Abstract: The paper investigates the model problem of construction and analysis of the stability of the state of social security in terms of threats to the population of the world in our time in the uncertaintly of individual cities, the conditions of stabilization. As a tool, a modification of asymptotic methods and a model proposed by difference equations with random Markov coefficients are proposed, because they most adequately reflect the reality with jumps in the number of patients. The study is the beginning of a series of works devoted to a thorough study of all factors of social security with its presentation as a function of many variables. Each variable is a separate threat to social security. The ultimate goal of theoretical research in the narrow sense is to build integrated indicators of the components of social security threats for their measurement and evaluation. The purpose in a broad sense is its practical application on the basis of the received indicators, formulation in the form of algorithms of 57 recommendations to state institutions of ways of prevention of social shocks. To propose a concept of the ability of society to preserve its essence in the uncertain conditions of constant change and to maintain a stable state of society in some areas and as a whole. The apparatus of difference equations with random coefficients is very well suited for modeling almost all social security threats. We build and study mathematical models related to measuring the number of patients. The obtained conditions of stability explain the state of society. It is also possible to formulate courses of action for practitioners using sustainability conditions. Understanding the new conditions of coexistence in the world during a pandemic puts in the first place a variety of research for the sake of MAN. Therefore, the work is a relevant scientific study, has theoretical and practical value.