A point process model with stochastic intensities for a branching population of two dependent types

We study a point process model with stochastic intensities for a particular branching population of individuals of two types. Type-I individuals immigrate into the population at the times of a Poisson process. During their lives they generate type-II individuals according to a random age dependent birth rate, which themselves may multiply and die. Living type-II descendants increase the death intensity of their type-I ancestor, and conversely, the multiplication and dying intensities of type-II individuals may depend on the life situation of their type-I ancestor. We show that the probability generating function of the marginal distribution of a type-I individual's life process, conditioned on its individual infection and death risk, satisfies an initial value problem of a partial differential equation, and derive its solution. This allows for the determination of additional distributions of observable random variables as well as for describing the complete population process.