Spatial dynamics and optimization method for a network propagation model in a shifting environment

In this paper, a reaction-diffusion \begin{document}$ ISCT $\end{document} rumor propagation model with general incidence rate is proposed in a spatially heterogeneous environment. We first summarize the well-posedness of global solutions. Then the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} is introduced for the model which contains the spatial homogeneity as a special case. The threshold-type dynamics are also established in terms of \begin{document}$ \mathcal{R}_0 $\end{document} , including the global asymptotic stability of the rumor-free steady state and the uniform persistence of all positive solutions. Furthermore, by applying a controller to this model, we investigate the optimal control problem. Employing the operator semigroup theory, we prove the existence, uniqueness and some estimates of the positive strong solution to the controlled system. Subsequently, the existence of the optimal control strategy is established with the aid of minimal sequence techniques and the first order necessary optimality conditions for the optimal control is deduced. Finally, some numerical simulations are performed to validate the main analysis. The results of our study can theoretically promote the regulation of rumor propagation on the Internet.