Until recently, geographical factors that have a significant impact on the spread of diseases have been studied relatively little. The validity of the assumption of homogeneous mixing of the population in a small town or village has long been questioned, although it is quite acceptable as a first approximation to accept that the movements of sources of infection are random and in many ways resemble the movement of particles in a colloidal solution. Nevertheless, it is necessary, of course, to have some idea of what effect the presence of a large number of susceptible individuals at sites quite long distances from any given source infections.
The deterministic model, due to D. Kendall, assumes the existence of an infinite two-dimensional continuum of the population, in which there are about 0 individuals per unit area. Consider the area surrounding the point P, and assume that the numbers of susceptible, infected and removed from the collective individuals are equal, respectively. The x, y, and z values can be functions of time and position, but their sum must be equal to one. The basic equations of motion, similar to system (9.18), have the form
where is the spatially weighted mean
Let and be constants, be an area element surrounding the point Q, and be a non-negative weighting factor.
Let us assume that the initial concentration of diseases is evenly distributed in some small area surrounding the initial focus. Note also that the factor o is explicitly introduced into the Rohu product so that the infection rate remains independent of the population density. If y remained constant on the plane, then the integral (9.53) would surely converge. In this case, it would be convenient to require that
The described model makes it possible to advance mathematical research quite far. It can be shown (with one or two caveats) that a pandemic will cover the entire plane if and only if the population density exceeds a threshold value . If a pandemic has occurred, then its intensity is determined by the single positive root of the equation
The meaning of this expression is that the proportion of individuals who eventually fall ill in any area, no matter how far it is from the original epidemic focus, will be no less?. Obviously, this Kendall pandemic threshold theorem is similar to the Kermack and McKendrick threshold theorem, in which the spatial factor was not taken into account.
It is also possible to build a model for the following special case. Let x and y be the spatial densities of susceptible and infected individuals, respectively. If we assume that the infection is local and isotropic, then it is easy to show that the equations corresponding to the first two equations of system (9.18) can be written as
where are not spatial coordinates] and
For the initial period, when it can be approximately considered a constant value, the second equation of the system (9.56) takes the form
This is the standard diffusion equation, the solution of which is
where the constant C depends on the initial conditions.
The total number of infected individuals outside the circle of radius R is
Hence,
and if , then . The radius corresponding to any selected value grows at a rate of . This value can be considered as the rate of spread of the epidemic, and its limiting value for large t is equal to . In one case of a measles epidemic in Glasgow for almost half a year, the spread rate was about 135 m per week.
Equations (9.56) can easily be modified to take into account the migration of susceptible and infected individuals, as well as the emergence of new susceptible individuals. As in the case of recurring epidemics discussed in Sect. 9.4, an equilibrium solution is possible here, but small oscillations decay just as quickly or even faster than in the non-spatial model. Thus, it is clear that in this case the deterministic approach has certain limitations. In principle, one should, of course, prefer stochastic models, but usually their analysis is associated with enormous difficulties, at least if it is carried out in a purely mathematical way.
Several works have been done to model these processes. Thus, Bartlett used computers to study several successive artificial epidemics. The spatial factor was taken into account by the introduction of the cell grid. Within each cell, typical non-spatial models were used for continuous or discrete time, and random migration of infected individuals between cells sharing a common boundary was allowed. Information was obtained on the critical volume of the population, below which the epidemic process attenuates. The main parameters of the model were derived from actual epidemiological and demographic data.
Recently, the author of this book undertook a number of similar studies in which an attempt was made to construct a spatial generalization of stochastic models for the simple and general cases considered in Sec. 9.2 and 9.3. Suppose we have a square lattice, each node of which is occupied by one receptive individual. The source of infection is placed in the center of the square and such a process of the chain-binomial type for discrete time is considered, in which only individuals directly adjacent to any source of infection are exposed to the risk of infection. These can be either only four nearest neighbors (Scheme 1), or also individuals located diagonally (Scheme 2); in the second case, there will be a total of eight individuals lying on the sides of the square, the center of which is occupied by the source of infection.
