  We study a dynamics of the epidemiological infection spreading at different
values of the risk factor $\beta$ (a control parameter) with the using of
dynamic Monte Carlo approach (DMC). In our toy model, the infection transmits
due to contacts of randomly moving individuals. We show that the behavior of
recovereds critically depends on the $\beta$ value. For sub-critical values
$\beta<\beta_{c}\sim 0.6$, the number of infected cases asymptotically
converges to zero, such that for a moderate risk factor the infection may
disappear with time. Our simulations shown that over time, the properties of
such a system asymptotically become close to the critical transition in 2D
percolation system. We also analyzed an extended system, which includes two
additional parameters: the limits of taking on/off quarantine state. It is
found that the early quarantine off does result in the irregular (with positive
Lyapunov exponent) oscillatory dynamics of infection. If the lower limit of the
quarantine off is small enough, the recovery dynamics acquirers a
characteristic nonmonotonic shape with several damped peaks. The dynamics of
infection spreading in case of the individuals with immunity is studied too.
