Geodesic
On a Model of Interacting Particles of Two Types Generalizing the Bartlett--McKendrick Epidemic Process
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 463-482

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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.
Mots-clés : particles
Keywords: interaction, change of type, non-Markov models, order statistics, boundary problems, limit theorems. 
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     author = {A. N. Startsev},
     title = {On a {Model} of {Interacting} {Particles} of {Two} {Types} {Generalizing} the {Bartlett--McKendrick} {Epidemic} {Process}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {463--482},
     publisher = {mathdoc},
     volume = {46},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2001_46_3_a3/}
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A. N. Startsev. On a Model of Interacting Particles of Two Types Generalizing the Bartlett--McKendrick Epidemic Process. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 3, pp. 463-482. http://geodesic.mathdoc.fr/item/TVP_2001_46_3_a3/