Epidemics on Hypergraphs: Spectral Thresholds for Extinction

The paper’s Theorem 8.1 proves the non-extinction probability bound by constructing a stochastically dominating auxiliary process Y with linear rates and then solving the corresponding linear ODE for its expectations, yielding P(∑ Xi(t) > 0) ≤ n i0 exp((β f′(0) λ(W) − δ) t) and exponential die-out when β f′(0) λ(W) < δ. The candidate solution derives a differential inequality directly for p(t)=E[X(t)] using the generator and the global tangent bound f(x) ≤ f′(0) x (concavity with f(0)=0), then controls ||p(t)||2 and converts to the same probability bound. Both are logically valid; the techniques differ (stochastic domination vs. direct expectation drift inequality).