Rumor model on social networks contemplating self-awareness and saturated transmission rate

Hui Wang; Shuzhen Yu; Haijun Jiang; Hui Wang; Shuzhen Yu; Haijun Jiang

doi:10.3934/math.20241246

AIMS Mathematics

2024, Volume 9, Issue 9: 25513-25531. doi: 10.3934/math.20241246

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Rumor model on social networks contemplating self-awareness and saturated transmission rate

1.
School of Mathematics Science, Xinjiang Normal University, Urumqi 830017, China
2.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China
3.
School of Mathematics and Statistics, Yili Normal University, Yining 835000, China

Received: 12 June 2024 Revised: 08 August 2024 Accepted: 22 August 2024 Published: 02 September 2024
MSC : 34A34, 34D05, 34D20

The propagation of rumors indisputably inflicts profound negative impacts on society and individuals. This article introduces a new unaware ignorants-aware ignorants-spreaders-recovereds $ (2ISR) $ rumor spreading model that combines individual vigilance self-awareness with nonlinear spreading rate. Initially, the positivity of the system solutions and the existence of its positive invariant set are rigorously proved, and the rumor propagation threshold is solved using the next-generation matrix method. Next, a comprehensive analysis is conducted on the existence of equilibrium points of the system and the occurrence of backward bifurcation. Afterward, the stability of the system is validated at both the rumor-free equilibrium and the rumor equilibrium, employing the Jacobian matrix approach as well as the Lyapunov stability theory. To enhance the efficacy of rumor propagation management, a targeted optimal control strategy is formulated, drawing upon the Pontryagin's Maximum principle as a guiding framework. Finally, through sensitivity analyses, numerical simulations, and tests of real cases, we verify the reliability of the theoretical results and further consolidate the solid foundation of the above theoretical arguments.
- rumor propagation,
- backward bifurcation,
- stability,
- optimal control
Citation: Hui Wang, Shuzhen Yu, Haijun Jiang. Rumor model on social networks contemplating self-awareness and saturated transmission rate[J]. AIMS Mathematics, 2024, 9(9): 25513-25531. doi: 10.3934/math.20241246

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Abstract

The propagation of rumors indisputably inflicts profound negative impacts on society and individuals. This article introduces a new unaware ignorants-aware ignorants-spreaders-recovereds $ (2ISR) $ rumor spreading model that combines individual vigilance self-awareness with nonlinear spreading rate. Initially, the positivity of the system solutions and the existence of its positive invariant set are rigorously proved, and the rumor propagation threshold is solved using the next-generation matrix method. Next, a comprehensive analysis is conducted on the existence of equilibrium points of the system and the occurrence of backward bifurcation. Afterward, the stability of the system is validated at both the rumor-free equilibrium and the rumor equilibrium, employing the Jacobian matrix approach as well as the Lyapunov stability theory. To enhance the efficacy of rumor propagation management, a targeted optimal control strategy is formulated, drawing upon the Pontryagin's Maximum principle as a guiding framework. Finally, through sensitivity analyses, numerical simulations, and tests of real cases, we verify the reliability of the theoretical results and further consolidate the solid foundation of the above theoretical arguments.

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