It is obvious that the choice of scheme is arbitrary, however, in our work, the latter arrangement was used.
At first, a simple epidemic with no cases of recovery was considered. For convenience, a grid of limited size was used, and information about each individual's condition (i.e., whether they are susceptible to or a source of infection) was stored on a computer. The modeling process kept a running record of changes in the status of all individuals and counted the total number of new cases in all squares with the original source of infection in the center. The machine's memory also recorded the current values of the sum and the sum of the squares of the number of cases. This made it fairly easy to calculate mean values and standard errors. The details of this study will be published in a separate article, but here we will note only one or two particular features of this work. For example, it is clear that with a very high probability of sufficient contact, an almost deterministic spread of the epidemic will take place, in which at each new stage in the development of the epidemic a new square with sources of infection will be added.
At lower probabilities, there will be a truly stochastic spread of the epidemic. Since each source of infection can infect only eight of its nearest neighbors, and not the entire population, one would expect that the epidemic curve for the entire lattice would not increase as sharply as if the entire population were homogeneously mixed. This prediction does indeed come true, and the number of new cases increases more or less linearly over time until edge effects start to kick in (because the lattice has a limited extent).
Table 9. Spatial stochastic model of a simple epidemic built on a 21x21 lattice
In table. 9 shows the results obtained for a lattice with one initial source of infection and a probability of sufficient contact equal to 0.6. It can be seen that between the first and tenth stages of the epidemic, the average number of new cases increases by about 7.5 each time. After that, the edge effect begins to dominate, and the epidemic curve drops sharply down.
One can also determine the average number of new cases for any given grid point and thus find the epidemic curve for that point. It is convenient to average over all points lying on the border of the square in the center of which the source of infection is located, although the symmetry in this case will not be complete. Comparing the results for squares of different sizes gives a picture of an epidemic wave moving away from the original source of infection.
Here we have a sequence of distributions whose modes increase in a linear progression and the variance increases continuously.
A more detailed study of the general type of epidemic was also carried out, with the removal of infected individuals. Of course, these are all very simplified models. However, it is important to understand that they can be significantly improved. To account for population mobility, it must be assumed that susceptible individuals also become infected from sources of infection that are not their immediate neighbors. You may need to use some kind of weighting factor here, depending on the distance. The modifications that will need to be introduced into the computer program in this case are relatively small. At the next stage, it may be possible to describe in this way real or typical populations with the most diverse structure. This will open up the possibility of assessing the epidemiological state of real populations in terms of the risk of various types of epidemics.
Model classification
Paragraph teaching elements:
1. Appointment of models. The way the models are implemented.
2. Abstract model. Real model.
3. Model description language. Model building method.
4. Likeness. direct likeness. indirect similarity. conditional likeness.
5. Text model. Graphic model. Mathematical model.
6. Analytical model. Experimental model. Spatial model.
7. Conformity of models to the original. The finiteness of the models is the simplification, the proximity of the models.
The purpose of the models allows us to divide the entire diverse set of models into three main types according to their purpose: cognitive , pragmatic , sensual ), For various objects(Fig. 1.3).
Fig.1.3 Classification of models
cognitive models are a form of organization and representation of knowledge, a means of connecting new knowledge with existing ones. Therefore, when a discrepancy between the model and reality is detected, the task of eliminating this discrepancy by changing the model arises. Cognitive activity is based on the approximation of the model and reality (Fig. 1.4a).
Pragmatic models are a means of organizing practical actions, a means of management, a way of presenting exemplary actions or their results.
Rice. 1.4. Differences between cognitive (a) and pragmatic model (b)
The use of pragmatic models is to, when discrepancies are found between the model and reality, to direct efforts to change reality in such a way as to bring reality closer to the model.
Examples of pragmatic models are plans, programs, examination requirements, instructions, manuals, etc. (Fig. 1.4b).
sensual models serve to satisfy the aesthetic needs of a person (work of art).
