We study a stochastic model for the spread of two pathogen strains—termed type 1 and type 2—among a homogeneously mixing community consisting of a finite number of individuals. In the model, we assume partial cross‐immunity, exogenous streams of infection, and that the degree of severity of a newly infective individual depends on who this infective individual was infected by. The aim is to characterize the joint probability distribution of the numbers M1 and M2 of type‐1 and type‐2 infections suffered by a focal individual during an outbreak of the disease. We present iterative procedures for computing the probability mass function of (M1,M2) under the assumption that the initial state of the focal individual is known, and a numerical study of the model is performed to investigate the influence of certain key parameters on the spread of resistant bacteria in hospitals.

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A Markov-modulated framework is presented to incorporate correlated inter-event times into the stochastic susceptible-infectious-recovered (SIR) epidemic model for a closed finite community. The resulting process allows us to deal with non-exponential distributional assumptions on the contact process between the compartment of infectives and the compartment of susceptible individuals, and the recovery process of infected individuals, but keeping the dimensionality of the underlying Markov chain model tractable. The variability between SIR-models with distinct level of correlation is discussed in terms of extinction times, the final size of the epidemic, and the basic reproduction number, which is defined here as a random variable rather than an expected value.1. Introduction. The SIR-model is a well-studied epidemic model that, together with its generalizations, has been widely applied to infectious diseases such as measles, chickenpox, or mumps, among other situations where infection confers (typically lifelong) immunity; a good discussion about SIR-models analyzed from deterministic and stochastic perspectives can be found in the lucid texts by Allen [1, 2], Andersson and Britton [8], Bailey [16] and Keeling and Rohani [36]. The SIR-model is first analyzed in depth by Kermack and McKendrick [37] in 1927 in order to study the evolution of a disease in a closed community of finite size where, at time t, individuals are classified into three categories: S(t) susceptibles, I(t) infectives, and R(t) removed individuals. The general description in [37] is related to an homogeneously mixed population (i.e., any infective can infect any susceptible with equal probability), where the infection and recovery rates of a given infective depend on the total time that this individual has been infected for. The analytical