Abstract
We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number \({\mathcal{R}_{0}}\): There is a unique positive solution giving the size of the epidemic if and only if \({\mathcal{R}_{0}}\) exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number \({\mathcal{R}_{0}}\). For other mixing patterns, the relation may be reversed.
Article PDF
Similar content being viewed by others
References
Anderson, R. M., Donnelly, C. A., Ferguson, N. M., Woolhouse, M. E. J., Watt, C. J. et al. (1996). Transmission dynamics and epidemiology of BSE in British cattle. Nature (London), 382, 779–788.
Andreasen, V. (1995). Instability in an SIR-model with age-dependent susceptibility. In O. Arino, D. Axelrod, M. Kimmel, & M. Langlais (Eds.), Mathematical population dynamics (Vol. 1, pp. 3–14). Winnipeg: Wuerz Publ.
Andreasen, V. (2003). Dynamics of annual influenza A epidemics with immuno-selection. J. Math. Biol., 46, 504–536.
Andreasen, V., & Frommelt, T. (2005). A school-oriented, age-structured epidemic model. SIAM J. Appl. Math., 65, 1870–1887.
Andreasen, V., & Sasaki, A. (2006). Shaping the phylogenetic tree of influenza by cross-immunity. Theor. Popul. Biol., 70, 164–173.
Arinaminpathy, N., & McLean, A. R. (2008). Antiviral treatment for the control of pandemic influenza: some logistical constraints. J. Roy. Soc. Interfaces, 5, 545–553.
Arino, J., Brauer, F., van den Driessche, P., Watmough, J., & Wu, J. (2007). A final size relation for epidemic models. Math. Biosci. Eng., 4, 159–175.
Ball, F. (1985). Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Probab., 17, 1–22.
Boni, M. F., Gog, J. R., Andreasen, V., & Christiansen, F. B. (2004). Influenza drift and epidemic size: the race between generating and escaping immunity. Theor. Popul. Biol., 65, 179–191.
Brauer, F. (2008). Epidemic models with heterogeneous mixing and treatment. Bull. Math. Biol., 70, 1869–1885.
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert W function. Adv. Comput. Math., 5, 329–359.
Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases. Chichester: Wiley.
Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R0 in models for infectious-diseases in heterogeneous populations. J. Math. Biol., 28, 365–382.
Diekmann, O., Heesterbeek, J. A. P., & Roberts, M. G. (2010). The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface, 7(47), 873–885.
Dwyer, G., Dushoff, J., Elkinton, J. S., & Levin, S. A. (2000). Pathogen driven outbreaks in forest defoliators revisited: Building models from experimental data. Am. Nat., 156, 105–120.
Eames, K. T. D., & Keeling, M. J. (2002). Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc. Natl. Acad. Sci. USA, 99, 13330–13335.
Elveback, L. R., Fox, J. P., Ackerman, E., Langworthy, A., Boyd, M., & Gatewood, L. (1976). Influenza simulation-model for immunization studies. Am. J. Epidemiol., 103, 152–165.
Eubank, S., Guclu, H., Kumar, V. S. A., Marathe, M. V., Srinivasan, A. et al. (2004). Modelling disease outbreaks in realistic urban social networks. Nature (London), 429, 180–184.
Ferguson, N. M., Donnelly, C. A., & Anderson, R. M. (2001). Transmission intensity and impact of control policies on the foot and mouth epidemic in Great Britain. Nature (London), 413, 542–548.
Ferguson, N. M., Cummings, D. A. T., Cauchemez, S., Fraser, C., Riley, S. et al. (2005). Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature (London), 437, 209–214.
Gart, J. J. (1968). The mathematical analysis of an epidemic with two kinds of susceptibles. Biometrics, 24, 557–565.
Getz, W. M., & Pickering, J. (1983). Epidemic models—thresholds and population regulation. Am. Nat., 121, 892–898.
Gillespie, J. H. (1975). Natural selection for resistance to epidemics. Ecology, 56, 493–495.
Hethcote, H. W., & Yorke, J. A. (1984). Gonorrhea transmission dynamics and control. Berlin: Springer.
Keeling, M. J. (1999). The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. Lond. B, 266, 859–867.
Kermack, W. O., & McKendrick, A. G. (1927). Contributions to the mathematical theory of epidemics, 1. Proc. R.. Soc. A, 115, 700–721. (Reprinted in Bull. Math. Biol., 53, 33–55, 1991)
Kiss, I. Z., Green, D. M., & Kao, R. R. (2006). The effect of contact heterogeneity and multiple routes of transmission on final epidemic size. Math. Biosci., 203, 124–136.
Kretzschmar, M., & Mikolajczyk, R. T. (2009). Contact profiles in eight European countries and implications for modelling the spread of airborne infectious diseases. PLOS One, 4.
Lloyd, A. L., & May, R. M. (2001). How viruses spread among computers and people. Science, 292, 1316–1317.
Longini, I. M., Nizam, A., Xu, S. F., Ungchusak, K., Hanshaoworakul, W. et al. (2005). Containing pandemic influenza at the source. Science, 309, 1083–1087.
Ma, J. L., & Earn, D. J. D. (2006). Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull. Math. Biol., 68, 679–702.
May, R. M. (1985). Regulation of populations with nonoverlapping generations by microparasites—a purely chaotic system. Am. Nat., 125, 573–584.
Mills, C. E., Robins, J. M., & Lipsitch, M. (2004). Transmissibility of 1918 pandemic influenza. Nature (London), 432, 904–906.
Mossong, J., Hens, N., Jit, M., Beutels, P., Auranen, K. et al. (2008). Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Med., 5, 381–391.
Newman, M. E. J. (2002). Spread of epidemic disease on networks. Phys. Rev. E, 66, 016128.
Pastor-Satorras, R., & Vespignani, A. (2001). Epidemic spreading in scale-free networks. Phys. Rev. Lett., 86(14), 3200–3203.
Riley, S., Fraser, C., Donnelly, C. A., Ghani, A. C., Abu-Raddad, L. J. et al. (2003). Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions. Science, 300, 1961–1966.
Tildesley, M. J., & Keeling, M. J. (2009). Is R-0 a good predictor of final epidemic size: Foot-and-mouth disease in the UK. J. Theor. Biol., 258, 623–629.
Volz, E. (2008a). Susceptible-infected-recovered epidemics in populations with heterogeneous contact rates. Eur. Phys. J. B, 63, 381–386.
Volz, E. (2008b). SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol., 56, 293–310.
Wallinga, J., Teunis, P., & Kretzschmar, M. (2006). Using data on social contacts to estimate age-specific transmission parameters for respiratory-spread infectious agents. Am. J. Epidemiol., 164, 936–944.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Andreasen, V. The Final Size of an Epidemic and Its Relation to the Basic Reproduction Number. Bull Math Biol 73, 2305–2321 (2011). https://doi.org/10.1007/s11538-010-9623-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-010-9623-3