Another principle for classifying the goals of modeling is the division of models into static and dynamic.
Static models reflect the specific state of an object ( snapshot). If you need to study the differences between the states of the system, dynamic models are built.
Models consciously created by the subject (man) are embodied from two types of materials suitable for their construction - the means of the surrounding world and the means of the human consciousness itself.
On this basis, the models are divided into abstract (ideal, mental, symbolic) and real (material, real).
Abstract models are ideal constructions built by the means of thinking. They are distinguished by the language of description and the method of construction (Fig. 1.3).
According to the method of construction, abstract models are divided into analytical (theoretical), formal (experimental) and combined . Analytical models are built on the basis of data on internal structure object and on the basis of physical laws that describe the processes occurring in it.
Formal models are built on the basis of data from experimental studies, during which relationships are established between input actions and (output) parameters of the state of the object.
Combined models use the principle of refinement in the experiment of the parameters of the structure and regularities of the analytical model.
According to the type of description language, symbolic models are divided into text (verbal) graphic (drawings, diagrams), mathematical And combined .
So that some material construction can be a mapping, i.e. replaced in some respect the original, between the model and the original must be established similarity relation .
We will distinguish three types of similarity: direct, indirect and conditional (Fig. 1.3).
direct likeness May be spatial (models of ships, aircraft, dummies, etc.) and physical . Physical similarity refers to phenomena in geometrically similar systems, in which, in the process of their functioning, the ratios of the same-named physical quantities characterizing them at similar points are a constant value (similarity criteria). Example physical model- Testing a mock-up aircraft in a wind tunnel.
The second type of similarity, in contrast to direct similarity, is called indirect . Indirect similarity between the original and the model is established not as a result of their physical interaction, but objectively exists in nature, is found in the form of a match or sufficient proximity of their abstract models, and after that they are used in the practice of real modeling. An example of indirect similarity is analogy between physical (phase) variables (Table 1.1).
Table 1.1
The laws of mechanical, thermal, electrical processes are described by the same equations: the difference lies only in the different physical interpretation of the variables included in the equations.
As a result, it becomes possible not only to replace cumbersome experimentation with a mechanical or thermal system, but with simple experiments with electric circuit (R, L, C- circuits) or electronic model (AVM).
The role of models that have an indirect similarity to the original is very great. Clock is analogous to time. Analog and digital computing moments (material object) allows you to find a solution to any differential equation.
The third special class of real models is formed by models whose similarity to the original is neither direct nor indirect, but is established as a result of agreement. This similarity is called conditional .
Examples of conditional similarity are money (value model), traffic signs (message model), etc.
You have to deal with conditional similarity models very often. They are a way of material embodiment of abstract models, a material form in which abstract models can be transferred from one person to another, is stored until the moment of their use, i.e. to be alienated from consciousness and still retain the possibility of returning to an abstract form. This is achieved by agreement on which state of the real object is assigned to a given element of the abstract model. Such an agreement takes the form of a set of rules for constructing conditional similarity models and rules for using them.
The object model can be characterized by several features (Tables 1.2 and 1.3).
Table 1.2
Table 1.3
Thus, we have considered questions about what the model displays, from what and how it can be built, what are the external conditions for the implementation of the model's functions. But the question of the value of the modeling itself is also important, i.e. the relationship of models with the reality they display: how do models and simulated objects or phenomena differ, in what sense, and to what extent can a model be identified with the original.
There are the following main differences between the model and the original: finiteness, simplicity and approximation (adequacy).
Model finite, since it displays the original only in finite number of relationships with limited resources.
Model always simplistically displays the original due to the finiteness of the model; displaying only the main essential properties and relationships; limited means of operating with the model. Simplicity characterizes quality differences between model and original.
The model displays the original approximately. This aspect allows quantitative assessment of the difference (“more - less”, “better - worse”). The concept of model approximation is related to adequacy .
A model with the help of which the goal is successfully achieved is called adequate to this goal.
