The Impact of Temperature and Isolation on COVID-19 in India : A Mathematical Modelling approach

The paper’s stability and Hopf-bifurcation analysis around the disease-free equilibrium E0 factorizes the characteristic equation into a scalar quarantine block and a 2×2 (E,I)-block, exactly as in the candidate solution. The paper states χ(λ)=(λ+µ)^3(λ+µ+δα+ρ(1−α)e^{−κλ})(λ^2+d1λ+d2+e^{−λτ}(e1λ+e2)) with d1=2µ+ε+γ, d2=(γ+µ)(ε+µ)−εβ, e1=pe^{−γτ}, e2=pe^{−γτ}(µ+ε) (matching the model) . For the (E,I)-block, both derive the same quasi-polynomial λ^2+d1λ+d2+(e1λ+e2)e^{−λτ}=0, reduce the imaginary-root condition to ω^4+f1ω^2+f2=0 with f1=d1^2−2d2−e1^2, f2=d2^2−e2^2, and give the same critical delay τ* via arccos(⋯) . Both compute the eigenvalue crossing speed via implicit differentiation and conclude transversality using the same x, y, z expressions; the paper presents Re((dλ/dτ)^{-1}) with the condition (x+y)>0 and (x+y)>z, which is equivalent in sign to Re(dλ/dτ)>0, while the model writes Re(dλ/dτ)=e1[(x+y)−z]/H with H>0 . The only issues in the paper are minor: (i) a typographical inconsistency in the theorem statement (“x>y” appears where later the proof uses (x+y)>z and (x+y)>0), and (ii) the scalar quarantine factor’s delay-independent stability is argued informally via the absence of imaginary roots, whereas the model correctly invokes the standard a>|b| criterion λ+a+be^{−κλ}=0 ⇒ asymptotic stability for all κ (a rigorous and widely known result). Overall, the candidate solution matches the paper’s derivations and conclusions and slightly strengthens the scalar-block argument.