Reaction-diffusion spatial modeling of COVID-19: Greece and Andalusia as case examples

Abstract

We examine the spatial modeling of the outbreak of COVID-19 in two regions: the autonomous community of Andalusia in Spain and the mainland of Greece. We start with a zero-dimensional (0D; ordinary-differentialequation-level) compartmental epidemiological model consisting of Susceptible, Exposed, Asymptomatic, (symptomatically) Infected, Hospitalized, Recovered, and deceased populations (SEAIHR model). We emphasize the importance of the viral latent period (reflected in the exposed population) and the key role of an asymptomatic population. We optimize model parameters for both regions by comparing predictions to the cumulative number of infected and total number of deaths, the reported data we found to be most reliable, via minimizing the 2 norm of the difference between predictions and observed data. We consider the sensitivity of model predictions on reasonable variations of model parameters and initial conditions, and we address issues of parameter identifiability. We model both the prequarantine and postquarantine evolution of the epidemic by a time-dependent change of the viral transmission rates that arises in response to containment measures. Subsequently, a spatially distributed version of the 0D model in the form of reaction-diffusion equations is developed. We consider that, after an initial localized seeding of the infection, its spread is governed by the diffusion (and 0D model "reactions") of the asymptomatic and symptomatically infected populations, which decrease with the imposed restrictive measures. We inserted the maps of the two regions, and we imported population-density data into the finite-element software package COMSOL Multiphysics , which was subsequently used to numerically solve the model partial differential equations. Upon discussing how to adapt the 0D model to this spatial setting, we show that these models bear significant potential towards capturing both the well-mixed, zero-dimensional description and the spatial expansion of the pandemic in the two regions. Veins of potential refinement of the model assumptions towards future work are also explored.

Modelling of spatial infection spread through heterogeneous population: from lattice to PDE models

“…It is also relevant to mention that both for reasons of concreteness, but also for practical ones related to the identifiability of the model [26] (which does not escape us as a central issue and a consistent source of concern about complex models), we opt within the present seed study to focus on the prototypical SIR-type model. Generalizations to more detailed models with a higher number of compartments will be evident, including also in connection to earlier work of some of the authors [15,27].…”

“…Such models have been used for a diverse host of countries including China [20,21] and Spain [22,23], while a comparison of different models developed, e.g., for the US can be found in the so-called COVID-19 Forecast Hub 1 . On the other hand, there exist also models that develop a PDE perspective such as [24,25], in addition to earlier work by the present authors such as [15,16] (see also references within these works).…”

“…At the heart of many epidemiology models and in the frame of this study as well, are the so-called compartmental models, consisting of various classes of individuals and their interactions. Among the many possibilities that have arisen not only in the context of COVID-19, but also earlier, we note the formulation of ODE models [6][7][8][9], statistical models [7,10], stochastic models [11], agent-based models [12,13], spatial network models [10,14] and partial differential equation (PDE) models [15,16]; see also [12,17] for reviews. Some of these works turn out to have a very deep influence on public thinking and policy [8,13].…”

Backcasting COVID-19: A Physics-Informed Estimate for Early Case Incidence

“…Assuming the accuracy of fatalities (or hospitalizations) the aim is to backcast the incidences during the first wave, i.e., during the early stages of the pandemic, based on the trends between cases and fatalities. The relatively small number of parameters and hyperparameters sets the current methodology apart from other ODE or PDE based approaches, for which a significant effort is required to resolve parameter identifiability issues [9,24].…”

“…While we may use the terms "parameter" and "hyperparameter" interchangeably in this text, we note that their choice does not raise identifiability issues, since it is indeed possible to recover the same accuracy of a resulting estimator with different choices of hyperparameters, and parameter uniqueness is not required by the nature of our problem (the pandemic). This is in contrast to closed form ODE/PDE models in which the values of corresponding parameters are fundamentally significant in interpreting results of such a model [24,31,51].…”