Extinction and Persistence in a Stochastic Mpox Model with Hawkes-type Self-Excitation

We develop a stochastic human-rodent compartment model for Mpox transmission that combines diffusion noise with Hawkes self-exciting jumps in the human infection dynamics. Including Hawkes processes allows, for instance, to model the short but significant spikes in transmission happening after crowded events. For the coupled human-rodent system, we prove global existence, uniqueness and positivity of solutions, derive a basic reproduction number R_0 that guarantees almost sure extinction when R_0 < 1, and obtain explicit persistence-in-the-mean conditions for both infected rodents and humans, which define persistence thresholds for the joint dynamics. Numerical experiments show how clustered human transmission events, environmental variability and control measures shift these thresholds and shape the frequency and size of Mpox outbreaks.

💡 Research Summary

The paper introduces a novel stochastic compartmental model for monkeypox (Mpox) transmission that simultaneously captures human‑rodent zoonotic dynamics and the clustered nature of human outbreaks. The human population is divided into four compartments—susceptible (S_h), infected (I_h), quarantined (Q_h), and recovered (R_h)—while the rodent reservoir is modeled with susceptible (S_r) and infected (I_r) classes. The deterministic backbone of the model follows classic mass‑action incidence with recruitment, natural death, disease‑induced mortality, quarantine, and recovery rates.

To reflect real‑world variability, the authors augment the deterministic system with two distinct stochastic perturbations. First, continuous environmental fluctuations are modeled by eight independent standard Brownian motions (B_1,…,B_8) with volatility coefficients σ_i, affecting each compartment’s drift. Second, and most innovatively, the human infection dynamics (particularly I_h and Q_h) are driven by self‑exciting Hawkes jump processes. Unlike memoryless Poisson jumps, Hawkes processes increase their conditional intensity after each jump, thereby reproducing the rapid, short‑lived spikes observed after mass‑gathering events, super‑spreader incidents, or sudden changes in human behavior. The jump terms are represented by compensated Hawkes random measures (\tilde H_i(dt,dy)) with jump‑size functions ε_i(y). No jumps are introduced for rodents, as their behavior is assumed not to be directly influenced by human control measures.

Mathematically, the authors prove global existence, uniqueness, and positivity of the solution in the non‑negative orthant (\mathbb{R}_+^6). The proof relies on constructing a Lyapunov function, applying Itô‑Lévy calculus, and exploiting the martingale property of the compensated Hawkes measures. Boundedness assumptions on total human population (N_h ≤ M) and a time‑varying ratio bound for rodents (N_r ≤ k(t) N_h) are used to control drift and jump coefficients.

A basic reproduction number (R_0) is derived by linearizing the system around the disease‑free equilibrium and computing the spectral radius of the next‑generation matrix that incorporates both human‑human, rodent‑human, and rodent‑rodent transmission pathways. The expression involves recruitment rates (θ_h, θ_r), contact rates (η_1, η_2, η_3), natural and disease‑induced death rates (μ_h, δ_h, μ_r, δ_r), quarantine and treatment effectiveness (p, θ), and the expected jump intensity of the Hawkes processes. The authors prove an extinction theorem: if (R_0 < 1), then almost surely the infected compartments (I_h(t)) and (I_r(t)) converge to zero as t → ∞. This result mirrors deterministic threshold behavior but holds in the stochastic setting despite the presence of jumps.

Conversely, the paper establishes persistence‑in‑the‑mean conditions for both human and rodent infections. By analyzing the expected dynamics and using the law of large numbers for Hawkes processes, the authors obtain sufficient inequalities that guarantee (\liminf_{t\to\infty}\mathbb{E}