# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $t\to \infty$. 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 ${R}_{0}:=r\left(\stackrel{˜}{T}\right)$ for some positive operator $\stackrel{˜}{T}$ 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