The adequacy of the model does not guarantee the requirements for the completeness, accuracy and truth of the model, but means that they are met to the extent that is sufficient to achieve the goal. Simplification and approximation of the model are necessary, inevitable, but the remarkable property of the world and ourselves is that this is sufficient for human practice.
Between the model and the original, in addition to differences, there are similarities .
The similarity is expressed, first of all, in the truth of the model. Degree truth the model can be clarified only in its practical relationship with the nature it reflects. At the same time, changing the conditions under which the comparison is carried out has a very significant effect on the result: it is precisely because of this that the existence of two contradictory, but “equally” true models of one object is possible. A striking example of this is the wave and corpuscular models of the electron.
The similarity of the model and the original depends on the combination of the true and false model types. In addition to, of course, the true content in the model, there is: 1) conditionally true (that is, true only under certain conditions); 2) presumably true (i.e. conditionally true under unknown conditions), and therefore logical. At the same time, in each specific condition, it is not known exactly what is the actual ratio of true and false in this model. The answer to this question is only practice.
However, in any case, the model is fundamentally poorer than the original, this is its fundamental property.
Concluding the consideration of the concept of “modeling”, it should be emphasized that when planning to create a system model, you need to keep in mind the following scheme (Fig. 1.5):
Fig.1.5. Evaluation of the simulation situation
Widespread in research technical systems received a method of mathematical modeling, which we consider in more detail.
Questions
1. What features form a family of models by purpose?
2. What features form a family of models according to the method of implementation?
3. What features form the types of models in similarity?
4. What is the difference between the pragmatic model and the cognitive model?
5. In what languages can models be submitted?
6. What are the types of direct similarity of material models?
7. What is the difference between real models of indirect and conditional similarity?
8. What are the signs of the difference between the model and the original?
9. What questions can be used to evaluate the simulation situation?
§ 1.1. 4. Objects of modeling and their classification
Paragraph teaching elements:
1. Signs of classification modeling objects.
2. Type, properties and methods of object research.
3. Continuous - discrete objects.
4. Stationary - non-stationary objects.
5. Concentrated - distributed objects.
6. One-dimensional, multidimensional objects.
7. Deterministic - stochastic objects.
8. Dynamic - static objects.
9. Linear, non-linear objects.
10. Analytical, identifiable, combined research methods.
11. Mathematical model.
12. Math modeling.
13. Parameters and phase model variables.
14. Model Specifications(universality, accuracy, adequacy and economy).
15. Signs of MM classification:
16. Structural - functional models;
17. Complete - macro models;
18. Analytical - algorithmic models;
Properties stationarity – non-stationarity characterize the degree of variability of the object in time.
Properties concentration – distribution characterizes objects from the point of view of the role played in their model description by the spatial extension and the final velocity of propagation in the space of physical processes.
If the spatial extension can be neglected and we can assume that the independent variable characteristic of the object is only time, then saying
t about the object with lumped parameters .
In spatially extended objects (gases, deforming bodies), it is necessary to take into account the dependence of characteristics on coordinates.
All real-life objects have the property stochasticity . Definition determinism means only the fact that, according to the conditions of the problem being solved and in relation to the properties of a particular object, random factors can be ignored.
concept dynamic the object reflects the change in the parameters of the object over time. This is due to the finite rate of accumulation of matter and energy reserves accumulated by the object.
In a static object, the connection between input and output parameters does not take into account dynamic effects.
It is very important to divide objects into linear And non-linear . The difference between them lies in the fact that the principle of superposition (position) is valid for the former, when each of the outputs of the object is characterized linear dependence from the corresponding input variables.
Objects with one output are called one-dimensional , but with several multidimensional .
The division of research methods of modeling objects into analytical ones, which are based on previously studied and described in mathematical form regularities of the object, and identifiable ones, which are built on the basis of a special experimental study, is associated with the degree of complexity of the object.
Questions for self-control and preparation for MC:
On what grounds are the objects of modeling classified?
What is the difference between deterministic objects and stochastic ones?
How can you distinguish a dynamic object from a static one?
