Stability Analysis of a Rumor-Spreading Model with Two Time Delays and Saturation Effect

Chunfeng Wei, Chunlong Fu, Xiaofan Yang (), Yang Qin and Luxing Yang
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Chunfeng Wei: Department of Computer Science, Sichuan University Jinjiang College, Meishan 620860, China
Chunlong Fu: Department of Computer Science, Sichuan University Jinjiang College, Meishan 620860, China
Xiaofan Yang: Department of Computer Science, Sichuan University Jinjiang College, Meishan 620860, China
Yang Qin: School of Big Data and Software Engineering, Chongqing University, Chongqing 400044, China
Luxing Yang: School of Information Technology, Deakin University, Melbourne, VIC 3125, Australia

Mathematics, 2025, vol. 13, issue 11, 1-27

Abstract: Time delay and nonlinear incidence functions have a significant effect on rumor-spreading. In this article, a rumor-spreading model with two unequal time delays and a saturation effect is proposed. The existence, uniqueness, and non-negativity of the solution to this model are shown. The basic reproduction number is determined. A criterion for the existence of a rumor-endemic equilibrium is derived. It is found that there is an interesting conditional forward bifurcation. As a consequence, a complex bifurcation phenomenon is exhibited. A collection of criteria for the asymptotic stability of the rumor-free equilibrium are outlined. In the absence of a time delay, a criterion for the local asymptotic stability of the rumor-endemic equilibrium is presented. In the presence of small time delays, a criterion for the local asymptotic stability of the rumor-endemic equilibrium is established by applying our recently developed technique. Finally, a rumor-spreading control problem is reduced to an optimal control model, which is tackled in the framework of optimal control theory. This work facilitates the understanding of the influence of time delays and the saturation effect on rumor-spreading.

Keywords: rumor-spreading model; time delay; nonlinear incidence function; equilibrium; local asymptotic stability; global asymptotic stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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