Acta Biotheoretica

, Volume 62, Issue 3, pp 417–427 | Cite as

The Basic Reproduction Number for Cellular SIR Networks

Regular Article


The basic reproduction number \(R_0\) is the average number of new infections produced by a typical infective individual in the early stage of an infectious disease, following the introduction of few infective individuals in a completely susceptible population. If \(R_0<1\), then the disease dies, whereas for \(R_0>1\) the infection can invade the host population and persist. This threshold quantity is well studied for SIR compartmental or mean field models based on ordinary differential equations, and a general method for its computation has been proposed by van den Driessche and Watmough. We concentrate here on SIR epidemiological models that take into account the contact network N underlying the transmission of the disease. In this context, it is generally admitted that \(R_{0}\) can be approximated by the average number \(R_{2,3}\) of infective individuals of generation three produced by an infective of generation two. We give here a simple analytic formula of \(R_{2,3}\) for SIR cellular networks. Simulations on two-dimensional cellular networks with von Neumann and Moore neighborhoods show that \(R_{2,3}\) can be used to capture a threshold phenomenon related the dynamics of SIR cellular network and confirm the good quality of the simple approach proposed recently by Aparicio and Pascual for the particular case of Moore neighborhood.


Basic reproduction number Infectious disease SIR compartmental model Cellular network 


  1. Andersson H (1997) Epidemics in a population with social structures. Math Biosci 140:79–84CrossRefGoogle Scholar
  2. Aparicio JP, Pascual M (2007) Building epidemiological models from r0: an implicit treatment of transmission in networks. Proc R Soc B 274:505–512CrossRefGoogle Scholar
  3. Auger P, Kouokam E, Sallet G, Tchuente M, Tsanou B (2008) The ross-macdonald model in a patchy environment. Mathl Biosci 216:123–131CrossRefGoogle Scholar
  4. Demongeot J, Goles E, Tchuente M (eds) (1985) Dynamical systems and cellular automata. Academic Press, LondonGoogle Scholar
  5. Fogelman F, Goles E, Weisbuch G (1983) Transient length in sequential iteration of threshold functions. Discrete Appl Math 6:95–98CrossRefGoogle Scholar
  6. Fortunato S (2010) Community detection in graphs. Phys Rep 486:75–174CrossRefGoogle Scholar
  7. Goles E, Olivos J (1982) Comportement priodique des fonctions seuil binaires et applications. Discrete Appl Math 3:93–105CrossRefGoogle Scholar
  8. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 4(4):599–653CrossRefGoogle Scholar
  9. Kermack WO, McKendrick AG (1927) Contributions to the mathematical theory of epidemics, part 1. Proc R Soc London Ser A 115:700–721CrossRefGoogle Scholar
  10. Newman MEJ (2003) The structured and function of complex networks. Stat Mech 45:167–256Google Scholar
  11. Ngonmang B, Tchuente M, Viennet E (2012) Local communities identification in social networks. Parallel Process Lett 22(1):1240004Google Scholar
  12. Pastor-Satorras R, Vespignani A (2003) Handbook of graphs and networks: from the genome to the internet, chapter 5. WILEY, AmsterdamGoogle Scholar
  13. Piccardi C, Casagrandi R (2008) Inefficient epidemic spreading in scale-free networks. Phys Rev E77(026113):1–4Google Scholar
  14. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmenal models of disease transmission. Math Biosci 180:29–48CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Département d’Informatique, Faculté des SciencesIRD UMI 209 UMMISCOYaoundéCameroun
  2. 2.LIRIMA, Equipe IDASCOUniversité de Yaoundé IYaoundéCameroun

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