An age-structured epidemic model with boosting and waning of immune status

Kento Okuwa; Hisashi Inaba; Toshikazu Kuniya; Kento Okuwa; Hisashi Inaba; Toshikazu Kuniya

doi:10.3934/mbe.2021289

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 5707-5736. doi: 10.3934/mbe.2021289

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Research article

An age-structured epidemic model with boosting and waning of immune status

1.
Manulife Life Insurance Company, 3-20-2 Nishishinjuku Shinjuku-ku Tokyo 163-1430, Japan
2.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan
3.
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho Nada-ku Kobe 657-8501, Japan

Received: 10 April 2021 Accepted: 03 June 2021 Published: 24 June 2021

In this paper, we developed an age-structured epidemic model that takes into account boosting and waning of immune status of host individuals. For many infectious diseases, the immunity of recovered individuals may be waning as time evolves, so reinfection could occur, but also their immune status could be boosted if they have contact with infective agent. According to the idea of the Aron's malaria model, we incorporate a boosting mechanism expressed by reset of recovery-age (immunity clock) into the SIRS epidemic model. We established the mathematical well-posedness of our formulation and showed that the initial invasion condition and the endemicity can be characterized by the basic reproduction number $ R_0 $. Our focus is to investigate the condition to determine the direction of bifurcation of endemic steady states bifurcated from the disease-free steady state, because it is a crucial point for disease prevention strategy whether there exist subcritical endemic steady states. Based on a recent result by Martcheva and Inaba ^[1], we have determined the direction of bifurcation that endemic steady states bifurcate from the disease-free steady state when the basic reproduction number passes through the unity. Finally, we have given a necessary and sufficient condition for backward bifurcation to occur.
- boosting and waning,
- immunity clock,
- epidemic model,
- Aron model,
- age-structure,
- basic reproduction number
Citation: Kento Okuwa, Hisashi Inaba, Toshikazu Kuniya. An age-structured epidemic model with boosting and waning of immune status[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 5707-5736. doi: 10.3934/mbe.2021289

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Abstract

In this paper, we developed an age-structured epidemic model that takes into account boosting and waning of immune status of host individuals. For many infectious diseases, the immunity of recovered individuals may be waning as time evolves, so reinfection could occur, but also their immune status could be boosted if they have contact with infective agent. According to the idea of the Aron's malaria model, we incorporate a boosting mechanism expressed by reset of recovery-age (immunity clock) into the SIRS epidemic model. We established the mathematical well-posedness of our formulation and showed that the initial invasion condition and the endemicity can be characterized by the basic reproduction number $ R_0 $. Our focus is to investigate the condition to determine the direction of bifurcation of endemic steady states bifurcated from the disease-free steady state, because it is a crucial point for disease prevention strategy whether there exist subcritical endemic steady states. Based on a recent result by Martcheva and Inaba ^[1], we have determined the direction of bifurcation that endemic steady states bifurcate from the disease-free steady state when the basic reproduction number passes through the unity. Finally, we have given a necessary and sufficient condition for backward bifurcation to occur.

References

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