Cramér–Lundberg Risk Process Driven by Compartmental Epidemic Models: Jacobian Triangularity, Volterra Bounds and Competing Claim Laws

We study the ruin of an insurer whose claims are driven by an epidemic. Claims arrive as a Poisson process whose rate is an affine function of the infectious population level I(t) of a deterministic epidemic model; this is the deterministic occupation-time specialization of the individual model of Guérin, Mandjes, Renaud and Zotsa Ngoufack. The insurer's risk then depends on the epidemic only through the cumulative infectiousness, the time integral of I(t), along the whole path. Our results convey six ideas. (1) The basic reproduction number R0 separates three insurance regimes: the cumulative infectiousness remains bounded when R0 < 1, grows logarithmically when R0 = 1, and grows linearly when R0 > 1. (2) Lyapunov functions, used in epidemiology to prove stability, become quantitative actuarial tools: they bound the peak and the cumulative infectiousness for all times, and this implies premium and reserve rules which bound the probability of ruin over every horizon. (3) When premiums exceed the expected claims in the long run and claim sizes are phase-type, the ruin probability decays exponentially in the initial reserve, at a rate fixed by the long-run frequency of claims (the baseline frequency plus the claim intensity caused by the endemic level I* of the infectious population, which is zero when \( R_0\le1 \)). At the epidemic threshold R0 = 1 the decay carries an exact polynomial correction: for exponential claims the ruin probability is asymptotically a constant times \( u^{b_0}e^{-\kappa_0u} \), with the power b0 and the constant both determined explicitly by the critical epidemic path, for the SIR model and for the SIH hospitalisation model alike. (4) The vaccination needed for a positive long-run surplus drift is less than that required for eliminating the disease (which is 1 - 1/R0). Between the two thresholds the disease persists, but the insurer remains solvent on average. (5) Competing strains with different claim costs are controlled jointly by one Volterra function: the claim cumulant is maximized over its sublevel set, which gives ruin bounds that retain every claim law, also for tied endemic strains and maternal immunity. (6) Random epidemic duration can change the exponent: for a stochastic epidemic, an exponential moment of the total infectiousness preserves the baseline exponent and gives the exact constant, but almost-sure extinction alone does not; the exponent is then the smaller of the baseline exponent and a pre-extinction exponent.