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A Dual-age Structured Epidemiological Model with Waning Immunity and Reinfection.

Raimund M Kovacevic1, Nikolaos I Stilianakis2,3, Vladimir M Veliov4

  • 1Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8, 1040, Vienna, Austria. Raimund.Kovacevic@tuwien.ac.at.

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|June 23, 2026
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Summary

This study introduces a new dual-age model to understand infectious disease dynamics with waning immunity and reinfection. It reveals that vaccination is most effective when targeting intermediate immunity ages, not just those with fully lost immunity.

Keywords:
Dual-age structured modelEpidemic dynamicsReinfectionVaccination strategyWaning immunity

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Infectious Disease Dynamics

Background:

  • Waning immunity and reinfection are key in infectious diseases but often oversimplified in models.
  • Existing epidemiological models struggle to integrate individual immunity history with current infection status.

Purpose of the Study:

  • To develop a novel dual-age structured model accounting for immunity age and infection age.
  • To analyze epidemic dynamics considering waning immunity and reinfection.
  • To explore the impact of vaccination strategies, particularly booster vaccinations, based on immunity age.

Main Methods:

  • Formulated a system of age-structured partial differential equations for susceptible and infected populations.
  • Stratified populations by immunity age (time since recovery) and infection age (time since infection).
  • Incorporated immunity and infection age-dependent parameters (contact, mortality, recovery, susceptibility, pathogen load).
  • Derived basic reproduction numbers and used a second-order Runge-Kutta scheme for numerical solutions.

Main Results:

  • Higher contact rates lead to larger oscillation amplitudes and longer inter-epidemic periods.
  • Long-term epidemic outcomes showed non-monotonic dependence on the initial infected population size.
  • Vaccination efficiency is highly dependent on pathogen load distribution; more concentrated loads require higher vaccination rates.
  • Optimal vaccination strategies target intermediate immunity ages, not solely individuals with fully waned immunity.

Conclusions:

  • The dual-age model provides a more nuanced understanding of infectious disease dynamics with waning immunity and reinfection.
  • Booster vaccination strategies can be effectively designed by targeting specific immunity age groups.
  • Understanding the interplay of immunity history and infection status is crucial for accurate epidemic forecasting and control.