Age-Dependent Branching Processes for Surveillance of Vaccine-Preventable Diseases with Incubation Period

The purpose of this paper is to review the recent results of the authors in the area of infectious disease modeling by means of branching stochastic processes. This is a new approach involving age-dependent branching models, which turned out to be more appropriate and flexible for describing the spread of an infection in a given population, than discrete time ones. Concretely, Bellman–Harris and Sevast'yanov's branching processes are investigated. It is justified that the proposed models are proper candidates as models of infectious diseases with incubation period like measles, mumps, avian flu, etc. It is worth to notice that in general the developed methodology is applicable to the diseases that follow the so-called SIR (susceptible–infected–removed) scheme in terms of epidemiological models. Two policies of extra-vaccination level are proposed and compared on the ground of simulation examples

Poisson Random Measures and Noncritical Multitype Markov Branching Processes

We investigate noncritical multi-type Markov branching processes with immigration generated by Poisson measures. Limiting distributions are obtained when the rates of the Poisson measures are asymptotically equivalent to exponential or regularly varying functions. In particular, results analogous to a strong LLN are presented, and limiting normal distributions are obtained when the rates increase. When the rates decrease, then conditional limiting distributions are established. A stationary limiting distribution is obtained when the mean Poisson measure grows linearly. The asymptotic behaviour of the first and second moments of the processes is also investigated.NFSR of the Ministry of Education and Science of Bulgaria, grant No. KP–06-H22/3

Limit Theorems for Age-Dependent Branching Processes with State-Dependent Immigration

[Slavtchova-Bojkova Maroussia N.; Славчова-Божкова Маруся Н.]; [Yanev Nickolay M.; Янев Николай М.]We consider a model of Bellman-Harris branching processes with immigration only in the state zero (BHIO) which permits in addition an immigration component of i.i.d. BHIO processes entering the population at i.i.d. times of an independent renewal process (BHIOR). Asymptotic properties and limit theorems are proved in поп-critical cases. 60J8