Modeling Transmission Intensity in SI Epidemics via CIR and Jacobi Processes: Asymptotic Results and Preliminary Intervention Strategies

TL;DR

This paper models epidemic transmission rates using stochastic processes, analyzing their asymptotic behavior and exploring intervention strategies through numerical simulations of CIR and Jacobi processes.

Contribution

It introduces a stochastic modeling framework for transmission intensity with asymptotic analysis and preliminary intervention strategies using CIR and Jacobi processes.

Findings

Asymptotic behavior is determined by the integrated intensity process.

Jacobi and CIR processes are suitable models with positive sample paths.

Numerical simulations illustrate intervention effects on transmission dynamics.

Abstract

This paper introduces a way of modeling the epidemic transmission rate using a stochastic process of the form $(\beta_t = \varphi(t)P_t : t \ge 0)$, where the positive deterministic function $\varphi(t)$ models the impact of a public health intervention and $P_t$ describes the stochastic evolution of the infection rate in the absence of any control measures. We establish general asymptotic results for an SI model governed by $(\beta_t : t \ge 0)$, showing that the asymptotic behavior is determined by the integrated intensity process $(H_t =\int_0^t \beta_s \, ds : t \ge 0)$. We study the intrinsically bounded Jacobi process and the Cox--Ingersoll--Ross (CIR) process as models for $(P_t : t \ge 0)$; both exhibit almost surely positive sample paths. We highlight that in the case of non-intervention $(\varphi \equiv 1)$, the process $(H_t : t \ge 0)$ is considerably more analytically tractable. Finally, we present numerical simulations for both models in two different scenarios: the case of non-intervention $(\varphi(t)=1)$ and the case of a successful intervention strategy (where $\int_0^\infty \varphi(t) \, dt < \infty$) modeled using exponential decay $\varphi(t) = e^{-\alpha t}$ for both models.