Journal of Mathematical Biology

, Volume 53, Issue 4, pp 719–746 | Cite as

An age-structured epidemic model of rotavirus with vaccination

  • E. Shim
  • Z. Feng
  • M. Martcheva
  • C. Castillo-Chavez
Article

Abstract

The recent approval of a rotavirus vaccine in Mexico motivates this study on the potential impact of the use of such a vaccine on rotavirus prevention and control. An age-structured model that describes the rotavirus transmission dynamics of infections is introduced. Conditions that guarantee the local and global stability analysis of the disease-free steady state distribution as well as the existence of an endemic steady state distribution are established. The impact of maternal antibodies on the implementation of vaccine is evaluated. Model results are used to identify optimal age-dependent vaccination strategies. A convergent numerical scheme for the model is introduced but not implemented. This paper is dedicated to Prof. K. P. Hadeler, who continues to push the frontier of knowledge in mathematical biology.

Keywords

Rotavirus Age-structure Vacccination 

Mathematics Subject Classification (2000)

92D30 65N06 65N12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anguelov R., Lubuma J.M.-S. (2001) Contributions to the mathematics of the nonstandard finite difference method and applications. Numer. Methods Part. Diff. Eq. 17(5): 518–543CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bernstein D.I., Sander D.S., Smith V.E., Schiff G.M., Ward RL. (1991) Protection from rotavirus reinfection: 2-year prospective study. J. Infect. Dis. 164(2): 277–283Google Scholar
  3. 3.
    Bishop R.F., Davidson G.P., Holmes I.H., Ruck B.J. (1973) Virus particles in epithelial cells of duodenal mucosa from children with viral gastroenteritis. Lancet 1, 1281–1283CrossRefGoogle Scholar
  4. 4.
    Bishop R.F., Barnes G.L., Cipriani E., Lund J.S. (1983) Clinical immunity after neonatal rotavirus infection. A prospective longitudinal study in young children. N. Engl. J. Med. 309(2): 72–76Google Scholar
  5. 5.
    Busenberg S., Castillo-Chavez C. (1991) A general solution of the problem of mixing subpopulations, and its application to risk- and age-structure epidemic models for the spread of AIDS. IMA. J. Math. Appl. Med. Biol. 8(1): 1–29CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Castillo-Chavez C., Feng Z. (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies. Math. Biosc. 151(2): 135–154CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Clark H.F., Lawley D., Shrager D., Jean-Guillaume D., Offit P.A., Whang S.Y., Eiden J.J., Bennett P.S., Kaplan K.M., Shaw A.R. (2004) Infant immune response to human rotavirus serotype G1 vaccine candidate reassortant WI79-9: different dose response patterns to virus surface proteins VP7 and VP4. Pediatr. Infect. Dis. J. 23(3): 206–211CrossRefGoogle Scholar
  8. 8.
    Cunliffe N.A., Bresee J.S., Hart C.A. (2002) Rotavirus vaccines: development, current issues and future prospects. J. Infect. 45(1): 1–9CrossRefGoogle Scholar
  9. 9.
    Dennehy P.H. (2005) Rotavirus vaccines: an update. Curr. Opin. Pediatr. 17(1): 88–92CrossRefGoogle Scholar
  10. 10.
    Dietz K., Schenzle D. (1985) Proportionate mixing models for age-dependent infection transmission. J. Math Biol. 22(1): 117–120CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Gripenberg G., Londen S.O., Staffans O.: Volterra Integral and Functional Equations. Series: Encyclopedia of Mathematics and its Applications (No. 34). Cambridge (1990)Google Scholar
  12. 12.
    Hadeler K.P., Müller J.(1993) Vaccination in age-structured populations II: optimal vaccination strategies. In: Isham V., Medley G., (ed) Models for Infectious Human Diseases: Their Structure and Relation to Data. Cambridge University Press, Cambridge, pp. 102–114Google Scholar
  13. 13.
    Hadeler K.P., Müller J.: Optimal harvesting and optimal vaccination, (In submission)Google Scholar
  14. 14.
    Hardy D. (1987) Epidemiology of rotaviral infection in adults. Rev. Infect. Dis. 9, 461–469Google Scholar
  15. 15.
    Hochwald C., Kivela L.: Rotavirus vaccine, live, oral, tetravalent (RotaShield). Pediatr. Nurs. 25(2), 203–204, 207 (1999)Google Scholar
  16. 16.
    Huang W., Castillo-Chavez C.: Age-structured core groups and their impact on HIV dynamics. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, IMA, vol. 126, pp. 261–273, Springer, Berlin Heidelberg New YorkGoogle Scholar
  17. 17.
