Assessing the Spatio-temporal Spread of COVID-19 via Compartmental Models with Diffusion in Italy, USA, and Brazil

Abstract

The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of an epidemic in space. Such research on PDE models within a Susceptible, Infected, Exposed, Recovered, and Deceased framework has led to promising results in reproducing COVID-19 contagion dynamics. In this paper, we assess the robustness of this modeling framework by considering different geometries over more extended periods than in other similar studies. We first validate our code by reproducing previously shown results for Lombardy, Italy. We then focus on the U.S. state of Georgia and on the Brazilian state of Rio de Janeiro, one of the most impacted areas in the world. Our results show good agreement with real-world epidemiological data in both time and space for all regions across major areas and across three different continents, suggesting that the modeling approach is both valid and robust.

Modeling Effects of Spatial Heterogeneities and Layered Exposure Interventions on the Spread of COVID-19 across New Jersey

“…It should be pointed out that this trend is consistent with a mechanistic nonlinear scaling function (dashed line in Figure 9) that was derived from a spatial contact model which consider contacts of individuals within a population and is analogous to models that are widely used in kinetic theory [82]. Our estimation for the county-level transmission rates provides evidence on the significance of distinguishing density-dependent and frequency-dependent conditions, for better assessing the spatiotemporal patterns of the COVID-19 dynamic with partial differential equation models [83]. The transmission rate is also related to geographic location.…”

“…First, this version of the stochastic SEIR model does not account explicitly for heterogeneities due to age disparities in susceptibility and transmission patterns [95]; incorporating age stratification should further reveal patterns in transmission/fatality rates (Figures 9 and S10) with respect to population density and relevant demographic factors. Though we have shown the benefits of prediction at finer spatial scales (Figure 10), in the present work, transmission of COVID-19 was modeled independently for each county without considering the effects of inter-county interactions; incorporating relevant information (such as population mobility) could have further improved the assessments of spatial transmission [64,83,96,97]; however, it should be recognized that the data-driven parameter estimation and optimization implicitly captures the effects of inter-county differences such as "differentials" in community levels. Second, the model calibration process considers key elements of transmission dynamics for different intervention measures and timing, while for simplicity, it assumes fixed transmission rates after the implementation of those mitigation measures.…”

Delay differential equations for the spatially-resolved simulation of epidemics with specific application to COVID-19

“…The COVID-19 PDE model presented in [26] and further analyzed and extended in [3,27,15,11,12] reads:…”

“…We briefly make a few additional remarks regarding this model. The first is that this model operates on the principle of mass-action, with the contact terms β i,e non-normalized and hence dependent on local population densities, reflected in their units of 1/(Time•Persons) [27,11,22,21,12]. The spatial dependence of contagion is further augmented by the addition of the Allee term A, which accounts for the tendency of COVID-19 cases to cluster in areas where n >> A.…”

“…In contrast to these approaches, in [26,27,15,11,3,12], the authors instead modeled the spatial diffusion of the disease using partial differential equation (PDE) models. The implemented models followed the compartmental framework, but also incorporated nonlinear heterogeneous diffusion terms, giving a reaction-diffusion system of equations.…”