Hopf bifurcation in a chronological age-structured SIR epidemic model with age-dependent infectivity

Toshikazu Kuniya; Hisashi Inaba; Toshikazu Kuniya; Hisashi Inaba

doi:10.3934/mbe.2023581

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 7: 13036-13060. doi: 10.3934/mbe.2023581

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Hopf bifurcation in a chronological age-structured SIR epidemic model with age-dependent infectivity

Toshikazu Kuniya ^{1
,
,},
Hisashi Inaba ²

1.
Graduate School of System Informatics, Kobe University, 1–1 Rokkodai-cho, Nada-ku, Kobe 657–8501, Japan
2.
Faculty of Education, Tokyo Gakugei University, 4–1–1 Nukuikita-machi, Koganei-shi, Tokyo 184–8501, Japan

Received: 02 February 2023 Revised: 11 May 2023 Accepted: 19 May 2023 Published: 05 June 2023

In this paper, we examine the stability of an endemic equilibrium in a chronological age-structured SIR (susceptible, infectious, removed) epidemic model with age-dependent infectivity. Under the assumption that the transmission rate is a shifted exponential function, we perform a Hopf bifurcation analysis for the endemic equilibrium, which uniquely exists if the basic reproduction number is greater than $ 1 $. We show that if the force of infection in the endemic equilibrium is equal to the removal rate, then there always exists a critical value such that a Hopf bifurcation occurs when the bifurcation parameter reaches the critical value. Moreover, even in the case where the force of infection in the endemic equilibrium is not equal to the removal rate, we show that if the distance between them is sufficiently small, then a similar Hopf bifurcation can occur. By numerical simulation, we confirm a special case where the stability switch of the endemic equilibrium occurs more than once.
- SIR epidemic model,
- chronological age,
- basic reproduction number,
- stability,
- Hopf bifurcation
Citation: Toshikazu Kuniya, Hisashi Inaba. Hopf bifurcation in a chronological age-structured SIR epidemic model with age-dependent infectivity[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 13036-13060. doi: 10.3934/mbe.2023581

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Abstract

In this paper, we examine the stability of an endemic equilibrium in a chronological age-structured SIR (susceptible, infectious, removed) epidemic model with age-dependent infectivity. Under the assumption that the transmission rate is a shifted exponential function, we perform a Hopf bifurcation analysis for the endemic equilibrium, which uniquely exists if the basic reproduction number is greater than $ 1 $. We show that if the force of infection in the endemic equilibrium is equal to the removal rate, then there always exists a critical value such that a Hopf bifurcation occurs when the bifurcation parameter reaches the critical value. Moreover, even in the case where the force of infection in the endemic equilibrium is not equal to the removal rate, we show that if the distance between them is sufficiently small, then a similar Hopf bifurcation can occur. By numerical simulation, we confirm a special case where the stability switch of the endemic equilibrium occurs more than once.

References

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