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 $ R_0 $ is introduced, which can be used to reveal the threshold-type dynamics of rumor propagation: if $ R_0 < 1 $, the rumor-free steady state is globally asymptotically stable, while $ R_0 > 1 $, 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.

中文翻译：

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一种了解空间异质环境中谣言传播动态的理论方法

以前有关谣言传播的大多数工作要么集中于具有时间维度的常微分方程，要么只考虑空间独立的参数，而偏微分方程（PDE）。时空异质环境中的谣言传播模型很少受到关注。本文致力于研究非均质和均质环境下具有一般非线性发生率的SCIR反应扩散谣言传播模型。在空间异构的情况下，首先建立全局解的适定性。引入了基本复制数$ R_0 $，可用于揭示谣言传播的阈值类型动态：如果$ R_0 <1 $，则无谣言的稳态是全局渐近稳定的，而$ R_0> 1 $，谣言一直持续存在。在空间均匀的情况下，引入时间延迟后，已经对稳定性性质进行了广泛的研究。最后，通过数值模拟说明了理论分析的有效性，并进一步证明了空间异质性对谣言传播的影响。