What is typical for a continuous modeling object?
Time series models that characterize the dependence of the resulting variable on time include:
a) the model of dependence of the resulting variable on the trend component or the trend model;
b) the result dependency model. variable from seasonal component or seasonality model;
c) the model of dependence of the resulting variable on the trend and seasonal components or the model of trend and seasonality.
If economic statements reflect a dynamic (depending on the time factor) relationship of the variables included in the model, then the values of such variables are dated and called dynamic or time series. If economic statements reflect a static (relating to one period of time) relationship of all variables included in the model, then the values of such variables are usually called spatial data. And there is no need to date them. Lag variables are exogenous or endogenous variables of the economic model, dated from previous points in time and in the equation with current variables. Models that include lag variables belong to the class of dynamic models. predetermined called lag and current exogenous variables, as well as lag endogenous variables
23. Trend and spatio-temporal EM in economic planning
Statistical observations in socio-economic studies are usually carried out regularly at regular intervals and are presented in the form of time series xt, where t = 1, 2, ..., p. regression models, the parameters of which are estimated from the available statistical base, and then the main trends (trends) are extrapolated to a given time interval.
Statistical forecasting methodology involves building and testing many models for each time series, comparing them based on statistical criteria, and selecting the best of them for forecasting.
When modeling seasonal phenomena in statistical studies, two types of fluctuations are distinguished: multiplicative and additive. In the multiplicative case, the range of seasonal fluctuations changes in time in proportion to the trend level and is reflected in the statistical model by a multiplier. With additive seasonality, it is assumed that the amplitude of seasonal deviations is constant and does not depend on the level of the trend, and the fluctuations themselves are represented in the model by a term.
The basis of most forecasting methods is extrapolation associated with the spread of patterns, relationships and relationships that operate in the period under study beyond its limits, or - in a broader sense of the word - this is obtaining ideas about the future based on information related to the past and present.
The most well-known and widely used are trend and adaptive forecasting methods. Among the latter, one can single out such methods as autoregression, moving average (Box-Jenkins and adaptive filtering), exponential smoothing methods (Holt, Brown and exponential average), etc.
To assess the quality of the forecast model under study, several statistical criteria are used.
When presenting a set of observation results in the form of time series, the assumption is actually used that the observed values belong to some distribution, the parameters of which and their change can be estimated. For these parameters (usually the mean and variance, although sometimes more Full description) one can construct one of the models of the probabilistic representation of the process. Another probabilistic representation is a frequency distribution model with parameters pj for the relative frequency of observations falling within j-th interval. In this case, if no change in the distribution is expected within the accepted lead time, then the decision is made on the basis of the available empirical frequency distribution.
When forecasting, it must be borne in mind that all factors influencing the behavior of the system in the base (investigated) and forecast periods must be unchanged or change according to a known law. The first case is implemented in single-factor forecasting, the second - in multi-factor forecasting.
Multifactorial dynamic models should take into account the spatial and temporal changes of factors (arguments), as well as (if necessary) the delay in the influence of these factors on the dependent variable (function). Multivariate forecasting makes it possible to take into account the development of interrelated processes and phenomena. Its basis is a systematic approach to the study of the phenomenon under study, as well as the process of comprehending the phenomenon, both in the past and in the future.
In multivariate forecasting, one of the main problems is the problem of choosing the factors that determine the behavior of the system, which cannot be solved purely statistically, but only with the help of a deep study of the essence of the phenomenon. Here it is necessary to emphasize the primacy of analysis (comprehension) over purely statistical (mathematical) methods of studying a phenomenon. In traditional methods (for example, in the method of least squares), it is considered that the observations are independent of each other (by the same argument). In reality, there is an autocorrelation and its neglect leads to non-optimal statistical estimates, makes it difficult to build confidence intervals for regression coefficients, as well as to check their significance. Autocorrelation is determined by deviations from trends. It can take place if the influence of a significant factor or several less significant factors, but directed “in one direction”, is not taken into account, or a model that establishes a relationship between factors and a function is incorrectly chosen. To detect the presence of autocorrelation, the Durbin-Watson test is used. To eliminate or reduce autocorrelation, a transition to a random component (trend elimination) or the introduction of time into the multiple regression equation as an argument is used.
