Белоусова Е. П.

The relevance of the topic. There are a considerable number of macroeconomics models that try to explain and predict long-term trends and dynamics of important economic indicators. One of these is the Richard Goodwin model, which was proposed in the early 1970s. It is based on the idea of the relationship between wages, unemployment and inflation. From a mathematical point of view, the Goodwin macroeconomic model is a system containing a small parameter. It is worth noting that such systems are very difficult to study. Here are a few more models for comparison. Exactly. The AS-AD model combines the movement of aggregated supply and demand for goods and services. It describes the relationship between the price level and real GDP and pays great attention to the welfare of consumers and companies, preferences and technology changes. The IS-LM model analyzes the relationship between the level of interest rates, investments and income in the economy. Unlike the Goodwin model, the IS-LM model focuses mainly on the financial and monetary aspects of the economy. The Solow growth model describes the process of economic growth in the long term and focuses on capital formation and technological progress. Classical economic theory assumes the cyclical nature of ongoing processes. This article presents their mathematical interpretation and shows how mathematical methods can be used to analyze and forecast the dynamics of economic indicators. In this context, the relevance of the task under consideration is obvious today. Goal. Determining the level of GDP that will ensure stability in the country's economy and in the lives of its citizens for a long time. Methodology. Application of the Poincare method to solve the formulated problem. Results and conclusions. The quantitative size of the limit cycle is determined, which reflects the periodic processes occurring in the Goodwin economic system, according to the input parameters. The stability of these processes has been proven. The sustainable cyclical nature of the economic process, which ensures the effective development of the state, is not always achieved. To do this, a number of rather stringent requirements must be met. In addition, it is also quite difficult to draw conclusions about the scope of this cycle from a practical point of view. But if it succeeds, then it is possible to make long-term forecasts regarding the development and the level of the main economic indicators that this development will ensure. The scope of application. Economic forecasting.

Mathematical modeling of diseases is a powerful tool for investigating the mechanisms of disease spread. Epidemiological models serve as a basis for predicting and estimating the spread of the disease. To mitigate and oversee the epidemic, it is important to study and develop qualitative and adequate mathematical models of the epidemic.

Since infectious diseases in a large population lead to epidemics, there is naturally a problem - how to combat this phenomenon by spending the least amount of effort and financial investment. The paper attempts to analyze this process for several types of diseases in the context of the growing epidemic by linear law.

The tasks of optimal control are choosing the most advantageous modes of managing complex dynamic systems. In this paper, we give a solution to the optimal control problem proposed by E. A. Andreeva and V. M. Tsirulova in the training manual «Calculus of Variations and Optimization Methods». To solve the problem, the Pontryagin maximum principle was applied.

Optimal control problem arise naturally in the economy when considering dynamic models of optimal allocation of resources. Often these tasks relate to the study of the processes of economic growth. In recent years, interest in tasks of optimal control in the economy has increased dramatically. This is called as the improvement of the general interest in dynamic models in economy and the emergence of new models of optimal economic growth - particularly models of economic development based on knowledge and models that take into account the global implications of the development of the world economy. In any economy, necessarily choose between current demand (consumption) and the future (capital investments). Despite the fact that higher consumption is always preferable to low, this level of consumption means lower capex, which entails a reduction in output in the future, and as a consequence of the downgrading of future consumption. It is therefore the task of choosing one or another policy in the field of consumption. One extreme is politics, in which as much as possible full met current needs, even if it threatens a catastrophe in the future due to the lowering of the level of consumption. At the other extreme is a policy, which seek to limit current consumption in order to increase capital and level of consumption in the future. Many functions of time for consumption, capital formation and output occurs when choosing between consumption and capital accumulation. From this multitude of possible trajectories of growth, you must select one. First, you should get an estimate of the ratio of the current and future level of consumption. Once the assessment is completed, immediately raises the task of choosing optimal growth path, that is, the problem of optimal economic growth.

The article studies the problem of influence of an advertising campaign on the processes of production and realization of a homogeneous product with a fixed-price sale. Compiled and examined the corresponding mathematical model, which describes how the effectiveness of the advertisement, and the effect of it annoy.