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Duality for some models of epidemic spreading

Chiara Franceschini, Ellen Saada, Gunter M. Schütz, Sonia Velasco

TL;DR

This work develops and applies duality techniques to three non-conservative, one-dimensional epidemic-spreading models: the DCP, the GDCP, and the SIR process. It shows that open boundary reservoirs qualitatively alter duality by eliminating self-duality and creating absorbing duals, while preserving a factorized duality structure in the DCP and GDCP and revealing a nonlocal, cluster-type duality for the SIR model. Correlation functions in the open DCP and GDCP are linked to dual absorption events, yielding explicit one-point results in special GDCP cases and time-dependent ODE systems for one-point values that recover open-SSEP structures under suitable parameter mappings. For the SIR model, duality manifests as a two-layer biased random walk with a trap, enabling explicit expressions for time evolution of cluster functions and extending results to non-translation-invariant initial conditions. Overall, the paper broadens the scope of duality methods in non-conservative particle systems and provides concrete, computable formulas for stationary and dynamic correlation structures with open boundaries.

Abstract

We examine the role of boundaries and the structure of nontrivial duality functions for three non conservative interacting particle systems in one dimension that model epidemic spreading: (i) the diffusive contact process (DCP), (ii) a model that we introduce and call generalized diffusive contact process (GDCP), both in finite volume in contact with boundary reservoirs, i.e., with open boundaries, and (iii) the susceptible-infectious-recovered (SIR) model on $\Z$. We establish duality relations for each system through an analytical approach. It turns out that with open boundaries self-duality breaks down and qualitatively different properties compared to closed boundaries (i.e., finite volume without reservoirs) arise: both the DCP and GDCP are ergodic but no longer absorbing, while the respective dual processes are absorbing but not ergodic. We provide expressions for the stationary correlation functions in terms of the dual absorption probabilities. We perform explicit computations for a small sized DCP, and for arbitrary size in a particular setting of the GDCP. The duality function is factorized for the DCP and GDCP, contrary to the SIR model for which the duality relation is nonlocal and yields an explicit expression of the time evolution of some specific correlation functions, describing the time decay of the sizes of clusters of susceptible individuals.

Duality for some models of epidemic spreading

TL;DR

This work develops and applies duality techniques to three non-conservative, one-dimensional epidemic-spreading models: the DCP, the GDCP, and the SIR process. It shows that open boundary reservoirs qualitatively alter duality by eliminating self-duality and creating absorbing duals, while preserving a factorized duality structure in the DCP and GDCP and revealing a nonlocal, cluster-type duality for the SIR model. Correlation functions in the open DCP and GDCP are linked to dual absorption events, yielding explicit one-point results in special GDCP cases and time-dependent ODE systems for one-point values that recover open-SSEP structures under suitable parameter mappings. For the SIR model, duality manifests as a two-layer biased random walk with a trap, enabling explicit expressions for time evolution of cluster functions and extending results to non-translation-invariant initial conditions. Overall, the paper broadens the scope of duality methods in non-conservative particle systems and provides concrete, computable formulas for stationary and dynamic correlation structures with open boundaries.

Abstract

We examine the role of boundaries and the structure of nontrivial duality functions for three non conservative interacting particle systems in one dimension that model epidemic spreading: (i) the diffusive contact process (DCP), (ii) a model that we introduce and call generalized diffusive contact process (GDCP), both in finite volume in contact with boundary reservoirs, i.e., with open boundaries, and (iii) the susceptible-infectious-recovered (SIR) model on . We establish duality relations for each system through an analytical approach. It turns out that with open boundaries self-duality breaks down and qualitatively different properties compared to closed boundaries (i.e., finite volume without reservoirs) arise: both the DCP and GDCP are ergodic but no longer absorbing, while the respective dual processes are absorbing but not ergodic. We provide expressions for the stationary correlation functions in terms of the dual absorption probabilities. We perform explicit computations for a small sized DCP, and for arbitrary size in a particular setting of the GDCP. The duality function is factorized for the DCP and GDCP, contrary to the SIR model for which the duality relation is nonlocal and yields an explicit expression of the time evolution of some specific correlation functions, describing the time decay of the sizes of clusters of susceptible individuals.
Paper Structure (31 sections, 15 theorems, 180 equations, 4 figures)

This paper contains 31 sections, 15 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The diffusive contact process (DCP)
  3. The model
  4. Duality for the DCP
  5. Correlation functions via duality
  6. One-point function
  7. Higher order correlation functions
  8. Fast stirring limit
  9. Proofs for Section \ref{['dcp section']}
  10. Proofs for Section \ref{['subsec:dcp-duality']}
  11. Proofs for Section \ref{['subsec:corfu']}
  12. Generalized diffusive contact process (GDCP)
  13. The model
  14. Duality results for the GDCP
  15. Application of duality: computing correlations
  16. ...and 16 more sections

Key Result

Theorem 1

The open DCP $(\eta_t)_{t\geq 0}$ with generator generator_opendiffusivecontact is dual to a purely absorbing contact process $(\xi_t)_{t\geq 0}$ with duality function $D:\Omega_N \times \Omega_N^{\rm dual} \rightarrow \mathbb{R}$, given by where the bulk duality function is: with $A(\xi) = \{y\in \{1,...,N\},~\xi_y=1 \}$. The dual generator is given by where $\mathcal{L}^{\rm CP}$ is the gener

Figures (4)

  • Figure 1: Contact + exclusion process on $\Lambda_N$ with reservoirs.
  • Figure 2: Dual diffusive contact process with sinks at sites $0$ and $N+1$.
  • Figure 3: Ordering of configurations
  • Figure 4: Dual dynamics for the SIR model

Theorems & Definitions (37)

  • Definition 1: Definition 3.1 in Chapter II of IPS
  • Remark 1: Countable state space
  • Theorem 1
  • Remark 2
  • Proposition 1
  • Lemma 1
  • Remark 3
  • Proposition 2
  • Proposition 3
  • proof : Proof of Theorem \ref{['Dual']}
  • ...and 27 more