A simplistic reaction-diffusion model is undertaken in the present work to mathematically explore the spatio-temporal development of concentration of indoor aerosols containing infectious COVID-19 respiratory virus nuclei. Extracting exact solutions of concentration field under the influence of several physical parameters is preferred rather than adopting a more realistic complex model requiring time-consuming numerical simulations. Even though the proposed model is not sophisticated, the analytical solutions can provide quick prediction of the probability of contracting the virus in a ventilated closed room. Moreover, from the obtained elementary solutions of the viral concentration field, it is easy to analyze its spatio-temporal evolution and final equilibrium state. Formulae enable us to estimate the time to get infected and the risk of getting infected within an elapsed time under various physical operative situations involving a uniform infectious particle mixture ejection into the medium, wearing a face mask with a well-defined efficiency parameter and taking into account a localized source of infection. One of the essential conclusion from the current research is that less aerosols carrying COVID-19 particles are as a result of good indoor ventilation conditions, of removing the medium air through windows (or other exits) and of wearing masks of high efficiency. Moreover, the risk and probability of being caught by the indoor COVID-19 disease increases in time, particularly in the downstream of a localized infectious person. The results can be beneficial to understand and take necessary safety considerations against the infection risk in closed public or governmental environments.