A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment

Abstract

<p style='text-indent:20px;'>Most of the previous work on rumor propagation either focus on ordinary differential equations with temporal dimension or partial differential equations (PDE) with only consideration of spatially independent parameters. Little attention has been given to rumor propagation models in a spatiotemporally heterogeneous environment. This paper is dedicated to investigating a SCIR reaction-diffusion rumor propagation model with a general nonlinear incidence rate in both heterogeneous and homogeneous environments. In spatially heterogeneous case, the well-posedness of global solutions is established first. The basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula> is introduced, which can be used to reveal the threshold-type dynamics of rumor propagation: if <inline-formula><tex-math id="M2">\begin{document}$ R_0 &lt; 1 $\end{document}</tex-math></inline-formula>, the rumor-free steady state is globally asymptotically stable, while <inline-formula><tex-math id="M3">\begin{document}$ R_0 &gt; 1 $\end{document}</tex-math></inline-formula>, the rumor is uniformly persistent. In spatially homogeneous case, after introducing the time delay, the stability properties have been extensively studied. Finally, numerical simulations are presented to illustrate the validity of the theoretical analysis and the influence of spatial heterogeneity on rumor propagation is further demonstrated.