[Paper Review] A reaction-diffusion system to better comprehend the unlockdown: Application of SEIR-type model with diffusion to the spatial spread of COVID-19 in France

Youcef Mammeri|arXiv (Cornell University)|May 6, 2020

COVID-19 epidemiological studies24 references47 citations

TL;DR

This paper develops a reaction-diffusion SEIR-type model with asymptomatic and moving populations to study spatial spread of COVID-19 in France, estimates parameters from data, computes R0, and analyzes unlockdown scenarios.

ABSTRACT

A reaction-diffusion model was developed describing the spread of the COVID-19 virus considering the mean daily movement of susceptible, exposed and asymptomatic individuals. The model was calibrated using data on the confirmed infection and death from France as well as their initial spatial distribution. First, the system of partial differential equations is studied, then the basic reproduction number, R0 is derived. Second, numerical simulations, based on a combination of level-set and finite differences, shown the spatial spread of COVID-19 from March 16 to June 16. Finally, scenarios of unlockdown are compared according to variation of distancing, or partially spatial lockdown.

Motivation & Objective

• Motivate understanding of COVID-19 spatial spread and the impact of mobility and distancing.
• Develop a coupled PDE/ODE SEIR-type framework incorporating moving susceptible, exposed, and asymptomatic individuals.
• Calibrate the model to French confirmed cases and deaths data and derive the basic reproduction number R0.
• Analyze spatial propagation and evaluate unlockdown strategies through numerical simulations.

Proposed method

• Formulate a six-compartment SEI_aI_sUR model with diffusion for S, E, and I_a compartments.
• Incorporate time-varying contact rate omega(t) and diffusion d(t) to model lockdown/unlockdown effects.
• Derive R0 from the next-generation matrix without diffusion and establish conditions for exponential growth.
• Prove global well-posedness and analyze asymptotic disease-free behavior under suitable parameter regimes.
• Use level-set and finite-difference methods with Runge-Kutta time integration for numerical simulations.
• Calibrate six parameters (rho, beta_e, beta_s, beta_a, p, mu) using nonlinear least squares via Approximate Bayesian Computation with a quasi-Newton step.

Experimental results

Research questions

• RQ1 How does the spatial diffusion of susceptible, exposed, and asymptomatic individuals influence the spread of COVID-19 in France?
• RQ2 What is the impact of lockdown and unlockdown policies on the basic reproduction number and spatial distribution of cases?
• RQ3 Can the model reproduce observed infection and death trajectories and provide a map-based unlockdown strategy?
• RQ4 What parameter values best fit the data and what do these imply about transmission from exposed and asymptomatic individuals?

Key findings

• The model yields a basic reproduction number R0 = 3.425257 under calibration.
• Lockdown reduces the effective reproduction number from about 3.42 to 0.38 in simulations.
• A distancing adherence of at least 63% can control the number of symptomatic infections and keep Reff below 1 in the modeled regions.
• Unlockdown scenarios show spatial heterogeneity, with eastern regions remaining more affected under certain strategies.
• Without interventions, the model predicts widespread and higher infection levels across France, highlighting the role of mobility and regional controls.
• The effective reproduction number Reff(x,t) depends on S(x,t)/N(x,t) and contact rate omega(t), and its mean over the domain tracks the epidemic’s control.