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Simulating the spread of COVID-19 via spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion
Alex Viguerie, Guillermo Lorenzo, Ferdinando Auricchio, Davide Baroli, Thomas J. R. Hughes, Alessia Patton, Alessandro Reali, Thomas E. Yankeelov, Alessandro Veneziani
TL;DR
The paper addresses the need for reliable tools that capture the spatio-temporal spread of COVID-19 and help geo-localize outbreaks. It develops a heterogeneous-diffusion PDE SEIRD model and tests it with finite-element simulations in Lombardy. The simulations show qualitative agreement with epidemiological data, while reopening scenarios highlight the importance of local population densities and contagion dynamics.
Problem
Reliable information about COVID-19 transmission and outbreak geography was limited, motivating models that can support spatially informed decisions and medical-resource allocation.
Method
The authors combine a compartmental SEIRD model with PDE-based heterogeneous diffusion whose parameters can represent geographic and human-related inhomogeneities.
Results
The Lombardy simulations show strong qualitative agreement with reported epidemiological data, including infected-case R2 values of 0.997 for Lombardy, 0.977 for Bergamo, 0.976 for Brescia, and 0.998 for Milan.
Takeaways & Limitations
Alternative reopening simulations suggest that strategies should account for local population densities and the specific dynamics of contagion.
Takeaways & Limitations
Infections are overpredicted relative to reported data, reflecting calibration to more reliable deceased-case data and probable underreporting caused by limited testing.
Abstract
from arXiv · showhide
We present an early version of a Susceptible-Exposed-Infected-Recovered-Deceased (SEIRD) mathematical model based on partial differential equations coupled with a heterogeneous diffusion model. The model describes the spatio-temporal spread of the COVID-19 pandemic, and aims to capture dynamics also based on human habits and geographical features. To test the model, we compare the outputs generated by a finite-element solver with measured data over the Italian region of Lombardy, which has been heavily impacted by this crisis between February and April 2020. Our results show a strong qualitative agreement between the simulated forecast of the spatio-temporal COVID-19 spread in Lombardy and epidemiological data collected at the municipality level. Additional simulations exploring alternative scenarios for the relaxation of lockdown restrictions suggest that reopening strategies should account for local population densities and the specific dynamics of the contagion. Thus, we argue that data-driven simulations of our model could ultimately inform health authorities to design effective pandemic-arresting measures and anticipate the geographical allocation of crucial medical resources.
1. Introduction
The paper proposes a continuous PDE-based SEIRD model with heterogeneous diffusion to represent COVID-19 spread across space and time. Lombardy simulations qualitatively agree with epidemiological data and show that reopening scenarios depend on local density and contagion dynamics.
- The model addresses limited information about COVID-19’s spatio-temporal spread and the need to geo-localize outbreaks for medical-resource allocation.
- The authors propose a PDE model combining a compartmental SEIRD structure with inhomogeneous diffusion terms.
- The diffusion operator represents local movement while accommodating geographical and social inhomogeneities such as mountains, rivers, and highways.
- The approach targets continuous spatio-temporal dynamics at mesoscales, including regions within Italy.
- Lombardy simulations show strong qualitative agreement with reported epidemiological data and produce contrasting reopening scenarios.
- The paper concludes that reopening strategies should consider local population densities and contagion dynamics.
2. Model
The model represents susceptible, exposed, infected, recovered, and deceased population densities with coupled PDEs. It combines transmission and progression flows with population-dependent heterogeneous diffusion and assumes recovered individuals are immune over the modeled timescale.
- The model defines densities s, e, i, r, and d for susceptible, exposed, infected, recovered, and deceased populations, with n representing the total living population.
- Coupled PDEs model births, transmission, incubation, recovery, mortality, and diffusion across the population compartments.
- Model parameters may vary across time, space, or compartments, while the Allee effect clusters outbreaks toward large population centers.
- Asymptomatic exposed patients can transmit infection and may move directly into the recovered compartment.
- The model assumes recovered patients are immune and includes no return flow from recovery to susceptibility.
- Inhomogeneous random-walk diffusion yields a second-order differential operator whose coefficient scales with population and can encode local geographic or human-related variation.
