The currently active COVID-19 pandemic has increased, among others, public interest in the computational techniques enabling the study of disease-spreading processes. Thus far, numerous approaches have been used to study the development of epidemics, with special attention paid to the identification of crucial elements that can strengthen or weaken the dynamics of the process. The main thread of this research is associated with the use of the ordinary differential equations method. There also exist several approaches based on the analysis of flows in the Cellular Automata (CA) approach.In this paper, we propose a new approach to disease-spread modeling. We start by creating a network that reproduces contacts between individuals in a community. This assumption makes the presented model significantly different from the ones currently dominant in the field. It also changes the approach to the act of infection. Usually, some parameters that describe the rate of new infections by taking into account those infected in the previous time slot are considered. With our model, we can individualize this process, considering each contact individually.The typical output from calculations of a similar type are epidemic curves. In our model, except of presenting the average curves, we show the deviations or ranges for particular results obtained in different simulation runs, which usually lead to significantly different results. This observation is the effect of the probabilistic character of the infection process, which can impact, in different runs, individuals with different significance to the community. We can also easily present the effects of different types of intervention. The effects are studied for different methods used to create the graph representing a community, which can correspond to different social bonds.We see the potential usefulness of the proposition in the detailed study of epidemic development for specific environments and communities. The ease of entering new parameters enables the analysis of several specific scenarios for different contagious diseases.