Skip to main content

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


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. 1.

    Agusto, F.B.: Mathematical model of Ebola transmission dynamics with relapse and reinfection. Math. Biosci. 287, 48–59 (2017a)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Agusto, F.B., Bewick, S., Fagan, W.F.: Mathematical model for Zika virus dynamics with sexual transmission route. Ecol. Complex. 29, 61–81 (2017b)

    Article  Google Scholar 

  3. 3.

    Allen, L.J.S., Bolker, B.M., Lou, Y., Nevai, A.L.: Asymptotic profiles of the steady states for an SIS epidemic patch model. SIAM J. Appl. Math. 67(5), 1283–1309 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Allen, L.J.S., Brauer, F., Van den Driessche, P., Wu, J.: Mathematical Epidemiology, vol. 1945. Springer, Berlin (2008)

    Book  Google Scholar 

  5. 5.

    Allen, L.J.S., Bolker, B.M., Lou, Y., Nevai, A.L.: Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discret. Contin. Dyn.Syst. 21(1), 1–20 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Andersson, H., Britton, T.: Stochastic Epidemic Models and Their Statistical Analysis. Springer Lecture Notes in Statistics. Springer, New York (2000)

    Book  Google Scholar 

  7. 7.

    Arnold, L., Theodosopulu, M.: Deterministic limit of the stochastic model of chemical reactions with diffusion. Adv. Appl. Prob. 12(2), 367–379 (1980)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Blount, D.: Law of large numbers in the supremum norm for a chemical reaction with diffusion. Ann. Appl. Probab. 2(1), 131–141 (1992)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Britton, T., Pardoux, E.: Stochastic epidemic in a homogeneous community (2019). arxiv:1808.05350

  10. 10.

    Debussche, A., Nankep, M.J.N.: A law of large numbers in the supremum norm for a multiscale stochastic spatial gene network (2017). arXiv:1711.06010

  11. 11.

    Du, Z., Peng, R.: A priori \(L^{\infty }\) estimates for solutions of a class of reaction-diffusion systems. J. Math. Biol. 72, 1429–1439 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  Google Scholar 

  13. 13.

    Kermack, W.O., McKendrick, A.G.: Proc. R. Soc. A 115, 700 (1927). Reprinted in Bull. Math. Biol. 53, 33 (1991)

  14. 14.

    Kotelenez, P.: Gaussian approximation to the nonlinear reaction-diffusion equation. Report 146, Universität Bremen Forschungsschwerpunkt Dynamische Systemes (1986)

  15. 15.

    Smith, H.L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems (No. 41). American Mathematical Society (1995)

  16. 16.

    Yamazaki, K.: Threshold dynamics of reaction-diffusion partial differential equations model of Ebola virus disease. Int. J. Biomath. 11, 1850108 (2018a)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Yamazaki, K.: Global well-posedness of infectious disease models without life-timme immunity: the cases of cholera and avian influenza. Math. Med. Biol. 35, 428–445 (2018b)

    Article  Google Scholar 

  18. 18.

    Yamazaki, K., Wang, X.: Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discret. Contin. Dyn. Syst. Ser. B 21, 1297–1316 (2016).

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Webb, G.F.: A reaction-diffusion model for a deterministic epidemic. J. Math. Anal. Appl. 84, 150–161 (1981)

    MathSciNet  Article  Google Scholar 

Download references


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


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.

Author information



Corresponding author

Correspondence to T. Yeo.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


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