Contact rate epidemic control of COVID-19: An equilibrium view

Romuald Elie, Emma Hubert, Gabriel Turinici

Research output: Contribution to journalArticlepeer-review

40 Scopus citations


We consider the control of the COVID-19 pandemic through a standard SIR compartmental model. This control is induced by the aggregation of individuals' decisions to limit their social interactions: when the epidemic is ongoing, an individual can diminish his/her contact rate in order to avoid getting infected, but this effort comes at a social cost. If each individual lowers his/her contact rate, the epidemic vanishes faster, but the effort cost may be high. A Mean Field Nash equilibrium at the population level is formed, resulting in a lower effective transmission rate of the virus. We prove theoretically that equilibrium exists and compute it numerically. However, this equilibrium selects a sub-optimal solution in comparison to the societal optimum (a centralized decision respected fully by all individuals), meaning that the cost of anarchy is strictly positive. We provide numerical examples and a sensitivity analysis, as well as an extension to a SEIR compartmental model to account for the relatively long latent phase of the COVID-19 disease. In all the scenario considered, the divergence between the individual and societal strategies happens both before the peak of the epidemic, due to individuals' fears, and after, when a significant propagation is still underway.

Original languageEnglish (US)
Article number2020022
JournalMathematical Modelling of Natural Phenomena
StatePublished - 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Applied Mathematics


  • COVID-19
  • Epidemic control
  • Mean Field Games
  • SARS-CoV-2
  • SEIR model


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