Global stability for a respiratory disease model with distributed or discrete delay on complex network

Lei Shi; Jiaying Zhou; Lei Shi; Jiaying Zhou

doi:10.3934/era.2025323

Electronic Research Archive

2025, Volume 33, Issue 12: 7310-7330. doi: 10.3934/era.2025323

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Research article

Global stability for a respiratory disease model with distributed or discrete delay on complex network

Lei Shi ^{1
,
,},
Jiaying Zhou ²

1.
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
2.
College of Science, Nanjing Forestry University, Nanjing 210037, China

Received: 14 October 2025 Revised: 18 November 2025 Accepted: 01 December 2025 Published: 04 December 2025

Distributed delay is a core concept in time-delay systems and it has been incorporated into the networked respiratory disease model that elucidates the occurrence of respiratory diseases induced by air pollution. Next we studied a respiratory disease model with distributed delay and discrete delay. By analyzing the linearized system, we showed that if the disease-free equilibrium $ E_{0} $ exists, exhibits global asymptotic stability without any constraint on the variable space. In addition, we proved the global stability of the endemic equilibrium $ E_{*} $ by constructing the Liapunov functional. Our findings contributed networked reaction-diffusion models with distributed delay and discrete delay to the existing body of knowledge. Our research found that distributed delay altered the transmission rhythm of respiratory diseases, which weakened local stability and disrupted global stability, which leads to disease recurrence. Discrete delay could disrupt the "synchrony" of respiratory disease transmission, thereby inducing Hopf branches that lead to periodic disease outbreaks and undermined global stability, making it impossible to completely eradicate the disease.
- network,
- respiratory diseases,
- distributed delay,
- discrete delay,
- Lyapunov functional
Citation: Lei Shi, Jiaying Zhou. Global stability for a respiratory disease model with distributed or discrete delay on complex network[J]. Electronic Research Archive, 2025, 33(12): 7310-7330. doi: 10.3934/era.2025323

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Abstract

Distributed delay is a core concept in time-delay systems and it has been incorporated into the networked respiratory disease model that elucidates the occurrence of respiratory diseases induced by air pollution. Next we studied a respiratory disease model with distributed delay and discrete delay. By analyzing the linearized system, we showed that if the disease-free equilibrium $ E_{0} $ exists, exhibits global asymptotic stability without any constraint on the variable space. In addition, we proved the global stability of the endemic equilibrium $ E_{*} $ by constructing the Liapunov functional. Our findings contributed networked reaction-diffusion models with distributed delay and discrete delay to the existing body of knowledge. Our research found that distributed delay altered the transmission rhythm of respiratory diseases, which weakened local stability and disrupted global stability, which leads to disease recurrence. Discrete delay could disrupt the "synchrony" of respiratory disease transmission, thereby inducing Hopf branches that lead to periodic disease outbreaks and undermined global stability, making it impossible to completely eradicate the disease.

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