In the last decade of April 2020, the second coronavirus epidemic wave in Iran has bloomed. The new wave has started in the vicinity of the critical point, marked by approximately 44,000 infections, where the rate of increase of the primary epidemic that appeared in Iran in mid-February 2020 was the highest. Today, this secondary wave almost has doubled the peak of the primary, and, passing the epidemic threshold of about 70,000 total cases in early June, generated the new third epidemic wave developing unpredictably and dynamically. The purpose of this work was to call into use a simple dynamical system represented by the discrete logistic equation with unknown parameters to predict secondary waves using the official statistical data. The mathematical modelling reveals the secondary epidemic waves in Sweden, the United States, Ukraine, Serbia, Romania, Czech Republic, Portugal, Luxembourg, Poland, and Ecuador. Also, the second waves appear in Russia and other countries. Despite many individual differences in the epidemic spread in different countries, we have traced regularity in the rise of secondary waves. The beginning of each new wave, if focusing on the number of total cases, practically coincides with the time of the maximum growth rate of the previous early epidemic. Thus, the passing through the threshold of the current wave should be the most responsible for strict observance of the rules of self-isolation and other sanitary standards.