Dynamics and optimal control of a fractional-order propagation model with time-varying intervention costs

Li Wang; Zongmin Yue; Li Wang; Zongmin Yue

doi:10.3934/nhm.2026051

Networks and Heterogeneous Media

2026, Volume 21, Issue 4: 1334-1375. doi: 10.3934/nhm.2026051

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Dynamics and optimal control of a fractional-order propagation model with time-varying intervention costs

Li Wang ,
Zongmin Yue ^,

School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China

Received: 19 March 2026 Revised: 13 June 2026 Accepted: 26 June 2026 Published: 03 July 2026

The growing role of artificial intelligence (AI) in information dissemination has made algorithmic recommendation a significant pathway for rumor spread. To address this, we developed a dual-path rumor propagation model that integrates interpersonal communication with AI-driven algorithmic diffusion. The model employs Caputo fractional derivatives to capture memory effects and historical dependencies in the spreading process. For intervention, we proposed a coordinated control framework combining traditional measures (e.g., legal regulation and science education) with dynamic AI algorithmic control. A key innovation was the design of a time-varying cost function and the introduction of a time-sensitive factor to model the realistic constraint of rising AI control costs over time, forming a novel fractional-order optimal control problem. The trade-off between control cost and governance effectiveness was analyzed via Pareto frontier optimization. Theoretical analysis confirmed that the equilibrium and its global stability are governed by the basic reproduction number. Numerical simulations revealed the existence of a critical regime for the time-sensitive factor ($ \rho = 0.8 $), in which the optimal strategy exhibits a distinct staged control behavior: intensive AI-based intervention is emphasized during the early phase, followed by a gradual transition toward traditional measures for sustained long-term governance. This strategy effectively reduces the peak and overall scale of rumor propagation, demonstrates robustness across parameters, and achieves a balanced trade-off between control cost and social benefit. The study provides a dynamic modeling framework for rumor governance in the AI era, highlighting the importance of integrating algorithmic and traditional controls while accounting for time-varying costs.
- fractional-order optimal control,
- time-varying cost,
- time-sensitive factor,
- Pareto frontier,
- rumor propagation control
Citation: Li Wang, Zongmin Yue. Dynamics and optimal control of a fractional-order propagation model with time-varying intervention costs[J]. Networks and Heterogeneous Media, 2026, 21(4): 1334-1375. doi: 10.3934/nhm.2026051

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Abstract

The growing role of artificial intelligence (AI) in information dissemination has made algorithmic recommendation a significant pathway for rumor spread. To address this, we developed a dual-path rumor propagation model that integrates interpersonal communication with AI-driven algorithmic diffusion. The model employs Caputo fractional derivatives to capture memory effects and historical dependencies in the spreading process. For intervention, we proposed a coordinated control framework combining traditional measures (e.g., legal regulation and science education) with dynamic AI algorithmic control. A key innovation was the design of a time-varying cost function and the introduction of a time-sensitive factor to model the realistic constraint of rising AI control costs over time, forming a novel fractional-order optimal control problem. The trade-off between control cost and governance effectiveness was analyzed via Pareto frontier optimization. Theoretical analysis confirmed that the equilibrium and its global stability are governed by the basic reproduction number. Numerical simulations revealed the existence of a critical regime for the time-sensitive factor ($ \rho = 0.8 $), in which the optimal strategy exhibits a distinct staged control behavior: intensive AI-based intervention is emphasized during the early phase, followed by a gradual transition toward traditional measures for sustained long-term governance. This strategy effectively reduces the peak and overall scale of rumor propagation, demonstrates robustness across parameters, and achieves a balanced trade-off between control cost and social benefit. The study provides a dynamic modeling framework for rumor governance in the AI era, highlighting the importance of integrating algorithmic and traditional controls while accounting for time-varying costs.

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