Adaptive mesh refinement and coarsening for diffusion–reaction epidemiological models

Abstract

The outbreak of COVID-19 in 2020 has led to a surge in the interest in the mathematical modeling of infectious diseases. Disease transmission may be modeled as compartmental models, in which the population under study is divided into compartments and has assumptions about the nature and time rate of transfer from one compartment to another. Usually, they are composed of a system of ordinary differential equations in time. A class of such models considers the Susceptible, Exposed, Infected, Recovered, and Deceased populations, the SEIRD model. However, these models do not always account for the movement of individuals from one region to another. In this work, we extend the formulation of SEIRD compartmental models to diffusion–reaction systems of partial differential equations to capture the continuous spatio-temporal dynamics of COVID-19. Since the virus spread is not only through diffusion, we introduce a source term to the equation system, representing exposed people who return from travel. We also add the possibility of anisotropic non-homogeneous diffusion. We implement the whole model in libMesh, an open finite element library that provides a framework for multiphysics, considering adaptive mesh refinement and coarsening. Therefore, the model can represent several spatial scales, adapting the resolution to the disease dynamics. We verify our model with standard SEIRD models and show several examples highlighting the present model’s new capabilities.

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

“…To adequately define a model of COVID-19 contagion, we make the following assumptions, as discussed in [ 19 , 42 ]: We consider only mortality due to COVID-19; We do not consider new births; We assume that a portion of exposed individuals will never develop symptoms (asymptomatic cases), and hence will move directly from the exposed compartment to the recovered compartment; We assume that both pre-symptomatic and asymptomatic (the exposed compartment), as well as symptomatic (the infected compartment) individuals may spread the disease; There is a latency period after exposure and before the development of symptoms; Movement is proportional to the population size; i.e., we expect greater diffusion to occur in heavily populated regions; There is no movement among the deceased population. Under the above assumptions, the frequency-dependent system of equations reads: where and denote the contact rates between symptomatic and susceptible individuals and asymptomatic and susceptible individuals, respectively (units days ), denotes the incubation period (units days ), corresponds to the asymptomatic recovery rate (units days ), the symptomatic recovery rate (units days ), represents the mortality rate (units days ), and , , , are the diffusion parameters of the different population groups as denoted by the sub-scripted letters (units km persons days ).…”

“…We additionally make use of an adaptive mesh refinement and coarsening strategy (AMR/C), allowing us to resolve multiple scales. One may find more details about the adopted methods in [ 19 , 42 ].…”

Dynamic mode decomposition in adaptive mesh refinement and coarsening simulations

“…The equations are considered using a Galerkin finite element discretization and are solved using libMesh [ 42 ], a high-performance C++ finite element library. The error estimators, refinement/coarsening strategies built-in on libMesh can be seen on [ 34 , 63 ]. Also, the -projection algorithm is embedded in libMesh.…”

“…Also, the -projection algorithm is embedded in libMesh. We explore the results in one and two spatial dimensions, where the 1D case is a hypothetical example, and the 2D case describes the COVID-19 evolution in the Lombardy region in Italy [ 34 , 63 , 64 ]. The simulation obtains the results for 44 and 60 days for the 1D and 2D cases, respectively.…”

“…Disease transmission may be modeled as compartmental models , in which the population under study is divided into compartments and has assumptions about the nature and time rate of transfer from one compartment to another [ 11 ]. Here, we work with a spatio-temporal SEIRD model, presented in [ 34 , 63 , 64 ], and given by, where , , , , and denote the densities of the susceptible , exposed , infected , recovered , and deceased populations, respectively. The sum of all the compartments with the exception of is represented by which is the total living population.…”