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A new stochastic SIS-type modelling framework for analysing epidemic dynamics in continuous space

Apolline Louvet, Bastian Wiederhold

TL;DR

The paper develops a rigorous stochastic SIS-type framework in a continuous space by embedding SIS dynamics into a spatial Λ-Fleming-Viot (SLFV) process, yielding the EpiSLFV model. It constructs the forward process via a well-posed martingale problem, establishes a measure-valued dual (the ancestral process) and a corresponding duality that links epidemic extinction/survival to genealogical dynamics; a reproduction-number analogue $R_0(\gamma,\nu)$ is defined and shown to drive a threshold behavior with extinction when $R_0<1$ and persistence when $R_0>1$. A quenched construction is developed to enable monotonicity analyses, coupling with branching processes, and inference-friendly approaches (e.g., ABC) leveraging duality. The work further demonstrates invariances under time/space rescaling and clarifies regime equivalences (endemic/pandemic/epidemic initial conditions) through duality, offering a computationally tractable, space-continuous toolkit for inferring spatial epidemic dynamics from demographic and genetic data. Altogether, the EpiSLFV framework provides a mathematically rigorous, scalable pathway to analyze and infer spatially structured stochastic epidemics in a continuum, with clear criteria for extinction, survival, and threshold behavior grounded in $R_0(\gamma,\nu)$.

Abstract

We propose a new stochastic epidemiological model defined in a continuous space of arbitrary dimension, based on SIS dynamics implemented in a spatial $Λ$-Fleming-Viot (SLFV) process. The model can be described by as little as three parameters, and is dual to a spatial branching process with competition linked to genealogies of infected individuals. Therefore, it is a possible modelling framework to develop computationally tractable inference tools for epidemics in a continuous space using demographic and genetic data.We provide mathematical constructions of the process based on well-posed martingale problems as well as driving space-time Poisson point processes. With these devices and the duality relation in hand, we unveil some of the drivers of the transition between extinction and survival of the epidemic. In particular, we show that extinction is in large parts independent of the initial condition, and identify a strong candidate for the reproduction number R 0 of the epidemic in such a model.

A new stochastic SIS-type modelling framework for analysing epidemic dynamics in continuous space

TL;DR

The paper develops a rigorous stochastic SIS-type framework in a continuous space by embedding SIS dynamics into a spatial Λ-Fleming-Viot (SLFV) process, yielding the EpiSLFV model. It constructs the forward process via a well-posed martingale problem, establishes a measure-valued dual (the ancestral process) and a corresponding duality that links epidemic extinction/survival to genealogical dynamics; a reproduction-number analogue is defined and shown to drive a threshold behavior with extinction when and persistence when . A quenched construction is developed to enable monotonicity analyses, coupling with branching processes, and inference-friendly approaches (e.g., ABC) leveraging duality. The work further demonstrates invariances under time/space rescaling and clarifies regime equivalences (endemic/pandemic/epidemic initial conditions) through duality, offering a computationally tractable, space-continuous toolkit for inferring spatial epidemic dynamics from demographic and genetic data. Altogether, the EpiSLFV framework provides a mathematically rigorous, scalable pathway to analyze and infer spatially structured stochastic epidemics in a continuum, with clear criteria for extinction, survival, and threshold behavior grounded in .

Abstract

We propose a new stochastic epidemiological model defined in a continuous space of arbitrary dimension, based on SIS dynamics implemented in a spatial -Fleming-Viot (SLFV) process. The model can be described by as little as three parameters, and is dual to a spatial branching process with competition linked to genealogies of infected individuals. Therefore, it is a possible modelling framework to develop computationally tractable inference tools for epidemics in a continuous space using demographic and genetic data.We provide mathematical constructions of the process based on well-posed martingale problems as well as driving space-time Poisson point processes. With these devices and the duality relation in hand, we unveil some of the drivers of the transition between extinction and survival of the epidemic. In particular, we show that extinction is in large parts independent of the initial condition, and identify a strong candidate for the reproduction number R 0 of the epidemic in such a model.

Paper Structure

This paper contains 42 sections, 46 theorems, 288 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spatial -Fleming Viot processes
  3. A new epidemiological model
  4. Construction of the EpiSLFV process and applications to inference
  5. A reproduction number for the EpiSLFV process
  6. Outline
  7. Acknowledgements
  8. The -EpiSLFV process - Definition and results
  9. Definition of the -EpiSLFV process
  10. State space
  11. Test functions
  12. Martingale problem
  13. Initial condition
  14. Extinction of the epidemic
  15. Survival regimes and partial equivalences
  16. ...and 27 more sections

Key Result

Theorem 2.1

For all $M^{0} \in \mathcal{M}_{\lambda}$, the martingale problem $(\mathcal{G}^{(\gamma,\nu)},\delta_{M^{0}})$ is well-posed.

Figures (2)

  • Figure 1: Snapshot of the spatial repartition of infected individuals in a -EpiSLFV process with initial density of healthy individuals . Here, and (that is, all reproduction events have radius and impact parameter 0.1). The snapshot was taken at time $t = 0.02$.
  • Figure 2: Transition between extinction and survival of the $(\gamma,\nu)$-EpiSLFV process, as a function of the reproduction number $\mathrm{R}_{0}(\gamma,\nu)$. Simulations were ran with $\gamma = 1$, from an initial density of healthy individuals $\omega^{0} = 1 - 0.9 \mathds{1}_{\mathcal{B}(0,50)}(\cdot)$, and with $\nu(dr,du) = \delta_{4}(dr) \delta_{0.03 + 0.0003x}(du)$ for $x = 0, ..., 8$. For each value of $x$, we ran $100$ simulations of the $(\gamma,\nu)$-EpiSLFV process on a $200 \times 200$ grid with edge length $1$, and recorded the average proportion of infected individuals at time $t = 100$. The two plots show the resulting median (dark blue line) and $90$-percentiles (light blue lines) of the proportion of infected individuals in the population, on standard and logarithmic scales and as a function of the reproduction number $\mathrm{R}_{0}(\gamma,\nu)$ (approximated by replacing the volume of $\mathcal{B}(0,4)$ by the number of locations on the grid covered by events with radius $4$). As a comparison, without any successful spreading event, the proportion of infected individuals would be around $6.5 \times 10^{-45}$. The vertical dotted grey line indicates the value of $\mathrm{R}_{0}(\gamma,\nu)$ at which the transition between extinction and survival is conjectured to occur.

Theorems & Definitions (102)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 92 more