Dynamics in a delayed rumor propagation model with logistic growth and saturation incidence

Rongrong Yin; Ahmadjan Muhammadhaji; Rongrong Yin; Ahmadjan Muhammadhaji

doi:10.3934/math.2024241

AIMS Mathematics

2024, Volume 9, Issue 2: 4962-4989. doi: 10.3934/math.2024241

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Research article

Dynamics in a delayed rumor propagation model with logistic growth and saturation incidence

Rongrong Yin ^1,2,
Ahmadjan Muhammadhaji ^{1,2
,
,}

1.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China
2.
The Key Laboratory of Applied Mathematics of Xinjiang Uygur Autonomous Region, Xinjiang University, Urumqi 830017, China

Received: 10 November 2023 Revised: 13 January 2024 Accepted: 18 January 2024 Published: 23 January 2024
MSC : 34A34, 34D05, 34D23

This paper studies a delayed rumor propagation model with logistic growth and saturation incidence. The next generation matrix method, some inequality techniques, the Lyapunov-LaSalle invariance principle, and the Lyapunov method are used in this paper. Our results indicate that if the basic regeneration number (which is analogous to the basic reproduction number in disease transmission models) is less than 1, the rumor-free equilibrium point (which is analogous to the disease-free equilibrium point in disease transmission models) is globally stable. If the basic regeneration number is greater than 1, then the rumor is permanent, and some sufficient conditions are obtained for local and global asymptotic stability of the rumor prevailing equilibrium point (which is analogous to the endemic equilibrium point in disease transmission models). Finally, three examples with numerical simulations are presented to illustrate the obtained theoretical results.
- Lyapunov-LaSalle invariance principle,
- permanence,
- Lyapunov functional,
- global stability
Citation: Rongrong Yin, Ahmadjan Muhammadhaji. Dynamics in a delayed rumor propagation model with logistic growth and saturation incidence[J]. AIMS Mathematics, 2024, 9(2): 4962-4989. doi: 10.3934/math.2024241

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Abstract

This paper studies a delayed rumor propagation model with logistic growth and saturation incidence. The next generation matrix method, some inequality techniques, the Lyapunov-LaSalle invariance principle, and the Lyapunov method are used in this paper. Our results indicate that if the basic regeneration number (which is analogous to the basic reproduction number in disease transmission models) is less than 1, the rumor-free equilibrium point (which is analogous to the disease-free equilibrium point in disease transmission models) is globally stable. If the basic regeneration number is greater than 1, then the rumor is permanent, and some sufficient conditions are obtained for local and global asymptotic stability of the rumor prevailing equilibrium point (which is analogous to the endemic equilibrium point in disease transmission models). Finally, three examples with numerical simulations are presented to illustrate the obtained theoretical results.

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