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On a Model of Interacting Particles of Two Types Generalizing the Bartlett–McKendrick Epidemic Process
A. N. Startsev Romanovskii Mathematical Institute of the National Academy of Sciences of Uzbekistan
Abstract:
A closed system (with respect to the number of particles) of interacting particles of two types
$A$ and
$B$ is considered. Each particle of type
$B$ possesses an amount of “energy,” while particles of type
$A$ are able to absorb the energy at the moments of interaction (occurring with unit intensity) and have a susceptibility threshold. If the total amount of the absorbed “energy” by a particle of type
$A$ attains the susceptibility threshold, then the particle transforms into a particle of type
$B$. A particle of type
$B$ that has exhausted the reserve of its “energy” dies. The process terminates if the system consists of particles of a single type only. Under the condition that the system has initially a large number of particles of both types, a class of limit laws is described for the number of particles
$\nu$ which changed their type given that the susceptibility thresholds of particles of type
$A$ are specified by independent exponentially distributed random variables with parameter 1, and given that the moments when particles of type
$B$ lose “energy” are arbitrary identically distributed random variables being independent of the previous random variables.
Keywords:
particles, interaction, change of type, non-Markov models, order statistics, boundary problems, limit theorems. Received: 04.02.1999
DOI:
10.4213/tvp3896
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