A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes

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Abstract
An extended version, called collective, of the randomized Reed-Frost processes is considered where each infective during his survival time fails to transmit the infection within any given set of susceptibles with a probability depending only on the size of that set. Our purpose is to provide a unified analysis of the distribution of the final size and severity, the two main components of the cost generated by the infection process. The method developed relies on the construction of a family of martingales and the use of a family of polynomials studied recently by the authors (Lefèvre and Picard (1990)). The results generalize a number of earlier ones and are derived in a more direct and systematic way than before.