Bayesian point processes models with applications in the COVID-19 pandemic

Abstract

A point process is a set of points randomly located in a space, such as time or abstract spaces. Point process models have found numerous applications in epidemiology, ecology, geophysics, social networks and many other areas. The Poisson process is the most widely known point process. Poisson intensity estimation is a vital task in various applications including medical imaging, astrophysics and network traffic analysis. A Bayesian Additive Regression Trees (BART) scheme for estimating the intensity of inhomogeneous Poisson processes is introduced. The new approach enables full posterior inference of the intensity in a non-parametric regression setting. The performance of the novel scheme is demonstrated through simulation studies on synthetic and real datasets up to five dimensions, and the new scheme is compared with alternative approaches. A drawback of the proposed algorithm is its axis-alignment nature. We discuss this problem and suggest alternative approaches to remedy the drawback. The novel coronavirus disease (COVID-19) has been declared a Global Health Emergency of International Concern with over 557 million cases and 6.36 million deaths as of 3 August 2022 according to the World Health Organization. Understanding the spread of COVID-19 has been the subject of numerous studies, highlighting the significance of reliable epidemic models. We introduce a novel epidemic model using a latent Hawkes process with temporal covariates for modelling the infections. Unlike other Hawkes models, we model the reported cases via a probability distribution driven by the underlying Hawkes process. Modelling the infections via a Hawkes process allows us to estimate by whom an infected individual was infected. We propose a Kernel Density Particle Filter (KDPF) for inference of both latent cases and reproduction number and for predicting new cases in the near future. The computational effort is proportional to the number of infections making it possible to use particle filter-type algorithms, such as the KDPF. We demonstrate the performance of the proposed algorithm on synthetic data sets and COVID-19 reported cases in various local authorities in the UK, and benchmark our model to alternative approaches.

We extend the unstructured homogeneously mixing epidemic model considering a finite population stratified by age bands. We model the actual unobserved infections using a latent marked Hawkes process and the reported aggregated infections as random quantities driven by the underlying Hawkes process. We apply a Kernel Density Particle Filter (KDPF) to infer the marked counting process, the instantaneous reproduction number for each age group and forecast the epidemic’s future trajectory in the near future. We demonstrate the performance of the proposed inference algorithm on synthetic data sets and COVID-19 reported cases in various local authorities in the UK. Taking into account the individual heterogeneity in age provides a real-time measurement of interventions and behavioural changes.