[Paper Review] Large Deviation Principle for Reflected Poisson driven SDEs in Epidemic Models

Étienne Pardoux, Brice Samegni-Kepgnou|arXiv (Cornell University)|Aug 14, 2018

Mathematical and Theoretical Epidemiology and Ecology Models9 references3 citations

TL;DR

This paper establishes a large deviation principle (LDP) for reflected Poisson-driven stochastic differential equations (SDEs) in epidemic models, where the process is constrained to remain within a compact domain O. The authors extend prior LDP results for non-reflected SDEs to handle boundary reflection, particularly in cases where the deterministic limit (law of large numbers) exhibits a characteristic boundary (i.e., vector field tangent to the boundary), which is critical for studying exit from basins of attraction of endemic equilibria.

ABSTRACT

We establish a large deviation principle for a reflected Poisson driven SDE. Our motivation is to study in a forthcoming paper the problem of exit of such a process from the basin of attraction of a locally stable equilibrium associated with its law of large numbers. Two examples are described in which we verify the assumptions that we make to establish the large deviation principle.

Motivation & Objective

• To develop a large deviation principle for reflected Poisson-driven SDEs in epidemic models where the deterministic limit has a characteristic boundary (i.e., drift tangent to the boundary).
• To enable the study of the most probable exit paths of stochastic epidemic models from the basin of attraction of a locally stable endemic equilibrium.
• To verify the assumptions of the LDP in two concrete epidemic model examples, demonstrating applicability to real-world disease dynamics.
• To extend the LDP framework from non-reflected to reflected SDEs, particularly addressing the challenge of vanishing jump rates on the boundary.

Proposed method

• Define a reflected SDE via a Skorokhod-type reflection mechanism that ensures the process remains within the compact domain O at all times.
• Use a time-changed Poisson process formulation with rate functions βj that are Lipschitz continuous and may vanish on parts of the boundary ∂O.
• Construct the reflected process Z̃N(t) by modifying the jump dynamics to prevent excursions outside O, using indicator functions to block jumps that would exit the domain.
• Establish the law of large numbers (LLN) limit for the reflected process, showing convergence to a deterministic ODE with drift b(z) = ∑βj(z)hj.
• Prove the LDP using a variational approach, adapting the proof structure from non-reflected SDEs but modifying arguments to handle boundary reflection and vanishing rates.
• Verify the key assumptions (e.g., uniform inward pointing condition, uniform positive distance from boundary along rays from an interior point) in two epidemic model examples, including an SIR-type model with saturation.

Experimental results

Research questions

• RQ1 Does a large deviation principle hold for reflected Poisson-driven SDEs in epidemic models where the deterministic limit has a characteristic boundary?
• RQ2 How does the presence of boundary reflection affect the rate function and the most probable exit path from a basin of attraction?
• RQ3 Can the assumptions required for the LDP be verified in realistic epidemic models, such as SIR or SIS-type models with saturation effects?
• RQ4 What is the role of the vector field's tangency to the boundary in determining the exit behavior of the stochastic process?

Key findings

• The reflected Poisson-driven SDE satisfies a large deviation principle with the same good rate function as the non-reflected version, under appropriate assumptions on the domain and rate functions.
• The proof of the LDP is adapted from the non-reflected case, with modifications to handle the reflection mechanism and the vanishing of jump rates on the boundary.
• The assumptions of the LDP are verified in two epidemic models: an SIR-type model with saturation and a two-compartment model with a characteristic boundary.
• The existence of an interior point z₀ such that rays from z₀ to ∂O do not intersect ∂O again ensures uniform control over the distance to the boundary, which is crucial for the proof.
• For the SIR-type model, the vector field on the boundary ∂O points in the northwest direction (i.e., g₁ < 0, g₂ > 0), and the sum g₁ + g₂ > 0 on the relevant part of the boundary, ensuring the trajectory remains within the domain.
• The proof establishes that the trajectory from the boundary point B to the endemic equilibrium z̃ lies strictly below the line z₁ + z₂ = 1, which supports the geometric assumptions needed for the LDP.