<p>The proliferation of rumors has profoundly disrupted societal cohesion and stability. However, traditional models, relying on ordinary differential equations, often overlook the spatial dimension’s impact on rumor propagation dynamics. Consequently, they fail to capture the intricate complexity of information dissemination, especially amidst rapid advancements in mobile communication technology. In this article, using partial differential equations, we introduce a novel delayed model to examine the spatio-temporal dynamics of rumor propagation. This model accounts for the incubation period and considers the spatial influence on rumor dissemination. Subsequently, through an analysis of the corresponding characteristic equations, we determine the local stability conditions of the equilibrium points with respect to a critical value <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1229_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. Our findings demonstrate that when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1229_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0 < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the non-rumor equilibrium point is asymptotically stable, precluding the existence of a rumor-spreading equilibrium. Conversely, when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1229_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0 > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the non-rumor equilibrium becomes unstable, leading to the emergence of a distinct, asymptotically stable rumor-spreading equilibrium. Finally, we present numerical simulations to vividly illustrate the theoretical results we have obtained.</p>

Stability analysis in a delayed rumor propagation model with spatio-temporal diffusion terms

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发表日期：

2025年4月30日

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F. Mortaji,
A. Abta,
H. Laarabi,
M. Rachik

摘要

The proliferation of rumors has profoundly disrupted societal cohesion and stability. However, traditional models, relying on ordinary differential equations, often overlook the spatial dimension’s impact on rumor propagation dynamics. Consequently, they fail to capture the intricate complexity of information dissemination, especially amidst rapid advancements in mobile communication technology. In this article, using partial differential equations, we introduce a novel delayed model to examine the spatio-temporal dynamics of rumor propagation. This model accounts for the incubation period and considers the spatial influence on rumor dissemination. Subsequently, through an analysis of the corresponding characteristic equations, we determine the local stability conditions of the equilibrium points with respect to a critical value $R_0$ $R_{0}$ . Our findings demonstrate that when $R_0 < 1$ $R_{0} < 1$ , the non-rumor equilibrium point is asymptotically stable, precluding the existence of a rumor-spreading equilibrium. Conversely, when $R_0 > 1$ $R_{0} > 1$ , the non-rumor equilibrium becomes unstable, leading to the emergence of a distinct, asymptotically stable rumor-spreading equilibrium. Finally, we present numerical simulations to vividly illustrate the theoretical results we have obtained.