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

TL;DR

This paper develops a method to estimate parameters of a hidden linear birth-death process with immigration, modeling infectious disease spread from limited, periodic data, and demonstrates its effectiveness on synthetic and real typhoid data.

Contribution

It introduces a novel parameter estimation approach for a hidden birth-death process with immigration using an adapted Baum-Welch algorithm and analytic expressions for transition probabilities.

Findings

Accurate parameter estimates on synthetic data

Successful application to typhoid fever data

Enhanced understanding of disease dynamics from limited observations

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.