[Paper Review] Large deviation principle for Poisson driven SDEs in epidemic models

Étienne Pardoux, Brice Samegni-Kepgnou|arXiv (Cornell University)|Jun 6, 2016

Stochastic processes and financial applications7 references3 citations

TL;DR

This paper establishes a Large Deviation Principle (LDP) for a class of epidemic models driven by Poisson processes, using a novel approach that simplifies prior work by Pardoux and Kratz. By introducing a regularization assumption on the domain and rate functions, the authors derive a good rate function for the empirical process, enabling the analysis of rare events in stochastic epidemic dynamics with vanishing transition rates.

ABSTRACT

We consider a general class of epidemic models obtained by applying a random time change to a collection of Poisson processes and we show the large deviation principle for such models. We generalize to a more general situation the approach followed by Dolgoaschinnykh [3] in the case of the SIR epidemic model. Thanks to an assumption which is satisfied in many examples, we simplify the recent work by P. Kratz and E. Pardoux [8].

Motivation & Objective

• To establish a Large Deviation Principle (LDP) for a general class of epidemic models driven by Poisson processes with potentially vanishing transition rates.
• To simplify the recent LDP proof by Pardoux and Kratz by introducing a geometric regularization assumption on the state space.
• To provide a rigorous large deviation framework for stochastic epidemic models where transition rates may approach zero, such as in rare event analysis.
• To derive exponential estimates for exit times from domains of attraction, linking the LDP to first-passage time asymptotics.

Proposed method

• Model the epidemic process as a jump Markov process using random time changes of Poisson processes with state-dependent rates.
• Define the empirical process $ Z^N(t) $ as a normalized sum of Poisson increments, with jump vectors $ h_j $ and rate functions $ \beta_j(z) $.
• Introduce a domain regularization via a family of mappings $ \Phi_a(z) $ that map points into the interior of the simplex $ A $, ensuring uniform distance from the boundary.
• Establish the LDP by proving upper and lower bounds on the logarithmic probabilities of path events using exponential Chebyshev-type inequalities and path approximation.
• Use the rate function $ I_T(\phi) $ defined via the action integral of the controlled ODE, derived from the Hamiltonian of the system.
• Apply the contraction principle and compactness arguments to extend the LDP to open and closed sets in the Skorokhod space $ D_{T,A} $.

Experimental results

Research questions

• RQ1 Can a Large Deviation Principle be established for Poisson-driven SDEs in epidemic models when transition rates vanish?
• RQ2 How does the introduction of a domain regularization via $ \Phi_a $ simplify the LDP proof compared to prior approaches?
• RQ3 What is the asymptotic behavior of the exit time $ \tau^N_O $ from a domain of attraction in the large population limit?
• RQ4 How does the rate function $ I_T(\phi) $ relate to the deterministic ODE limit and rare path deviations?

Key findings

• The probability measures $ \mathbb{P}^N_z $ satisfy a Large Deviation Principle with a good rate function $ I_T $, ensuring exponential decay of rare event probabilities.
• The upper bound is established via a path approximation argument using the set $ F^s_\delta $, with exponential decay rate $ \exp\{-N(s - \eta)\} $.
• The lower bound is derived by constructing a controlled path that stays within the interior of the domain and satisfies the required rate function condition.
• The exit time $ \tau^N_O $ from a domain $ O $ satisfies $ \mathbb{P}_z(\exp\{N(V_{\widetilde{\partial O}} - \eta)\} < \tau^N_O < \exp\{N(V_{\widetilde{\partial O}} + \eta)\}) \to 1 $ as $ N \to \infty $.
• The mean exit time satisfies $ \mathbb{E}_z[\tau^N_O] \asymp \exp\{N V_{\widetilde{\partial O}}\} $, with $ V_{\widetilde{\partial O}} = \inf_{y \in \widetilde{\partial O}} V_{\bar{O}}(z^*, y) $.
• The rate function $ I_T(\phi) $ is finite only for absolutely continuous paths, and the LDP holds uniformly over compact subsets of the initial state space.