Skip to main content

Advertisement

SpringerLink
Log in

Some Elementary Properties of SIR Networks or, Can I Get Sick because You Got Vaccinated?

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We consider epidemics on social networks and address the question of whether administering a safe vaccine to one or more individuals can raise another individual’s chances of becoming infected. Surprisingly, this can happen if transmission probabilities vary over time. If transmission probabilities do not vary with time, we show that in the discrete SIR model vaccination cannot cause collateral damage. We phrase this question in terms of monotonicity properties and answer it using bond percolation methods. By passing to a covering graph we are able to extend these results to models with more complicated latent and infective states.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson, R.M., 1991. Discussion: the Kermack–McKendrick epidemic threshold theorem. Bull. Math. Biol. 53, 3–32.

    Article  Google Scholar 

  • Colizza, V., Barrat, A., Barthelemy, M., Valleron, A.-J., Vespignani, A., 2007. Modeling the Worldwide spread of pandemic influenza: baseline case and containment interventions. PLoS Medecine 4.

  • Dezsö, Z., Barabási, A.-L., 2002. Halting viruses in scale-free networks. Phys. Rev. E 65, 055103(R).

    Article  Google Scholar 

  • Eubank, S., Anil Kumar, V.S., Marathe, M.V., Srinivasan, A., Wang, N., 2006. Structure of social contact networks and their impact on epidemics. In: Abello, J., Cormode, G. (Eds.), Discrete Methods in Epidemiology. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 70, pp. 179–185.

  • Ferrari, M.J., Bansal, S., Meyers, L.A., Björnstad, O.N., 2006. Network frailty and the geometry of herd immunity. Proc. Roy. Soc. B 273, 2743–2748.

    Article  Google Scholar 

  • Grassberger, P., 1983. Critical behavior of the general epidemic process and dynamical percolation. Math. Biosci. 63, 157–172.

    Article  MATH  Google Scholar 

  • Hartvigsen, G., Dresch, J.M., Zielinski, A.L., Macula, A.J., Leary, C.C., 2007. Network structure, and vaccination strategy and effort interact to affect the dynamics of influenza epidemics. J. Theor. Biol. 246, 205–213.

    Article  MathSciNet  Google Scholar 

  • Kermack, W.O., McKendrick, A.G., 1927. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A 115, 700–721.

    Article  Google Scholar 

  • Meyers, L.A., 2007. Contact network epidemiology: bond percolation applied to infectious disease prediction and control. Bull. Am. Math. Soc. 44, 63–86.

    Article  MATH  MathSciNet  Google Scholar 

  • Newman, M.E.J., 2002. Spread of epidemic disease on networks. Phys. Rev. E 66, 016128.

    Article  MathSciNet  Google Scholar 

  • Sander, L.M., Warren, C.P., Sokolov, I.M., Simon, C., Koopman, J., 2002. Percolation on heterogeneous networks as a model for epidemics. Math. Biosci. 180, 293–205.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    William Floyd & Leslie Kay

  2. Department of Pathology, Tufts University, Boston, MA, 02111, USA

    Michael Shapiro

Authors
  1. William Floyd
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leslie Kay
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Shapiro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Floyd, W., Kay, L. & Shapiro, M. Some Elementary Properties of SIR Networks or, Can I Get Sick because You Got Vaccinated?. Bull. Math. Biol. 70, 713–727 (2008). https://doi.org/10.1007/s11538-007-9275-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-007-9275-0

Keywords

Search

Navigation