Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameter estimation for a hidden linear birth and death process with immigration

Ibrahim Bouzalmat, Benoîte de Saporta, Solym M. Manou-Abi

Abstract

In this paper, we use a linear birth and death process with immigration to model infectious disease propagation when contamination stems from both person-to-person contact and contact with the environment. Our aim is to estimate the parameters of the process. The main originality and difficulty comes from the observation scheme. Counts of infected population are hidden. The only data available are periodic cumulated new retired counts. Although very common in epidemiology, this observation scheme is mathematically challenging even for such a standard stochastic process. We first derive an analytic expression of the unknown parameters as functions of well-chosen discrete time transition probabilities. Second, we extend and adapt the standard Baum-Welch algorithm in order to estimate the said discrete time transition probabilities in our hidden data framework. The performance of our estimators is illustrated both on synthetic data and real data of typhoid fever in Mayotte.

Parameter estimation for a hidden linear birth and death process with immigration

Abstract

In this paper, we use a linear birth and death process with immigration to model infectious disease propagation when contamination stems from both person-to-person contact and contact with the environment. Our aim is to estimate the parameters of the process. The main originality and difficulty comes from the observation scheme. Counts of infected population are hidden. The only data available are periodic cumulated new retired counts. Although very common in epidemiology, this observation scheme is mathematically challenging even for such a standard stochastic process. We first derive an analytic expression of the unknown parameters as functions of well-chosen discrete time transition probabilities. Second, we extend and adapt the standard Baum-Welch algorithm in order to estimate the said discrete time transition probabilities in our hidden data framework. The performance of our estimators is illustrated both on synthetic data and real data of typhoid fever in Mayotte.
Paper Structure (27 sections, 7 theorems, 79 equations, 7 figures, 11 tables)

This paper contains 27 sections, 7 theorems, 79 equations, 7 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Model and observation framework
  3. Linear birth and death process with immigration
  4. Cumulated new isolated counts
  5. Estimation strategy
  6. Parameters estimation
  7. Analytical expression of the transition probabilities
  8. Hidden information framework
  9. Proofs
  10. Proof of Lemma \ref{['lem:KolmoY']}
  11. Proof of Theorem \ref{['main0']}
  12. Proof of Theorem \ref{['main1']}
  13. Proof of Theorem \ref{['th:HMMiter']}
  14. Numerical study
  15. Synthetic data
  16. ...and 12 more sections

Key Result

Lemma 1

For $t>0$ and $i,j,y\in\mathbb{N}$, one has for $j \leq i+y$, with $p_{i,(j, y)}(t) = 0$ whenever $j > i+y$ and with initial conditions $p_{i,(j, y)}(0) = \delta_{i=j} \delta_{y=0}$.

Figures (7)

  • Figure 1: Asymptotic standard deviations as a function of the time lapse $\Delta t$ for estimators $\hat{\lambda}$, $\hat{\mu}$ and $\hat{\nu}$ obtained from the maximum likelihood estimates of the transition probabilities (synthetic data, infected observed, true parameter values $\lambda=0.03$, $\mu=0.1$ and $\nu=0.01$).
  • Figure 2: Example of a sample path of the LBDI process in continuous time and the corresponding cumulated new isolated counts for a time lapse $\Delta t=1$ until a time horizon $H=5000$ with parameters $\lambda=0.03$, $\mu=0.1$ and $\nu=0.01$ (synthetic data)
  • Figure 3: Impact of the choice of the truncation parameter $N$ on the estimation with empirical $95\%$ confidence intervals ($100$ replications, synthetic data, infected hidden, true parameter values $\lambda=0.03$, $\mu=0.1$ and $\nu=0.01$)
  • Figure 4: Impact of the number of observations $n$ on the estimation on the estimation for a truncation parameter $N=5$ with empirical $95\%$ confidence intervals ($100$ replications, synthetic data, infected hidden, true parameter values $\lambda=0.03$, $\mu=0.1$ and $\nu=0.01$)
  • Figure 5: Impact of the number of observations $n$ on parameter estimations and their confidence intervals using HMM with least-squares approximation ($N=5$, $100$ replications, synthetic data, infected hidden, true parameter values $\lambda=0.03$, $\mu=0.1$ and $\nu=0.01$))
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 2
  • Theorem 4
  • proof
  • Lemma 3
  • proof
  • proof
  • ...and 2 more