3. Numerical implementation
The numerical study discretizes Lombardy with a finite-element mesh and initializes populations from municipality-level data. Parameters are calibrated in two stages, prioritizing deceased-case agreement because infection data are less reliable.
- The Lombardy domain is discretized with an unstructured finite-element mesh containing 30,407 triangles.Backward Euler, fully implicit Picard iterations, GMRES, and Jacobi preconditioning are used for time integration and linear solves.
- Initial subpopulation conditions use Gaussian functions centered on municipalities with at least 10,000 inhabitants, weighted by population and geographic area.The initialization represents the reported 27 February 2020 outbreak pattern, including severe activity in Lodi and moderate activity in Bergamo, Brescia, and Cremona.
- Births and non-COVID-19 mortality are omitted because the simulations cover months.The specified rates are α = 0 and µ = 0.
- Parameters are first fitted with a zero-dimensional SEIRD model and then refined through recursive spatial simulations against spatio-temporal epidemiological data.Goodness of fit is assessed with R2 and RMSE.
- Calibration prioritizes quantitative agreement in the deceased compartment because infection data are considered less reliable due to incomplete testing and asymptomatic cases.
- Transmission rates for symptomatic and asymptomatic groups are set equal because their relative infectivity is uncertain, while time-varying values represent escalating lockdown restrictions.Diffusion parameters also vary across lockdown phases, and symptomatic individuals are assumed largely immobile.
4. Results
The model reproduces Lombardy’s evolving outbreak pattern, including regional differences in spread and a strong qualitative agreement with reported data. Calibration to deceased-case data yields higher predicted infections than reported infections.
- The outbreak moved north from Lodi into Milan, while Lodi and Cremona improved and avoided the explosive growth seen in Milan, Bergamo, and Brescia.
- R2 values between model forecasts and infected-case data were 0.997 for Lombardy, 0.977 for Bergamo, 0.976 for Brescia, and 0.998 for Milan.
- Lockdowns appeared to halt spread in Bergamo and Brescia, while reducing but not stopping Milan’s growth.
- The model predicted more infections than reported because parameters were calibrated using comparatively more accurate deceased-case data.
- For the deceased subgroup, the model achieved R2=0.972 and range-normalized RMSE=7.6%.
A B C
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E F G
Figure 2 compares reported and simulated cumulative infections across Bergamo, Brescia, and Milan, while reopening simulations indicate that maintaining restrictions in Milan may prevent explosive growth.
- E F G: Figure 2 compares cumulative infections from reported data with simulations for Bergamo, Brescia, and Milan.Reported data are shown as dots, simulations as dashed lines, and scaled simulations as solid lines.
- E F G: The model predicts that restrictions reduce exposure faster in Brescia and Bergamo than in Milan.
- E F G: The model is calibrated to deceased-case data, producing a larger infection forecast than reported infection data; scaled results are shown only to highlight qualitative agreement.
- E F G: Scenario B assumes greater success in limiting contact through public awareness of measures such as mask-wearing and social distancing despite increased mobility.
- E F G: Relaxing restrictions across Lombardy may cause severe rapid growth around Milan, whereas maintaining restrictions there and relaxing them elsewhere produces more favorable dynamics.
5. Discussion
The paper presents a regional PDE model for COVID-19 spread and uses it to explore reopening strategies. Simulations show qualitative agreement with observed dynamics and support locally tailored restrictions, while the model remains an early-stage framework with important omissions.
- 5. Discussion: The paper introduces a compartmental PDE model for spatio-temporal disease propagation and applies it to Lombardy’s 2020 COVID-19 outbreak.
- 5. Discussion: Alternative reopening simulations suggest that restrictions should reflect local population and contagion dynamics rather than follow a one-size-fits-all approach.
- 5. Discussion: The simulations provide a proof-of-concept with good qualitative agreement, reproducing outbreak dynamics and transmission paths across areas and time.
- 5. Discussion: The model is in an early stage and may require data assimilation, more realistic regional boundary conditions, geographically informed diffusion, and non-local effects.
- 5. Discussion: The framework does not yet include hospitalizations, intensive-care patients, age or biological-sex structures, or the socioeconomic costs of lockdowns.