Abstract
Hawkes processes are a form of self-exciting process that has been used in numerous applications, including neuroscience, seismology, and terrorism. While these self-exciting processes have a simple formulation, they can model incredibly complex phenomena. Traditionally Hawkes processes are a continuous-time process, however we enable these models to be applied to a wider range of problems by considering a discrete-time variant of Hawkes processes. We illustrate this through the novel coronavirus disease (COVID-19) as a substantive case study. While alternative models, such as compartmental and growth curve models, have been widely applied to the COVID-19 epidemic, the use of discrete-time Hawkes processes allows us to gain alternative insights. This paper evaluates the capability of discrete-time Hawkes processes by modelling daily mortality counts as distinct phases in the COVID-19 outbreak. We first consider the initial stage of exponential growth and the subsequent decline as preventative measures become effective. We then explore subsequent phases with more recent data. Various countries that have been adversely affected by the epidemic are considered, namely, Brazil, China, France, Germany, India, Italy, Spain, Sweden, the United Kingdom and the United States. These countries are all unique concerning the spread of the virus and their corresponding response measures. However, we find that this simple model is useful in accurately capturing the dynamics of the process, despite hidden interactions that are not directly modelled due to their complexity, and differences both within and between countries. The utility of this model is not confined to the current COVID-19 epidemic, rather this model could explain many other complex phenomena. It is of interest to have simple models that adequately describe these complex processes with unknown dynamics. As models become more complex, a simpler representation of the process can be desirable for the sake of parsimony.
Highlights
The outbreak of the novel 2019 coronavirus disease (COVID-19) was declared a Global Health Emergency of International Concern on 30th January 2020, and pronounced a Pandemic on 11th March 2020
We propose a parsimonious model for COVID-19 deaths, namely discrete-time Hawkes processes (DTHP) [32, 33, 42], to describe the complicated dynamics of the COVID-19 epidemic
We develop a variant of the DTHP to model the distinct phases of the COVID-19 epidemic
Summary
The outbreak of the novel 2019 coronavirus disease (COVID-19) was declared a Global Health Emergency of International Concern on 30th January 2020, and pronounced a Pandemic on 11th March 2020. In the absence of a vaccine, countries implemented a range of non-pharmaceutical interventions and strategies to reduce the spread of the virus, from measures such as social distancing, mask-wearing and contact tracing, to complete city lockdowns and stay at home orders. These recommendations are guided by mathematical and statistical modelling to quantify the efficacy of these measures [2,3,4,5,6,7,8,9]. More detailed approaches such as agent-based modelling have been considered by numerous authors [24,25,26,27]
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