The influence of a transport process on the epidemic threshold

The paper’s first-order mean‑field model yields a transcritical bifurcation at χ(p∞)=1 with explicit endemic equilibrium (S*,I*,R*) and shows that transport lowers the threshold, while fractional (nonlocal) transport raises it by driving p∞ toward uniform; monotonicity in the fractional exponent can fail on certain topologies, but holds for stars (Appendix). These statements appear explicitly and/or are sketched in the paper’s Results and Appendix (definition of χ, equilibrium points, and bifurcation; fractional L^α, p∞(α)∝diag(L^α), α→0 uniform; non-monotone exceptions; star-graph monotonicity) . The candidate solution reproduces the same analysis: (i) the transport block ∂t p=−μΔᵀp converges to p∞, (ii) reduction to a closed SIRS with B(p∞)=(βc k+βt N||p∞||2)/N, (iii) DFE eigenvalues −σ and γ(χ−1), (iv) the same endemic equilibrium and local stability via negative trace/positive determinant, (v) a transcritical exchange of stability at χ=1, (vi) fractional L^α driving p∞ to uniform as α↓0 and thus raising the threshold; it also correctly notes possible non‑monotonicity and gives compliant examples (stars monotone), aligning with the paper. Thus, both are correct and follow substantially the same proof strategy.