In multifactorial models, the problem of multicollinearity also arises - the presence of a strong correlation between factors, which can exist without any dependence between the function and the factors. By identifying which factors are multicollinear, it is possible to determine the nature of the interdependence between the multicollinear elements of the set of independent variables.
In multivariate analysis, along with estimating the parameters of the smoothing (studied) function, it is necessary to build a forecast for each factor (based on some other functions or models). Naturally, the values of the factors obtained in the experiment in the base period do not coincide with the similar values found from the predictive models for the factors. This difference must be explained either by random deviations, the magnitude of which is revealed by the indicated differences and must be taken into account immediately when estimating the parameters of the smoothing function, or this difference is not accidental and no prediction can be made. That is, in the problem of multivariate forecasting, the initial values of the factors, as well as the values of the smoothing function, must be taken with the corresponding errors, the distribution law of which must be determined in the appropriate analysis preceding the forecasting procedure.
24. Essence and content of EM: structural and expanded
Econometric models are systems of interrelated equations, many of whose parameters are determined by methods of statistical data processing. To date, many hundreds of econometric systems have been developed and used abroad for analytical and forecasting purposes. Macroeconometric models, as a rule, are first presented in a natural, meaningful form, and then in a reduced, structural form. The natural form of econometric equations makes it possible to qualify their substantive side, to assess their economic meaning.
To build predictions of endogenous variables, it is necessary to express the current endogenous variables of the model as explicit functions of predefined variables. The last specification, obtained by including random perturbations, is obtained as a result of mathematical formalization of economic laws. This form of specification is called structural. In general, endogenous variables are not explicitly expressed in terms of predefined variables in a structural specification.
In the equilibrium market model, only the supply variable is explicitly expressed in terms of a predefined variable; therefore, to represent endogenous variables in terms of predefined ones, it is necessary to perform some transformations of the structural form. Let's solve the system of equations for the last specification with respect to endogenous variables.
Thus, the endogenous variables of the model are expressed explicitly in terms of predefined variables. This form of specification is called given. In a particular case, the structural and reduced forms of the model may coincide. With the correct specification of the model, the transition from the structural to the reduced form is always possible, the reverse transition is not always possible.
The system of joint, simultaneous equations (or the structural form of the model) usually contains endogenous and exogenous variables. Endogenous variables are denoted in the earlier system of simultaneous equations as y. These are dependent variables, the number of which is equal to the number of equations in the system. Exogenous variables are usually denoted as x. These are predefined variables that affect but do not depend on endogenous variables.
The simplest structural form of the model is: 
where y are endogenous variables; x are exogenous variables.
The classification of variables into endogenous and exogenous depends on the theoretical concept of the adopted model. Economic variables can act as endogenous variables in some models, and as exogenous variables in others. Non-economic variables (for example, climatic conditions) enter the system as exogenous variables. The values of endogenous variables for the previous period of time (lag variables) can be considered as exogenous variables.
Thus, the consumption of the current year (y t) may depend not only on a number of economic factors, but also on the level of consumption in the previous year (y t-1)
The structural form of the model allows you to see the impact of changes in any exogenous variable on the values of the endogenous variable. It is expedient to choose such variables as exogenous variables that can be the object of regulation. By changing and controlling them, it is possible to have target values of endogenous variables in advance.
The structural form of the model on the right side contains the coefficients b i and a j for endogenous and exogenous variables, (b i is the coefficient for the endogenous variable, a j is the coefficient for the exogenous variable), which are called the structural coefficients of the model. All variables in the model are expressed in deviations from the level, i.e., x means x- (and y, respectively, y- (. Therefore, there is no free term in each equation of the system.
The use of LSM for estimating the structural coefficients of the model gives, as is commonly believed in theory, the biased structural coefficients of the model.
