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Deterministic models for rumor transmission. (English) Zbl 1156.91460
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:
 91D10 Models of societies, social and urban evolution 92D30 Epidemiology 34C60 Qualitative investigation and simulation of models (ODE) 34K30 Functional-differential equations in abstract spaces 91D30 Social networks
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