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The probability generating function (pgf) B ⁢ (s) of the limiting conditional law (LCL) of a subcritical Markov branching process (Z t : t ≥ 0) (MBP) has a certain integral representation and it satisfies B ⁢ (0) = 0 and B ′ ⁢ (0) > 0 . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 (Z t = j) have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL. [ABSTRACT FROM AUTHOR]

The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, S (t) and R (t) , respectively. This feature allows a substantially complete elucidation of qualitative properties. The model exhibits three modes of behaviour classified in terms of the sign of − S ′ (0) , the initial value of the epidemic curve. Model behaviour is similar to that of the SIS model if S ′ (0) > 0 and to the SIR model if S ′ (0) < 0 . The separating case is completely soluble and S (t) is constant-valued. Long-term outcomes are determined for all cases, together with determination of the rate of convergence. Determining the shape of the epidemic curve motivates an investigation of curvature properties of all three state functions and quite complete results are obtained that are new, even for the SIR model. Finally, the second threshold theorem for the SIR model is extended in refined and generalised forms. [ABSTRACT FROM AUTHOR]

Concepts of infinitely divisible distributions are reviewed and applied to mutant number distributions derived from the Lea-Coulson and other models which describe the Luria-Delbrück fluctuation test. A key finding is that mutant number distributions arising from a generalised Lea-Coulson model for which normal cell growth is non-decreasing are unimodal. An integral criterion is given which separates the cases of a mode at the origin, or not. [ABSTRACT FROM AUTHOR]