Infection spreading in cell culture as a reaction-diffusion wave

¹ Laboratoire déquations aux dérivées partielles non linéaires et histoire des mathématiques, Ecole Normale Supérieure, B.P. 92, Vieux Kouba, 16050 Algiers, Algeria
² Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 199178 Saint Petersburg, Russia
³ Institute of Fundamental Technological Research Polish Academy of Sciences, Warsaw, Poland
⁴ Laboratoire de Mathématiques Raphaël Salem, Université de Rouen Normandie, CNRS UMR 6085, Avenue de l’Université, BP 12, F-76801 Saint-Étienne-du-Rouvray, France
⁵ Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
⁶ INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France
⁷ Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow 117198, Russian Federation

^* Corresponding author: volpert@math.univ-lyon1.fr

Received: 19 May 2021
Accepted: 14 February 2022

Abstract

Infection spreading in cell culture occurs due to virus replication in infected cells and its random motion in the extracellular space. Multiplicity of infection experiments in cell cultures are conventionally used for the characterization of viral infection by the number of viral plaques and the rate of their growth. We describe this process with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells, virus, and interferon. Time delay corresponds to the duration of viral replication inside infected cells. We show that infection propagates in cell culture as a reaction-diffusion wave, we determine the wave speed and prove its existence. Next, we carry out numerical simulations and identify three stages of infection progression: infection decay during time delay due to virus replication, explosive growth of viral load when infected cells begin to reproduce it, and finally, wave-like infection progression in cell culture characterized by a constant or slowly growing total viral load. The modelling results are in agreement with the experimental data for the coronavirus infection in a culture of epithelial cells and for some other experiments. The presence of interferon produced by infected cells decreases the viral load but does not change the speed of infection progression in cell culture. In the 2D modelling, the total viral load grows faster than in the 1D case due to the increase of plaque perimeter.