Stochastic dynamics with self-exciting factor for Monkeypox transmission

TL;DR

This work develops a stochastic human–rodent mpox model that combines diffusion noise with Hawkes self-exciting jumps to capture clustered transmission events. It establishes strong theoretical results: global existence and positivity of solutions under subcritical Hawkes excitation, an extinction threshold via a reproduction number $\mathcal{R}_0$ that includes Hawkes contributions, and persistence-in-mean conditions for both humans and rodents. The analysis reveals how clustering and environmental variability shape outbreak thresholds and long-run infection levels, with numerical insights into how control measures and event-driven spikes shift these thresholds. Overall, the framework provides a rigorous, jump-diffusion perspective on Mpox dynamics that highlights the impact of self-exciting events on cross-species transmission and outbreak risk.

Abstract

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.

Figures (8)

• Figure 1: Schematic diagram of Mpox virus transmission
• Figure 2: The first two graphs show some sample paths of linear Hawkes processes with exponential kernels and their corresponding conditional intensity function. The last two graphs display the plots of the associated variables.
• Figure 3: $\mathcal{R}_0$ as a function of the natural death rates $\mu_h$ and $\mu_r$. In the left-hand side the Hawkes jumps have exponential distribution with mean value $1$. The right-hand side describes the process in absence of jumps. The involved parameters are fixed according to Tables \ref{['tab:baseline']} and \ref{['model_params_main']}.
• Figure 4: $\mathcal{R}_0$ as a function of the disease-induced death rates $\delta_h$ and $\delta_r$. On the left-hand side, the Hawkes jumps are exponentially distributed with mean value $1$, whereas on the right-hand side, there is absence of jumps. The involved parameters are fixed according to Tables \ref{['tab:baseline']} and \ref{['model_params_main']}.
• Figure 5: Heat maps of $\mathcal{R}_0$ computed with baseline parameter values. (Left) Dependence on the combined rodent--human contact rate $\eta_1+\eta_2$ and the human disease-induced mortality $\delta_h$, with public enlightenment fixed at $p=0.3$. (Right) Dependence on $(\eta_1+\eta_2,p)$ for fixed $\delta_h=10^{-3}\,\text{day}^{-1}$. The white contour marks the threshold $\mathcal{R}_0=1$.

Theorems & Definitions (19)

• Definition 2.1
• Proposition 2.2: Itô Formula for Hawkes-Itô Processes bensoussan
• Theorem 3.1
• proof
• Remark 3.2
• Theorem 4.1
• proof
• Remark 4.2
• Definition 5.1
• Lemma 5.2