Multitype branching processes with disasters II: Total sojourn time and number of deaths

We consider a multitype Markov branching process under the influence of disasters occurring as a renewal process. For the branching particles in the system, the disasters may lead to possible death or mutation to other types of particles. For such a process, the total mean sojourn time of all the particles in the system is analysed using the regeneration point technique. Explicit expressions of the mean matrices of the total sojourn time are obtained for the decomposable process and the Markovian models. The analysis gains importance in studying the total duration of the toxic effects of drugs in cellular systems. Expression for the mean matrix of the number of particles that die without offspring during the time interval (0, t) is also obtained. An interesting relationship connecting the mean of the total sojourn time of all particles, the mean of the number of particles which die during the time interval (0, t) and the mean number of particles which are alive at time t, is established.