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We study a dynamics of the epidemiological infection spreading at differentvalues of the risk factor $\beta$ (a control parameter) with the using ofdynamic Monte Carlo approach (DMC). In our toy model, the infection transmitsdue to contacts of randomly moving individuals. We show that the behavior ofrecovereds critically depends on the $\beta$ value. For sub-critical values$\beta<\beta_{c}\sim 0.6$, the number of infected cases asymptoticallyconverges to zero, such that for a moderate risk factor the infection maydisappear with time. Our simulations shown that over time, the properties ofsuch a system asymptotically become close to the critical transition in 2Dpercolation system. We also analyzed an extended system, which includes twoadditional parameters: the limits of taking on/off quarantine state. It isfound that the early quarantine off does result in the irregular (with positiveLyapunov exponent) oscillatory dynamics of infection. If the lower limit of thequarantine off is small enough, the recovery dynamics acquirers acharacteristic nonmonotonic shape with several damped peaks. The dynamics ofinfection spreading in case of the individuals with immunity is studied too.

Is it possible to suspend the spread of an epidemic infection? The dynamic Monte Carlo approach