Journal of Mathematical Biology

, Volume 65, Issue 2, pp 237–262 | Cite as

The size of epidemics in populations with heterogeneous susceptibility

  • Guy KatrielEmail author


We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R 0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R 0 alone, and a sharp lower bound in terms of R 0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising ‘50% law’: if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R 0 > 1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R 0 is larger than a certain critical value greater than 1.


Epidemics Heterogeneous susceptibility 

Mathematics Subject Classification (2000)

92D30 92D25 92D40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andersson H, Britton T (1998) Heterogeneity in epidemic models and its effect on the spread of infection. J Appl Probab 35: 651–661MathSciNetzbMATHCrossRefGoogle Scholar
  2. Andreasen V (2011) The final size of an epidemic and its relation to the basic reproduction number. Bull Math Biol (Online First)Google Scholar
  3. Ball F (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv Appl Probab 17: 1–22zbMATHCrossRefGoogle Scholar
  4. Bansal S, Meyers LA (2008) The impact of past epidemics on future disease dynamics. Preprint, arxiv:0910.2008vGoogle Scholar
  5. Bellamy, R (ed) (2004) Susceptibility to infectious diseases: the importance of host genetics. Cambridge University Press, CambridgeGoogle Scholar
  6. Bonzi B, Fall AA, Iggidr A, Sallet G (2010) Stability of differential susceptibility and infectivity models, epidemic models. J Math Biol [Epub ahead of print]Google Scholar
  7. Brauer F (2008) Age-of-infection and the final size relation. Math Biosci Eng 5(2008): 681–690MathSciNetzbMATHCrossRefGoogle Scholar
  8. Coutinho FAB, Massad E, Lopez LF, Burattini MN, Struchiner CJ, Azevedo-Neto RS (1999) Modelling heterogeneities in individual frailties in epidemic models. Math Comput Model 30: 97–115MathSciNetzbMATHCrossRefGoogle Scholar
  9. Craig A, Scherf A (2003) Antigenic variation. Academic Press, AmsterdamGoogle Scholar
  10. Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. Wiley, New YorkGoogle Scholar
  11. Dwyer G, Elkinton JS, Buonaccorsi JP (1997) Host heterogeneity in susceptibility and disease dynamics: tests of a mathematical model. Am Nat 150: 685–707CrossRefGoogle Scholar
  12. Dwyer G, Dushoff J, Elkinton JS, Levin SA (2000) Pathogen-driven outbreaks in forest defoliators revisited: building models from experimental data. Am Nat 156: 105–120CrossRefGoogle Scholar
  13. Frank SA (2002) Immunology and evolution of infectious diseases. Princeton University Press, PrincetonGoogle Scholar
  14. Gart J (1972) The statistical analysis of chain-binomial epidemic models with several kinds of susceptibles. Biometrics 28: 921–930CrossRefGoogle Scholar
  15. Halloran ME, Longini IM, Struchiner CJ (2009) Design and analysis of vaccine studies. Springer, New YorkGoogle Scholar
  16. Hyman JM, Li J (2005) Differential susceptibility epidemic models. J Math Biol 50: 626–644MathSciNetzbMATHCrossRefGoogle Scholar
  17. Karev GP (2005) Dynamics of heterogeneous populations and communities and evolution of distributions. Discrete Contin Dyn Sys (Suppl): 487–496MathSciNetGoogle Scholar
  18. Lefévre C, Picard P (1995) Collective epidemic processes: a general modelling approach to the final outcome of SIR infectious diseases. In: Mollison J (ed) Epidemic models: their structure and relation to dataGoogle Scholar
  19. Ma J, Earn DJD (2006) Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull Math Biol 68: 679–702MathSciNetCrossRefGoogle Scholar
  20. May RM, Anderson RM, Irwin ME (1988) The transmission dynamics of human immunodeficiency virus (HIV). Philos Trans R Soc Lond B 321: 565–607CrossRefGoogle Scholar
  21. Novozhilov AS (2008) On the spread of epidemics in a closed heterogeneous population. Math Biosci 215: 177–185MathSciNetzbMATHCrossRefGoogle Scholar
  22. Pastor-Satorras R, Vespignani A (2001) Epidemic spreading in scale free networks. Phys Rev Lett 86: 3200–3203CrossRefGoogle Scholar
  23. Rass L, Radcliffe J (2003) Spatial deterministic epidemics. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  24. Reluga TC, Medlock J, Perelson AS (2008) Backward bifurcation and multiple equilibria in epidemic models with structured immunity. J Theor Biol 252: 155–165CrossRefGoogle Scholar
  25. Rodrigues P, Margheri A, Rebelo C, Gomes MGM (2009) Heterogeneity in susceptibility to infection can explain high reinfection rates. J Theor Biol 259: 280–290CrossRefGoogle Scholar
  26. Scalia-Tomba G (1986) Final-size distribution of the multitype Reed–Frost process. J Appl Probab 23: 563–584MathSciNetzbMATHCrossRefGoogle Scholar
  27. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New YorkzbMATHCrossRefGoogle Scholar
  28. Veliov VM (2005) On the effect of population heterogeneity on dynamics of epidemic diseases. J Math Biol 51: 124–143MathSciNetCrossRefGoogle Scholar
  29. White LJ, Medley GF (1998) Microparasite population dynamics and continuous immunity. Proc R Soc Lond B 265: 1977–1983CrossRefGoogle Scholar
  30. Yan P, Feng Z (2010) Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness. Math Biosci 224: 43–52MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Biomathematics Unit, Faculty of Life SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations