FLAT LIKELIHOODS: SIR-POISSON MODEL CASE

Abstract

Systems of differential equations are used as the basis to define mathematical structures for moments, like the mean and variance, of random variables probability distributions.  Nevertheless, the integration of a deterministic model and a probabilistic one, with the aim of describing a random phenomenon, and take advantage of the observed data for making inferences on certain population dynamic characteristics, can lead to parameter identifiability problems. Furthermore, approaches to deal with those problems are usually inappropriate. In this paper, the shape of the likelihood function of a SIR-Poisson model is used to describe the relationship between flat likelihoods and the identifiability parameter problem.   In particular, we show how a flattened shape for the profile likelihood of the basic reproductive number R0, arises as the observed sample (over time) becomes smaller, causing ambiguity regarding the shape of the average model behavior.  We conducted some simulation studies to analyze the flatness severity of the R0 likelihood, and the coverage frequency of the likelihood-confidence regions for the model parameters. Finally, we describe some approaches to deal the practical identifiability problem, showing the impact those can have on inferences.  We believe this work can help to raise awareness on the way statistical inferences can be affected by a priori parameter assumptions and the underlying relationship between them, as well as by model reparameterizations and incorrect model assumptions.