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We introduce a discrete delayed exponential depending on sequence of matrices. This discrete matrix gives a representation of a solution to the Cauchy problem for a discrete linear system with pure delay with sequence of matrices. We discard the commutativity condition used in recent works related to the representation of solutions for discrete delay linear systems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a heroin epidemic model with distributed time delays. The basic reproduction number for the model is identified and the threshold property of is established. It is shown that drug-free equilibrium is globally asymptotically stable if . When , there is a disease endemic equilibrium which is locally asymptotically stable, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained by which the eventual lower bound of the drug user individuals can be computed. [Copyright &y& Elsevier]