for S and N , and Proposition 7.1 including the derivative (106)–(108) .