Optimal control and parameter identification of a reaction–diffusion network propagation model

In the era of rapid development of the network, information security is a topic worthy of attention. This paper establishes a reaction–diffusion rumor propagation model with secondary transmission mechanism. We derive the necessary conditions for its Turing instability and then obtain that an increase in media refutation rate γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} can effectively suppress rumors through sensitivity analysis. By converting the parameter γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} to γx,t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \left( \textbf{x},t\right) $$\end{document} and using the Projected Gradient Method, rumors are controlled as the target propagation mode. By applying the method of optimal control, the parameter identification of the system is achieved through three algorithms. The Projected Gradient Method can effectively identify the patterns of two unknown parameters and has global convergence, but the convergence speed is relatively slow. The Barzilar-Borwein method and the BFGS Quasi-Newton Algorithm can effectively improve the convergence speed while ensuring the reliability of the results. The Barzilar-Borwein method is used to effectively identify six parameters of the system, with a relative error of only 0.3030%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\%$$\end{document}. Finally, by changing the parameter γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} to a spatial heterogeneity parameter γx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \left( \textbf{x}\right) $$\end{document}, we have achieved the reproduction of natural biological surface patterns through the Projected Gradient Method.