School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
Australian Research Council, Centre of Excellence for Mathematical and Statistical Frontiers, Brisbane, Australia.
PLoS One. 2021 Apr 9;16(4):e0250015. doi: 10.1371/journal.pone.0250015. eCollection 2021.
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.
Hawkes 过程是一种自激发过程，已在许多应用中得到应用，包括神经科学、地震学和恐怖主义。虽然这些自激发过程具有简单的公式，但它们可以模拟极其复杂的现象。传统上，Hawkes 过程是一个连续时间过程，但是我们通过考虑 Hawkes 过程的离散时间变体，使这些模型能够应用于更广泛的问题。我们通过新颖的冠状病毒病（COVID-19）作为实质性案例研究来说明这一点。虽然替代模型，如隔间和增长曲线模型，已广泛应用于 COVID-19 疫情，但离散时间 Hawkes 过程的使用允许我们获得替代见解。本文通过将 COVID-19 爆发的每日死亡率计数建模为不同阶段来评估离散时间 Hawkes 过程的能力。我们首先考虑指数增长的初始阶段，然后考虑随着预防措施生效而随后的下降阶段。然后，我们使用最近的数据探索后续阶段。考虑了受疫情严重影响的各个国家，即巴西、中国、法国、德国、印度、意大利、西班牙、瑞典、英国和美国。这些国家在病毒传播及其相应的应对措施方面都具有独特性。然而，我们发现，尽管由于其复杂性以及国家内部和国家之间的差异而未直接建模隐藏的相互作用，但该简单模型对于准确捕捉过程动态非常有用。该模型的用途不仅限于当前的 COVID-19 疫情，而是可以解释许多其他复杂现象。拥有简单模型来充分描述这些具有未知动态的复杂过程是很有意义的。随着模型变得越来越复杂，为了简洁起见，可能需要更简单的过程表示。
