We study an extended and modified SIR (Susceptible-Infectious-Recovered) model of epidemic spread in which susceptible agents during interactions with infectious neighbors are exposed to the disease and can consequently become infectious. The studied model is extended to include heterogeneity of interactions which is modeled assuming random character of the dose accumulated by susceptible agents in every interaction with infectious neighbors. When the accumulated exposition is larger than the individual's resistance, an agent becomes infectious and consequently introduces a new source of an epidemic which is capable of passing the disease further. We study statistical properties characterizing the course of an epidemic. The examination of the modified SIR model reveals a possible 'resonant activation'-like behavior of the system in the duration of the epidemic outbreak and a possible bistable behavior of the model with accumulated exposition. Furthermore, the linear scaling of the duration of the epidemic with the system size for a wide range of the model parameters is recorded. Within the model, the duration of the epidemics is defined as the maximal length of the epidemics experienced by a single individual because of the fact that epidemic outbreak ends when the last infectious individual moves to the recovered class. Consequently statistical properties of the epidemics are of the extreme statistics type. |