An integral renewal equation approach to behavioural epidemic models with information index

The paper proves that with weak Erlang memory K(τ)=a e^{-aτ}, the slope condition g′(Fe)β′(Me)Fe ≥ −2β(Me) ensures local asymptotic stability of the endemic equilibrium for any infectivity-with-demography kernel Aµ, via the characteristic equation (22) and the bound |1 + a g′β′Fe/(β(a+w))| ≤ 1 in Re w ≥ 0, hence no roots with nonnegative real part (Theorem 5.3) . The candidate’s solution derives an equivalent factorization 1 = B̂(z)G(z) (rearranging (22)) and uses |B̂(z)| ≤ 1 and |G(z)| < 1 in Re z ≥ 0 to exclude roots, which is the same frequency-domain argument in slightly different algebraic form. Both are correct and materially the same approach.