Abstract
With the unfolding of the COVID-19 pandemic, mathematical modelling of epidemics has been perceived and used as a central element in understanding, predicting, and governing the pandemic event. However, soon it became clear that long-term predictions were extremely challenging to address. In addition, it is still unclear which metric shall be used for a global description of the evolution of the outbreaks. Yet a robust modelling of pandemic dynamics and a consistent choice of the transmission metric is crucial for an in-depth understanding of the macroscopic phenomenology and better-informed mitigation strategies. In this study, we propose a Markovian stochastic framework designed for describing the evolution of entropy during the COVID-19 pandemic together with the instantaneous reproductive ratio. Then, we introduce and use entropy-based metrics of global transmission to measure the impact and the temporal evolution of a pandemic event. In the formulation of the model, the temporal evolution of the outbreak is modelled by an equation governing the probability distribution that describes a nonlinear Markov process of a statistically averaged individual, leading to a clear physical interpretation. The time-dependent parameters are formulated by adaptive basis functions, leading to a parsimonious representation. In addition, we provide a full Bayesian inversion scheme for calibration together with a coherent strategy to address data unreliability. The time evolution of the entropy rate, the absolute change in the system entropy, and the instantaneous reproductive ratio are natural and transparent outputs of this framework. The framework has the appealing property of being applicable to any compartmental epidemic model. As an illustration, we apply the proposed approach to a simple modification of the susceptible–exposed–infected–removed model. Applying the model to the Hubei region, South Korean, Italian, Spanish, German, and French COVID-19 datasets, we discover significant difference in the absolute change of entropy but highly regular trends for both the entropy evolution and the instantaneous reproductive ratio.
Highlights
Coronaviruses are one of the most significant threats to human society [1,2,3,4,5,6]
We propose a compartmental stochastic model that has the following characteristics. (i) Stochastic: the model describes a statistically averaged individual by a nonlinear Markov process with compartmental epidemic states. (ii) Time-dependent: the model parameters are decomposed onto generic basis functions. (iii) Parsimonious: instead of conventional orthogonal basis functions the adaptive basis functions are adopted to achieve a representation with minimum number of basis functions. (iv) Bayesian: the time-dependent parameters are assumed to be random and are calibrated by full Bayesian inversion
We have proposed a stochastic compartmental modelling framework of epidemics equipped with entropy-based metrics to measure both the impact and the evolution of a pandemic event
Summary
Coronaviruses are one of the most significant threats to human society [1,2,3,4,5,6]. They can have different data acquisition schemes (from a simple frequentist analysis of the single parameters to a complete Bayesian inversion scheme) They can macroscopically describe the pandemic evolution of a given location (top-down approach) or include a spatial topological description (including mobility) and/or a different degree of spreading among individuals by including adjacent matrices (bottom-up approach). The dynamics of entropy in the COVID-19 outbreaks developed by measuring the reproductive ratio (constant or time varying) and estimating the effectiveness of containment measures This metric allows for a comparison between different regions but does not provide a quantitative measure of the impact of the spread. We conclude the study by identifying the limitations, conclusions, and future research directions
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