Mathematical modelling of contagious disease outbreaks is discussed with the help of the SIR (susceptible, infected and recovered) model of the epidemic. In the analysis, governing equations of the SIR model are solved by a numerical technique called Euler’s method and physical interpretations are exhibited through tables and graphs. It is found that a disease outbreak is sudden and infection reached its peak in 13 days and consequence 78 percent of the total population is infected with the disease. This mathematical modelling is useful for predicting the disease outbreak caused by contagious diseases such as SARS, Zika, Flu, Covid-19, and monkeypox etc.