    Iannelli M., Milner F., Pugliese A. (1992) Analytical and numerical results for the age structured SIS epidemic model with mixed inter-intra-cohort transmission. SIAM J. Math. Anal. Publ. Soc. Indust. Appl. Math. 23, 662–688MathSciNetMATHGoogle Scholar
  18. 18.
    Kapikian A.Z., Kim H.W., Wyatt R.G., Cline W.L., Arrobio J.O., Brandt C.D., Rodriguez W.J., Sack D.A., Chanock R.M., Parrott R.H. (1976) Human reovirus-like agent as the major pathogen associated with “winter” gastroenteritis in hospitalized infants and young children. N. Engl. J. Med. 294, 965–972CrossRefGoogle Scholar
  19. 19.
    Kapikian A.Z., Wyatt R.G., Levine M.M. et al. (1983) Oral administration of human rotavirus to volunteers: induction of illness and correlates of resistance. J. Infect. Dis. 147, 95–106Google Scholar
  20. 20.
    Kribs-Zaleta C.M., Martcheva M. (2002) Vaccination strategies and backward bifurcation in an age-since-infection structured model. Math. Biosci. 177–178: 317–332CrossRefMathSciNetGoogle Scholar
  21. 21.
    Mastretta E., Longo P., Laccisaglia A., Balbo L., Russo R., Mazzaccara A., Gianino P. (2002) Effect of lactobacillus GG and breast-feeding in the prevention of rotavirus nosocomial infection. J. Pediat. Gastroenterol. Nutr. 35(4): 527–531CrossRefGoogle Scholar
  22. 22.
    Mickens R.E. (ed) (2005) Advances in the Applications of Nonstandard Finite Difference Schemes. World Scientific Publishing Company, SingaporeMATHGoogle Scholar
  23. 23.
    Nguyen T.V., Yuan L., P Azevedo M.S., Jeong K.I., Gonzalez A.M., Iosef C., Lovgren-Bengtsson K., Morein B., Lewis P., Saif L.J. (2006) Low titer maternal antibodies can both enhance and suppress B cell responses to a combined live attenuated human rotavirus and VLP-ISCOM vaccine. Vaccine 24(13): 2302–2316CrossRefGoogle Scholar
  24. 24.
    Parashar U.D., Bresee J.S., Gentsch J.R., Glass R.I. (1998) Rotavirus. Emerg. Infect. Dis. 4(4): 561–570CrossRefGoogle Scholar
  25. 25.
    Parashar U.D., Holman R.C., Clarke M.J., Bresee J.S., Glass R.I. (1998) Hospitalizations associated with rotavirus diarrhea in the United States, 1993 through 1995: surveillance based on the new ICD-9-CM rotavirus-specific diagnostic code. J. Infect. Dis. 177(1): 7–13CrossRefGoogle Scholar
  26. 26.
    Pérez-Schael I., Guntiñas M.J., Pérez M., Pagone V., Rojas A.M., González R., Cunto W., Hoshino Y., Kapikian A.Z. (1997) Efficacy of the Rhesus RotavirusĐBased Quadrivalent Vaccine in Infants and Young Children in Venezuela. N. Engl. J. Med. 337(17): 1181–1187CrossRefGoogle Scholar
  27. 27.
    Ramos, P D, Stefanelli C C, Linhares R E C et al. (1998) The infectivity of pig rotavirus in stools. Braz. J. Vet. Res. Anim. Sci. 35(2): 00–00CrossRefGoogle Scholar
  28. 28.
    Rorres C., Fair W. (1975) Optimal harvesting policy for an age-specific population. Math. Biosci. 24, 31–47CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Rotavirus Vaccine Program: A path affiliate. As found at http://www.rotavirusvaccine.org/ vaccine-facts.htmGoogle Scholar
  30. 30.
    Shim E., Banks T.B., Castillo-Chavez C.: Seasonality of rotavirus infection with its vaccination. In: Gumel A. (Chief Ed.), Castillo-Chavez C., Clemence D.P. and Mickens R.E (eds.) Modeling The Dynamics of Human Diseases: Emerging Paradigms and Challenges. AMS Cotempor. Math. Ser. (to appear).Google Scholar
  31. 31.
    Velazquez F.R., Matson D.O., Guerrero M.L., Shults J., Calva J.J., Morrow A.L., Glass R.I., Pickering L.K., Ruiz-Palacios G.M. (2000) Serum antibody as a marker of protection against natural rotavirus infection and disease. J. Infect. Dis. 182(6): 1602–1609CrossRefGoogle Scholar
  32. 32.
    Vesikari, T, Clark H.F., Offit P.A., et al.: The effect of dose and composition of a pentavalent rotavirus vaccine (RotaTeq) upon safety, efficacy, and immunogenicity in healthy infants. In: Presented at the 22nd Annual Meeting of the European Society for Pediatric Infectious Diseases (ESPID), Tampere, Finland, 26–28, (2004)Google Scholar
  33. 33.
    Vesikari T., Giaquinto C., Huppertz H.I. (2006) Clinical trials of rotavirus vaccines in europe. Pediatr. Infect. Dis. J. 25(1): S42–7Google Scholar
  34. 34.
    Ward R.L., Bernstein D.I. (1994) Protection against rotavirus disease after natural rotavirus infection. US Rotavirus Vaccine Efficacy Group. J. Infect. Dis. 169(4): 900–904Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • E. Shim
    • 1
  • Z. Feng
    • 2
  • M. Martcheva
    • 3
  • C. Castillo-Chavez
    • 1
  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempeUSA
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA

Personalised recommendations