On a Model of Interacting Particles of Two Types Generalizing the Bartlett–McKendrick Epidemic Process

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.