Journal of Mathematical Biology

, Volume 75, Issue 2, pp 327–339 | Cite as

Elementary proof of convergence to the mean-field model for the SIR process



The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (IMA J Appl Math, doi: 10.1093/imamat/hxw010, 2016) to the SIR process, we show convergence using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model.


Epidemic SIR Mean-field ODE Markov chain Mean-square convergence 

Mathematics Subject Classification

92D30 60J28 



We thank Peter L. Simon, Tom Britton, and two anonymous referees for helpful comments.


  1. Anderson RM, May RM (1991) Infectious diseases of humans dynamics and control. Oxford University Press, OxfordGoogle Scholar
  2. Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. Lecture Notes in Statistics, chapter 5. Springer, New YorkGoogle Scholar
  3. Armbruster B, Beck E (2016) An elementary proof of convergence to the mean-field equations for an epidemic model. IMA J Appl Math. doi: 10.1093/imamat/hxw010
  4. Benaïm M, Le Boudec J-Y (2008) A class of mean-field interaction models for computer and communication systems. Perform Eval 65(11–12):823–838CrossRefGoogle Scholar
  5. Bortolussi L, Hillston J, Latella D, Massink M (2013) Continuous approximation of collective system behavior: a tutorial. Perform Eval 70(5):317–349CrossRefGoogle Scholar
  6. Cardelli L (2008) From processes to ODEs by chemistry. TCNature, International Federation for Information Processing, vol 273. Springer, Boston, pp 261–281Google Scholar
  7. Daley D, Kendall D (1964) Epidemics and rumours. Nature 204(225):1118CrossRefGoogle Scholar
  8. Daley D, Kendall D (1965) Stochastic rumors. IMA 1(1):42–55Google Scholar
  9. Decreusefond L, Dhersin J-S, Moyal P, Chi Tran V (2012) Large graph limit for an SIR process in random network with heterogeneous connectivity. Ann Appl Probab 22(2):541–575MathSciNetCrossRefMATHGoogle Scholar
  10. Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence, chapter 11.2. Wiley series in probability and statistics. Wiley, HobokenGoogle Scholar
  11. Giraudo D (2014) Bound the variance of the product of two random varables. Mathematics Stack Exchange. (version: 2014-11-30)
  12. Hale J (2009) Ordinary Differential equations, chapter 1.6. Dover Books on Mathematics Series. Dover Publications, MineolaGoogle Scholar
  13. Keeling MJ (1999) The effects of local spatial structure on epidemiological invasions. Proc R Soc Lond B 266:859–867CrossRefGoogle Scholar
  14. Kephart J, White S (1993) Measuring and modeling computer virus prevalence. In: Proceedings, 1993 IEEE computer society symposium on research in security and privacy, pp 2–15Google Scholar
  15. Kermack W, McKendrick A (1927) A contribution to the mathematical theory of epidemics. In: Proceedings of the royal society of London. series a: containing papers of a mathematical and physical character, vol. 115, No. 772, pp. 700–721Google Scholar
  16. Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Probab 7:49–58MathSciNetCrossRefMATHGoogle Scholar
  17. Kurtz TG (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab 8(2):344–356MathSciNetCrossRefMATHGoogle Scholar
  18. May RM, Anderson RM (1983) Epidemiology and genetics in the coevolution of parasites and hosts. Proc R Soc Lond B Biol Sci 219(1216):281–313CrossRefMATHGoogle Scholar
  19. Rand DA (1999) Correlation equations and pair approximations for spatial ecologies. CWI Q 12(3&4):329–368MATHGoogle Scholar
  20. Ross S (2007) Introduction to probability models, chapter 6.4, 9th edn. Academic Press, San DiegoGoogle Scholar
  21. Simon PL, Kiss IZ (2012) From exact stochastic to mean-field ODE models: a case study of three different approaches to prove convergence results. IMA J Appl Math 78(5):945–964Google Scholar
  22. Volz E (2008) SIR dynamics in random networks with heterogeneous connectivity. J Math Biol 56(3):293–310MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanstonUSA

Personalised recommendations