On the algebraic structure in Markovian processes of death and epidemic types

This paper is concerned with the standard bivariate death process as well a s with some Markovian modifications and extensions of the process that are of interest especially in epidemic modeling. A new and powerful approach is developed that allows us to obtain the exact distribution of the populatio n state at any point in time, and to highlight the actual nature of the sol ution. Firstly, using a martingale technique, a central system of relations with two indices for the temporal state distribution will be derived. A re markable property is that for all the models under consideration, these rel ations exhibit a similar algebraic structure. Then, this structure will be exploited by having recourse to a theory of Abel-Gontcharoff pseudopolynomi als with two indices. This theory generalizes the univariate case examined in a preceding paper and is briefly introduced in the Appendix.