A comparative study of deterministic and stochastic dynamics of rumor propagation model with counter-rumor spreader

With the popularity of various social media, the propagation of rumors is becoming a social threat. Here, the proposed mathematical model signifies the dynamics of rumor propagation on social media with the influence of counter-rumor spreaders in regulating the transmission process as well as controlling its harmful effect. The total number of users is divided into four categories: (i) newcomer, (ii) spreaders, (iii) counter-rumor spreaders, iv) stiflers. The spreading threshold (R0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {R}_0)$$\end{document} of rumor transmission regulates the condition of the prevalence of rumor. (R0)<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {R}_0)<1$$\end{document} assures the stability of rumor-free state, while (R0)>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {R}_0)>1$$\end{document} assures that one prevailing state exists uniquely with stable nature. Condition for global stability of prevailing state for deterministic system is derived. Subsequently, the corresponding stochastic model demonstrates the effect of random external factors (Wiener process) on rumor propagation dynamics. The global existence and uniqueness of the solution are established to study the asymptotic behavior of that solution around the steady-states. We have also compared the persistence criterion of rumor propagation for the modified system with the deterministic system and derived the condition for the extinction of rumor. Furthermore, scatter plots indicate the significant impact of parameters and numerical simulations are presented to validate the analytical studies. Numerical results assure that environmental noise plays a significant role in suppressing rumor propagation.