Stochastic compartment model with mortality and its application to epidemic spreading in complex networks

The paper derives the endemic equilibrium under zero mortality via Laplace-transform persistence arguments, obtaining Jw = (R0−1)/R0 · [βw⟨twI⟩/(1+βw⟨twI⟩)] and Jn = (R0−1)/R0 · [βn⟨tnI⟩/(1+βn⟨tnI⟩)], existing uniquely for R0 = βwβn⟨twI⟩⟨tnI⟩ > 1; no interior solution exists for R0 ≤ 1 . With mortality affecting walkers, the paper defines RM = βwβn⟨tnI⟩⟨min(twI,tM)⟩ and proves RM ≤ R0, with equality only for zero mortality . The candidate solution mirrors these results using steady-state incidence×duration reasoning and the inequality E[min(X,Y)] ≤ E[X], giving the same conditions and explicit formulas. The logical steps and assumptions (bilinear rates, finite mean infectious durations, and independence) align with the paper’s model setup .