Bulletin of Mathematical Biology

, Volume 69, Issue 3, pp 1067–1091 | Cite as

Approximation of the Basic Reproduction Number R 0 for Vector-Borne Diseases with a Periodic Vector Population

  • Nicolas Bacaër
Original Article


The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R 0 of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p 0 (1+ε cos (ωt − φ)) with ε ≪ 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p 0. The maximum correction due to the second term is (ε2/8)% and always tends to decrease R 0. The basic reproduction number R 0 is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R 0 are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality.


Epidemics Basic reproduction number Seasonality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M., Rohani, P., 2006. Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9, 467–484.CrossRefGoogle Scholar
  2. Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans, Dynamics and Control. Oxford University Press, Oxford.Google Scholar
  3. Anita, S., Iannelli, M., Kim, M.Y., Park E.J., 1998. Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math. 58, 1648–1666.zbMATHCrossRefGoogle Scholar
  4. Aron, J.L., Schwartz, I.B., 1984. Seasonality and period-doubling bifurcations in an epidemic model. J. Theor. Biol. 110, 665–679.Google Scholar
  5. Bacaër, N., Guernaoui, S., 2006. The epidemic threshold of vector-borne diseases with seasonality—The case of cutaneous leishmaniasis in Chichaoua, Morocco. J. Math. Biol. 53, 421–436.zbMATHCrossRefGoogle Scholar
  6. Bailey, N.T.J., 1982. The Biomathematics of Malaria. Charles Griffin, London.zbMATHGoogle Scholar
  7. Bartlett, M.S., 1960. Stochastic Population Models in Ecology and Epidemiology. Methuen, London.zbMATHGoogle Scholar
  8. Coale, A.J., 1972. The Growth and Structure of Human Populations—A Mathematical Investigation. Princeton University Press, Princeton, NJ.Google Scholar
  9. Codeço, C.T., 2001. Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir. BMC Infect. Dis. 289, 2801–2810.Google Scholar
  10. Cohen-Tannoudji, C., Diu, B., Laloë, F., 1986. Mécanique Quantique, 3rd edn. Hermann, Paris.Google Scholar
  11. Cooke, K.L., Kaplan, J.L., 1976. A periodicity threshold theorem for epidemics and population growth. Math. Biosci. 31, 87–104.zbMATHCrossRefGoogle Scholar
  12. Diekmann, O., Heesterbeek, J.A.P., 2000. Mathematical Epidemiology of Infectious Diseases—Model Building, Analysis and Interpretation. Wiley, Chichester, UK.Google Scholar
  13. Dietz, K., 1976. The incidence of infectious diseases under the influence of seasonal fluctuations. In: Berger, J., Bühler, W., Repges, R., Tautu, P. (Eds), Mathematical Modelling in Medicine. Springer, Berlin, pp. 1–15.Google Scholar
  14. Duhamel, G., Gombert, D., Paupy, C., Quatresous, I., 2006. Mission d’appui à la lutte contre l’épidémie de chikungunya à la Réunion. Inspection générale des affaires sociales, Paris.
  15. Grossman, Z., 1980. Oscillatory phenomena in a model of infectious diseases. Theor. Popul. Biol. 18, 204–243.zbMATHCrossRefGoogle Scholar
  16. Grossman, Z., Gumowski, I., Dietz, K., 1977. The incidence of infectious diseases under the influence of seasonal fluctuations —Analytical approach. In: Lakshmikantham, V. (Ed.), Nonlinear Systems and Applications. Academic, New York, pp. 525–546.Google Scholar
  17. Hale, J.K., 1980. Ordinary Differential Equations. Krieger, New York.zbMATHGoogle Scholar
  18. Heesterbeek, J.A.P., 2002. A brief history of R 0 and a recipe for its calculation. Acta Biotheor. 50, 189–204.CrossRefGoogle Scholar
  19. Heesterbeek, J.A.P., Roberts, M.G., 1995a. Threshold quantities for helminth infections. J. Math. Biol. 33, 415–434.zbMATHCrossRefGoogle Scholar
  20. Heesterbeek, J.A.P., Roberts, M.G., 1995b. Threshold quantities for infectious diseases in periodic environments. J. Biol. Syst. 3, 779–787.CrossRefGoogle Scholar
  21. Hochstadt, H., 1973. Integral Equations. Wiley, New York.zbMATHGoogle Scholar
  22. Jagers, P., Nerman, O., 1985. Branching processes in periodically varying environment. Ann. Probl. 13, 254–268.zbMATHGoogle Scholar
  23. Kato, T., 1984. Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin.zbMATHGoogle Scholar
  24. Krasnosel’skij, M.A., Lifshits, Je.A., Sobolev, A.V., 1980. Positive Linear Systems: The Method of Positive Operators. Heldermann, Berlin.Google Scholar
  25. Kuznetsov, Yu.A., Piccardi, C., 1994. Bifurcation analysis of periodic SEIR and SIR epidemic models. J. Math. Biol. 32, 109–121.zbMATHCrossRefGoogle Scholar
  26. Ma, J., Ma, Z., 2006. Epidemic threshold conditions for seasonally forced SEIR models. Math. Biosci. Eng. 3, 161–172.zbMATHGoogle Scholar
  27. Moneim, I.A., Greenhalgh, D., 2005. Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate. Math. Biosci. Eng. 2, 591–611.zbMATHGoogle Scholar
  28. Nussbaum, R.D., 1977. Periodic solutions of some integral equations from the theory of epidemics. In: Lakshmikantham, V. (Ed.), Nonlinear Systems and Applications. Academic, New York, pp. 235–257.Google Scholar
  29. Nussbaum, R.D., 1978. A periodicity threshold theorem for some nonlinear integral equations. SIAM J. Math. Anal. 9, 356–376.zbMATHCrossRefGoogle Scholar
  30. Pierre, V., Thiria, J., Rachou, E., Sissoko, D., Lassalle, C., Renault, P., 2005. Epidémie de dengue 1 à la Réunion en 2004. Journées de veille sanitaire 2005, Poster # 13.
  31. Ross, R., 1911. The Prevention of Malaria, 2nd edn. John Murray, London.Google Scholar
  32. Schaefer, H.H., 1974. Banach Lattices and Positive Operators. Springer, New York.zbMATHGoogle Scholar
  33. Schwartz, I.B., Smith, H.L., 1983. Infinite subharmonic bifurcation in an SEIR epidemic model. J. Math. Biol. 18, 233–253.zbMATHCrossRefGoogle Scholar
  34. Smith, H.L., 1977, On periodic solutions of a delay integral equation modelling epidemics. J. Math. Biol. 4, 69–80.Google Scholar
  35. Thieme, H.R., 1984. Renewal theorems for linear periodic Volterra integral equations. J. Integr. Equ. 7, 253–277.zbMATHGoogle Scholar
  36. Williams, B.G., Dye, C., 1997. Infectious disease persistence when transmission varies seasonally. Math. Biosci. 145, 77–88.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Nicolas Bacaër
    • 1
  1. 1.Institut de Recherche pour le Développement (I.R.D.)Bondy CedexFrance

Personalised recommendations