Extinction and persistence in a delayed Hawkes-modulated SIQ epidemic model with Crowley–Martin incidence
Year: 2026
Type: article
Abstract: We formulate and analyse a delayed Hawkes-modulated SIQ epidemic model in which a Crowley–Martin incidence is driven by a self-exciting, marked shot-noise transmission coefficient, and in which the current infectious burden feeds back into the event intensity. Our aim is to determine how discrete delay, quarantine, disease-induced mortality, and prevalence-driven contact clustering jointly reshape the extinction–persistence boundary. Analytically, we establish positivity, boundedness, and global well-posedness, and we prove non-explosion of the state-dependent Hawkes driver by dominating it with a subcritical linear Hawkes process; we then derive a stochastic threshold R 0 sto from the stationary boundary environment via a characteristic-root analysis of the linearised random delay equation and ergodic averaging. Numerically, we use a delay-aware Euler scheme coupled to Ogata thinning of the Hawkes events, with Monte-Carlo estimation of the pathwise Lyapunov exponent and the persistence mean. We show that R 0 sto < 1 implies almost-sure exponential extinction, while R 0 sto > 1 , under explicit tightness and a stated irreducibility condition, implies persistence in the mean with an interior invariant measure; the delay does not move the threshold but governs the transient rate and oscillations, as confirmed by a sensitivity study. The model unifies delayed Crowley–Martin transmission, quarantine, mortality, and Hawkes feedback in one framework, and the analysis extends stochastic threshold methods to delayed, self-exciting random environments.
Source: Alexandria Engineering Journal
Authors: Guorong Wu, Ali Akgül, Fahad Sameer Alshammari
Language: English
Field: Medicine
Domain: Health Sciences
Open Access status: gold
Estimated APC paid: $1,900
Award: G7740345600
OpenAlex ID: W7204195401