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A filtering approach for statistical inference in a stochastic SIR model with an application to Covid-19 data

Katia Colaneri, Camilla Damian, Rüdiger Frey

TL;DR

A discrete-time stochastic SIR model, where the transmission rate and the number of infectious individuals are random and unobservable and this model accounts for random fluctuations in infectiousness and for non-detected infections is considered.

Abstract

In this paper, we consider a discrete-time stochastic SIR model, where the transmission rate and the true number of infectious individuals are random and unobservable. An advantage of this model is that it permits us to account for random fluctuations in infectiousness and for non-detected infections. However, a difficulty arises because statistical inference has to be done in a partial information setting. We adopt a nested particle filtering approach to estimate the reproduction rate and the model parameters. As a case study, we apply our methodology to Austrian Covid-19 infection data. Moreover, we discuss forecasts and model tests.

A filtering approach for statistical inference in a stochastic SIR model with an application to Covid-19 data

TL;DR

A discrete-time stochastic SIR model, where the transmission rate and the number of infectious individuals are random and unobservable and this model accounts for random fluctuations in infectiousness and for non-detected infections is considered.

Abstract

In this paper, we consider a discrete-time stochastic SIR model, where the transmission rate and the true number of infectious individuals are random and unobservable. An advantage of this model is that it permits us to account for random fluctuations in infectiousness and for non-detected infections. However, a difficulty arises because statistical inference has to be done in a partial information setting. We adopt a nested particle filtering approach to estimate the reproduction rate and the model parameters. As a case study, we apply our methodology to Austrian Covid-19 infection data. Moreover, we discuss forecasts and model tests.
Paper Structure (22 sections, 14 equations, 9 figures, 2 tables)

This paper contains 22 sections, 14 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model specification
  3. Step 1: an HMM for epidemics without quarantine
  4. The state variables
  5. The observations
  6. Step 2: an extension with quarantine
  7. Statistical Methodology
  8. Inner (State) Filter
  9. Parameter Estimation via Nested Particle Filtering
  10. Simulation results
  11. Empirical results
  12. Forecasting and model tests
  13. Methodology
  14. Empirical results
  15. Details on the nested particle filtering
  16. ...and 7 more sections

Figures (9)

  • Figure 1: Simulated trajectory of the number of positive tests (observation process).
  • Figure 2: True (black) and filtered (magenta) trajectory of the effective reproduction rate.
  • Figure 3: Posterior estimates for $\mu$, $\kappa$, and $\sigma$. The black line corresponds to the true value of the parameter, the two blue lines to the 5%- and 95%-quantile of the posterior distribution (as obtained by the nested particle filter) and the red line to its mean.
  • Figure 4: Mean relative errors for $\mu$ (top left panel), $\sigma^2/2\kappa$ (top right panel), $\kappa$ (bottom left panel) and $\sigma$ (bottom right panel), on a logarithmic scale.
  • Figure 5: Confirmed cases of Covid-19 in Austria (from May 1, 2020 to June 15, 2022).
  • ...and 4 more figures

Theorems & Definitions (1)

  • Remark 2.1