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On the duration of a Maki-Thompson epidemic
1Stockholm University, Department of Statistics, Sweden.
Abstract:
In a Maki-Thompson model for the spread of a rumor, it is assumed that a person tells a rumor to anyone he meets until he encounters another person who has already heard it. If the population is large, it has been proved that, in the end, approximately 80% of the population will know the rumor. In this paper we will derive the (asymptotic) mean of the time it takes until the rumor dies away. It is shown that this time grows rather slowly with population size. In fact, if the population consists of W persons, and if a person meets in the mean mu persons per time unit, then the asymptotic mean duration is approximately [2.68 ln(W) + 1.34]/mu time units.
Insights
In the Maki-Thompson rumor model, approximately 80% of a large population eventually hears a rumor. The study derives the average time for a rumor to spread and die out, finding it grows slowly with population size.
Area of Science:
- Mathematical modeling
- Epidemiology
- Social dynamics
Background:
- The Maki-Thompson model describes rumor propagation in a population.
- Previous work established that ~80% of a large population eventually learns a rumor.
Purpose of the Study:
- To derive the asymptotic mean duration of a rumor's spread in the Maki-Thompson model.
- To analyze the relationship between rumor duration and population size.
Main Methods:
- Mathematical derivation of the asymptotic mean time for rumor extinction.
- Analysis of the Maki-Thompson model under large population assumptions.
Main Results:
- The asymptotic mean duration a rumor persists is derived.
- Rumor duration grows slowly with population size (W).
- The formula for mean duration is approximately [2.68 ln(W) + 1.34]/μ time units, where μ is the average number of contacts per person per time unit.
Conclusions:
- The time it takes for a rumor to die out increases slowly with population size.
- The derived formula provides a quantitative estimate for rumor persistence duration in large populations.