Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2145–2163 | Cite as

Analysis of Age-Structured Pertussis Models with Multiple Infections During a Lifetime

  • Xiaojing Wang
  • Yangyang Shi
  • Jingan CuiEmail author
  • Zhilan FengEmail author


One of the unique features of bacterial diseases such as pertussis is the possibility of multiple infections during a lifetime. Immunity gained after each infection may play an important role in disease transmission dynamics. A PDE model with two infections for pertussis was considered in Feng et al. (J Theor Biol 356:123–132, 2014) and applied to the Swedish population to estimate the age-dependent probability of infection on contact. However, no detailed analysis of the dynamic properties of the model was considered in that study. Here we present the analysis including existence and stability of equilibrium solutions, which are shown to be determined by the effective (or basic) reproduction number \({\mathcal {R}}\) (or \({\mathcal {R}}_0\)). We also extend the model in Feng et al.  (2014) to allow three infections during a lifetime. And we derive the age-specific probability of infection during a lifetime, denoted by F(a), using two approaches: one uses the model solutions, and the other one is based on biological interpretations. Numerical explorations of the model suggest that, with reasonable assumptions, two infections are all that one needs to consider to account for the dynamics of this disease. This increases the importance of the analytic results from the 2-infection model. Nonetheless, the formula for F(a) in the 3-infection model may provide more accurate description for the probability of infection because it permits more realistic distributions for waiting times in disease stages.


Partial differential equations Age-structured epidemiological model Multiple infections of pertussis Global asymptotic stability Age-dependent probability of infection 



We thank the Reviewer for careful reading of the manuscript and for many helpful comments that have improved the presentation of the paper. The research is partially supported by the Natural Science Foundation of China (11371048 and 11701026), the BUCEA Post Graduate Teaching Quality Improvement Project (J2017008), the BUCEA Post Graduate Innovation Project (PG2017031), and the BUCEA Basic Research Operating Costs for 2018 Projects (ZC05, FZ03).


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Authors and Affiliations

  1. 1.School of ScienceBeijing University of Civil Engineering and ArchitectureBeijingChina
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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