Integral equation modeling of an SEIQHR epidemic system with non-exponentially distributed disease stages

Fang Liu; Zhen Jin; Fang Liu; Zhen Jin

doi:10.3934/math.2026504

AIMS Mathematics

2026, Volume 11, Issue 4: 12290-12322. doi: 10.3934/math.2026504

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Research article

Integral equation modeling of an SEIQHR epidemic system with non-exponentially distributed disease stages

Fang Liu ^1,2,
Zhen Jin ^{3,4,5
,
,}

1.
School of Data Science and Technology, North University of China, Taiyuan, Shanxi 030051, China
2.
Department of Mathematics, Xinzhou Normal University, Xinzhou, Shanxi 034000, China
3.
Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, China
4.
Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, Shanxi 030006, China
5.
Shanxi Key Laboratory for Mathematical Technology in Complex Systems, Shanxi University, Taiyuan, Shanxi 030006, China

Received: 01 February 2026 Revised: 08 April 2026 Accepted: 21 April 2026 Published: 30 April 2026
MSC : 45G15, 92D25, 92D30

Many infectious diseases, such as COVID-19 and tuberculosis, are characterized by latent and infectious stages, substantial transmission potential, and a non-negligible risk of severe outcomes, which necessitates integrated control strategies including early detection, quarantine, and hospitalization. Compared with exponential distributions, non-exponential distributions provide a more realistic description of disease transmission processes. In this paper, we incorporate general (non-exponential) distributions to describe stage durations associated with infection, recovery, quarantine, and hospital isolation, including Gamma distributions as a special case. Under suitable assumptions, we formulate an SEIQHR epidemic model as a system of integral equations, incorporating natural mortality, disease-induced mortality, and hospital-related mortality. We establish the existence and uniqueness of solutions. An explicit expression of the control reproduction number is derived and used as a threshold quantity for disease control. If the control reproduction number is less than one, the disease-free equilibrium is globally asymptotically stable, otherwise, the endemic equilibrium is globally asymptotically stable. Under the assumption of Gamma-distributed stage durations, we derive an equivalent system of ODEs. Numerical simulations indicate that the shape parameters of the Gamma distributions can significantly affect the control reproduction number.
- epidemiological model,
- SEIQHR model,
- integral equation,
- non-exponentially distributed disease stages,
- control reproduction number
Citation: Fang Liu, Zhen Jin. Integral equation modeling of an SEIQHR epidemic system with non-exponentially distributed disease stages[J]. AIMS Mathematics, 2026, 11(4): 12290-12322. doi: 10.3934/math.2026504

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Abstract

Many infectious diseases, such as COVID-19 and tuberculosis, are characterized by latent and infectious stages, substantial transmission potential, and a non-negligible risk of severe outcomes, which necessitates integrated control strategies including early detection, quarantine, and hospitalization. Compared with exponential distributions, non-exponential distributions provide a more realistic description of disease transmission processes. In this paper, we incorporate general (non-exponential) distributions to describe stage durations associated with infection, recovery, quarantine, and hospital isolation, including Gamma distributions as a special case. Under suitable assumptions, we formulate an SEIQHR epidemic model as a system of integral equations, incorporating natural mortality, disease-induced mortality, and hospital-related mortality. We establish the existence and uniqueness of solutions. An explicit expression of the control reproduction number is derived and used as a threshold quantity for disease control. If the control reproduction number is less than one, the disease-free equilibrium is globally asymptotically stable, otherwise, the endemic equilibrium is globally asymptotically stable. Under the assumption of Gamma-distributed stage durations, we derive an equivalent system of ODEs. Numerical simulations indicate that the shape parameters of the Gamma distributions can significantly affect the control reproduction number.

References

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