The conditions and the results of predicting epidemics of influenza in 1971-1985 years using a mathematical model of L.A. Rvachev and automated system developed at the Department of Epidemiology and General Cybernetics Institute of Influenza of the Ministry of Health of the USSR (head Yu. G. Ivannikov). It is shown that a forecast of daily epidemic influenza morbidity population of 100 cities in the country with a lead of up to 3 months was quite accurate both in terms of morbidity, and in time the peak of the epidemic. The causes of the difficulties inpredicting influenza epidemics in the end of 80s and 90s years, and gives recommendations for the resumption of the mathematical prediction of influenza epidemics in Russia at present