Parameter estimation in uncertain multiple-delay differential equations

Yilin Yang; Yin Gao; Yilin Yang; Yin Gao

doi:10.3934/jimo.2026009

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 1: 238-255. doi: 10.3934/jimo.2026009

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Research article

Parameter estimation in uncertain multiple-delay differential equations

Yilin Yang ,
Yin Gao ^,

College of Science, Beijing Forestry University, Beijing 100872, China

Received: 16 September 2025 Revised: 17 October 2025 Accepted: 30 October 2025 Published: 19 November 2025
34D20, 93D15

Uncertain multiple-delay differential equations (UMDDEs) driven by Liu process are critical for modeling systems with multiple-delay interactions and environmental noise. This paper proposes the method of moment estimation to estimate the parameters for UMDDEs with known or unknown delays. When the time interval of the observed data is particularly large, the parameters estimated by the above moment estimation are not very good. In order to overcome this shortcoming, the concept of residuals is introduced, and then we use the method of residual estimation to estimate the parameters for UMDDEs. Moreover, some numerical validations are investigated to show the effectiveness of the above methods for UMDDEs. Besides, the paradox of the stochastic multiple delay Logistic model is proved. Therefore, an uncertain multiple delay Logistic model is defined and applied to describe the dynamics of U.S. population dynamics.
- Uncertain multiple-delay differential equations,
- parameter estimation,
- liu processes
Citation: Yilin Yang, Yin Gao. Parameter estimation in uncertain multiple-delay differential equations[J]. Journal of Industrial and Management Optimization, 2026, 22(1): 238-255. doi: 10.3934/jimo.2026009

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Abstract

Uncertain multiple-delay differential equations (UMDDEs) driven by Liu process are critical for modeling systems with multiple-delay interactions and environmental noise. This paper proposes the method of moment estimation to estimate the parameters for UMDDEs with known or unknown delays. When the time interval of the observed data is particularly large, the parameters estimated by the above moment estimation are not very good. In order to overcome this shortcoming, the concept of residuals is introduced, and then we use the method of residual estimation to estimate the parameters for UMDDEs. Moreover, some numerical validations are investigated to show the effectiveness of the above methods for UMDDEs. Besides, the paradox of the stochastic multiple delay Logistic model is proved. Therefore, an uncertain multiple delay Logistic model is defined and applied to describe the dynamics of U.S. population dynamics.

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