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.

Fig. 1

Spread of the COVID-19 virus spread at Lombardy, Italy. Comparison between the results obtained in [41], in which the time integration was done using the Backward-Euler method with a fixed mesh, and the present work, in which we use the BDF2 method and AMR

Fig. 2

Map of the state of GA partitioned into 159 internal counties

Fig. 3

Initial susceptible population ($people/k{m}^2$)

Fig. 4

Initial exposed population ($people/k{m}^2$)

Fig. 5

Initial infected population ($people/k{m}^2$)

Fig. 6

Diffusion (left) and contact rate (right) in time for Georgia

Fig. 7

Comparison between simulation and real data of deaths in Georgia (total)

Fig. 8

Comparison between simulation and real data of deaths in Georgia (per county)—part 1

Fig. 9

Comparison between simulation and real data of deaths in Georgia (per county)—part 2

Fig. 10

Snapshots of Georgia simulation. Top to bottom: Susceptible, Exposed, Infected, Recovery and Deceased population (people/km²), followed by the mesh at different time-steps

Fig. 11

Map of the state of RJ partitioned into 92 internal counties

Fig. 12

Initial susceptible population ($people/k{m}^2$)

Fig. 13

Initial exposed population ($people/k{m}^2$)

Fig. 14

Initial infected population ($people/k{m}^2$)

Fig. 15

Functions that multiplies the diffusion (a) and contact rate (b) in time for Rio de Janeiro

Fig. 16

Comparison between simulation and real data of deaths in Rio de Janeiro state (total)

Fig. 17

Comparison between simulation and real data of deaths in RJ (per county)—part 1

Fig. 18

Comparison between simulation and real data of deaths in RJ (per county)—part 2

Fig. 19

Snapshots of Rio de Janeiro simulation. Top to bottom: Susceptible, Exposed, Infected, Recovery and Deceased population ($people/k{m}^2$), followed by the mesh at different time-steps