Journal of Mathematical Biology

, Volume 67, Issue 5, pp 1083–1110

Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis

  • Muntaser Safan
  • Mirjam Kretzschmar
  • Karl P. Hadeler

DOI: 10.1007/s00285-012-0582-1

Cite this article as:
Safan, M., Kretzschmar, M. & Hadeler, K.P. J. Math. Biol. (2013) 67: 1083. doi:10.1007/s00285-012-0582-1


The aim of this paper is to study the impact of introducing a partially protective vaccine on the dynamics of infection in SIRS models where primary and secondary infections are distinguished. We investigate whether a public health strategy based solely on vaccinating a proportion of newborns can lead to an effective control of the disease. In addition to carrying out the qualitative analysis, the findings are further explained by numerical simulations. The model exhibits backward bifurcation for certain values of the parameters. In these cases the standard basic reproduction number (obtained by inspection of the uninfected state) is not significant. The key threshold is the reinfection level which depends on the relative transmissibility (susceptibility) of secondary, with respect to primary, infected (susceptible) individuals and the relative loss of immunity of vaccinated, with respect to recovered, individuals. If one or all of these ratios decrease, then the threshold increases which increases the possibility to contain the infection by vaccination. The analysis shows further that symptomatic infections can be eliminated by vaccination solely.


Two stage SIRS modelBackward bifurcationVaccination coverageEradication effortReinfectionControllability

Mathematics Subject Classification (2010)

Primary 92B99Secondary 92D2592D30

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Muntaser Safan
    • 1
  • Mirjam Kretzschmar
    • 2
    • 3
  • Karl P. Hadeler
    • 4
  1. 1.Mathematics Department, Faculty of ScienceMansoura UniversityMansouraEgypt
  2. 2.Julius Centre for Health Sciences and Primary CareUniversity Medical Centre UtrechtUtrechtThe Netherlands
  3. 3.Centre for Infectious Disease ControlRIVMBilthovenThe Netherlands
  4. 4.Biomathematics, Department of MathematicsUniversity of TuebingenTuebingenGermany