This paper is concerned with the standard bivariate death process as well as 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 population state at any point in time, and to highlight the actual nature of the solution. Firstly, using a martingale technique, a central system of relations with two indices for the temporal state distribution will be derived. A remarkable property is that for all the models under consideration, these relations exhibit a similar algebraic structure. Then, this structure will be exploited by having recourse to a theory of Abel-Gontcharoff pseudopolynomials with two indices. This theory generalizes the univariate case examined in a preceding paper and is briefly introduced in the Appendix.

The paper is concerned with the final state and severity of a number of SIR epidemic models in finite populations. Two different classes of models are considered, namely the classical SIR Markovian models and the collective models introduced recently by the authors. First, by applying a simple martingale argument, it is shown that in both cases, there exists a common algebraic structure underlying the exact law of the final state and severity. Then, a unified approach to these statistics is developed by exploiting the theory of Abel-Gontcharoff pseudopolynomials (presented in a preceding paper).