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Zbl 1156.91460
Kawachi, Kazuki
Deterministic models for rumor transmission. (English)
[J] Nonlinear Anal., Real World Appl. 9, No. 5, 1989-2028 (2008). ISSN 1468-1218

Summary: We consider deterministic models for the transmission of a rumor. First, we investigate the age-independent case and introduce four models, which are classified according to whether the population is closed or not and whether the rumor is constant or variable. After formulating the models as finite-dimensional ODE systems, we show that the solutions converge to an equilibrium as $t\rightarrow \infty $. Next, we investigate a model for the transmission of a constant rumor in an age-structured population with age-dependent transmission coefficients. We formulate the model as an abstract Cauchy problem on an infinite-dimensional Banach space and show the existence and uniqueness of solutions. Then, under some appropriate assumptions, we examine the existence of its nontrivial equilibria and the stability of its trivial equilibrium. We show that the spectral radius $R_0:= r(\tilde T)$ for some positive operator $\tilde T$ is the threshold. We also show sufficient conditions for the local stability of the nontrivial equilibria. Finally, we show that the model is uniformly strongly persistent if $R_{0}>1$.

MSC 2000:: *91D10 Models of societies etc.
92D30 Epidemiology
34C60 Applications of qualitative theory of ODE
34K30 Functional-differential equations in abstract spaces
91D30 Social networks

Keywords: rumor transmission; threshold condition; age-structured population; rumor-free equilibrium; rumor-endemic equilibrium; global stability; local stability; uniform strong persistence

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Zentralblatt MATH Berlin [Germany]