Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Duvan Cataño, Raul Morán, Leon A. Valencia

Abstract

This paper introduces a way of modeling the epidemic transmission rate using a stochastic process of the form $(β_t = \varphi(t)P_t : t \ge 0)$, 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 $(β_t : t \ge 0)$, showing that the asymptotic behavior is determined by the integrated intensity process $(H_t =\int_0^t β_s \, ds : t \ge 0)$. 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 $\varphi(t) = e^{-αt}$ for both models.

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

Abstract

This paper introduces a way of modeling the epidemic transmission rate using a stochastic process of the form , where the positive deterministic function models the impact of a public health intervention and 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 , showing that the asymptotic behavior is determined by the integrated intensity process . We study the intrinsically bounded Jacobi process and the Cox--Ingersoll--Ross (CIR) process as models for ; both exhibit almost surely positive sample paths. We highlight that in the case of non-intervention , the process is considerably more analytically tractable. Finally, we present numerical simulations for both models in two different scenarios: the case of non-intervention and the case of a successful intervention strategy (where ) modeled using exponential decay for both models.

Paper Structure

This paper contains 23 sections, 9 theorems, 28 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. General Model Formulation and Asymptotic Results
  3. Model I: A Bounded Rate via the Jacobi Process
  4. Model II: A Non-Negative Rate via the CIR Process
  5. Random Time to Reach Infection Thresholds: The Non-Intervention Case
  6. Numerical studies
  7. The Jacobi Case: Simulations and Discussion
  8. Baseline Scenario: Non-intervention ($\varphi \equiv 1$).
  9. Scenario with Exponential Intervention.
  10. The CIR Case: Simulations and Discussion
  11. Baseline Scenario: Non-intervention ($\varphi \equiv 1$).
  12. Scenario with exponential Intervention.
  13. Simulation of Threshold Times Without Intervention Under the CIR Process
  14. Numerical Comparison of the Jacobi and CIR Processes for $P_t$
  15. Experimental Setup and Calibration
  16. ...and 8 more sections

Key Result

Proposition 2.1

Let $(\beta_t)_{t \ge 0}$ be a stochastic process with a.s. continuous sample paths. Equation eq:si-random admits a unique global solution given by $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Stochastic trajectories of the Jacobi transmission intensity $P^J_t$ (top) and the corresponding infected fraction $1-S_t$ (bottom) for the baseline case $\varphi \equiv 1$. The trajectories illustrate the inevitable saturation of the epidemic ($1-S_t \to 1$) as the cumulative intensity $H_t$ diverges.
  • Figure 2: System dynamics under exponential intervention $\varphi(t) = e^{-0.2t}$. The top panel shows the effective transmission rate $\beta_t = \varphi(t)P_t^J$ decaying towards zero, while the bottom panel illustrates how the infected fraction $1-S_t$ stabilizes at a value strictly less than one, preserving a susceptible fraction $S_\infty > 0$.
  • Figure 3: Temporal evolution of the CIR transmission rate $P_t^{C}$ (top) and the resulting infected fraction $1-S_t$ (bottom) for $\varphi \equiv 1$. As $t$ increases, the cumulative pressure $H_t$ leads the system to a state where $1-S_t \approx 1$ almost surely.
  • Figure 4: The epidemic dynamics under the intervention $\varphi(t) = e^{-0.2t}$. The top panel displays the effect of the intervention on the transmission rate, while the bottom panel shows the resulting impact on the fraction of infected individuals.
  • Figure 5: Sensitivity of the risk bound $f(\lambda)$ and corresponding probability bounds across different regimes of $M$. The legend displays the approximate value of the tightest Chernoff bound $f(\lambda^*)$, which acts as an upper bound for the probability of reaching the infection threshold before time $t=1$.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Proposition 2.1
  • Theorem 2.2: Asymptotic State under Successful Intervention
  • proof
  • Theorem 2.3: Asymptotic State without Intervention, $\varphi \equiv 1$
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Proposition 2.6
  • proof : Proof Sketch
  • ...and 5 more