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A SIR Model on a Refining Spatial Grid I: Law of Large Numbers

Abstract

We study in this paper a compartmental SIR model for a population distributed in a bounded domain D of \(\mathbb {R}^d\), \(\hbox {d}= 1\), 2 or 3. We describe a spatial model for the spread of a disease on a grid of D. We prove two laws of large numbers. On the one hand, we prove that the stochastic model converges to the corresponding deterministic patch model as the size of the population tends to infinity. On the other hand, by letting both the size of the population tend to infinity and the mesh of the grid go to zero, we obtain a law of large numbers in the supremum norm, where the limit is a diffusion SIR model in D.

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Acknowledgements

The authors are deeply indebted to the referee for a careful reading and several suggestions that greatly improved the paper.

Funding

Ténan Yeo was supported by a thesis scholarship from the government of Ivory Coast, and a salary as instructor at University of Aix–Marseille, and the two other authors by their respective university.

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Correspondence to T. Yeo.

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N’zi, M., Pardoux, E. & Yeo, T. A SIR Model on a Refining Spatial Grid I: Law of Large Numbers. Appl Math Optim 83, 1153–1189 (2021). https://doi.org/10.1007/s00245-019-09582-1

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Keywords

  • Spatial model
  • Deterministic
  • Stochastic
  • Law of large numbers