The reduced form of the model is a system of linear functions of endogenous variables from exogenous ones: 
In its appearance, the reduced form of the model does not differ in any way from the system of independent equations, the parameters of which are estimated by traditional least squares. Using the least squares method, one can estimate δ and then estimate the values of endogenous variables in terms of exogenous ones.
Deployed EM(her blocks)
There is a model that connects and harmonizes between two descriptions of a person, at first glance, far from each other - the psychophysical and the Transpersonal. This model has a centuries-old history and is based on deep research and practical experience, transmitted directly from the Teacher to the Student. In the language of Tradition, of which the authors of this book are representatives, this model is called the Volumetric-Spatial Model (which was repeatedly mentioned already in the first chapters). There are some parallels of the Volumetric - Spatial Model with other ancient descriptions of a person (the system of Chakras - "thin" bodies; "energy centers" - "planes of consciousness", etc.). Unfortunately, a serious study of these models now, in most cases, has been replaced by a widespread vulgar idea of Chakras as some kind of spatially localized formations, and about “thin” bodies, as a kind of “matryoshka doll”, consisting of some invisible to the naked eye entities. The authors know only a relatively small number of modern sober studies of this issue [see, for example, Yog No. 20 “Questions of the General Theory of the Chakras” St. Petersburg 1994.]
The current situation is extremely unfavorable: critically thinking specialists are skeptical about the model of the Chakras and “thin” bodies, while others (sometimes even despite a long experience as a psychologist or psychotherapist) become on a par with housewives (no offense to them), attending courses " psychics”, and replenish the army of bearers of legends about Chakras and “Bodies”, distributed by popular brochures. It sometimes comes to a comic turn. So, one of the authors of this book had a chance to attend a psychological training several years ago, with elements of “esotericism”, where a very authoritative leader gave approximately the following instructions for one of the exercises: “... And now, with your ethereal hand, put an “anchor” directly to the client in the lower Chakra...”, which most of those present immediately enthusiastically tried to implement (of course, no further than in their imagination).
Further we will not mention Chakras and Bodies, but will use the language of Volumes and Spaces. One should not, however, carry out an unambiguous correspondence between Volumes and Chakras, Spaces and Bodies; despite some similarities, these models differ; the differences, in turn, are connected not with a claim for greater or lesser correctness, but with convenience for the Practice that we present on the pages of this book.
Let's go back to the definitions of Volumes and Spaces that we gave in chapters 1 and 2:
So, Volumes are not parts of the physical body and not some localized areas. Each Volume is a Holistic psychophysical state, a formation that reflects a certain (congruent) set of certain qualities of an organism as a whole. Speaking in energy terms, Volume is a certain range of energy, which, when perception is focused on the physical world, manifests itself in a combination of tissues, organs, sections of the nervous system, etc. In a rather simplified version, it is possible to find for each Volume the most characteristic function and task that it performs in the body. . So, the functions of the Coccygeal Volume can be associated with the task of survival in all its forms (physical, social, spiritual), manifestation, birth, becoming... The functions of the Urogenital Volume are associated with prosperity, abundance, fertility, development and multiplication, diversity and prosperity.. For the Umbilical Volume, the main tasks (read - energy range) are ordering, structuring, managing and binding. And so on. For now, we will not be interested in specific functions of the Volumes. and general mechanisms for working with them.
Each experience, any experience is perceived by us mainly through this or that Volume. This applies to any experience - if we want to activate this or that experience, then this or that Volume is excited and we begin to perceive the World “through it”. In relation to psychotherapeutic work, when the therapist addresses some kind of client’s experience: “problematic” or “resourceful”, tries to work with a certain “part of the personality”, he, thereby, focuses the patient’s consciousness in some area of a particular Volume ( By the way, we briefly mentioned the functions of only the three lower Volumes because the real productive focusing of attention in the upper Volumes is an extraordinary phenomenon - everything is not as simple as it is described in books). The same applies to Spaces. Recall that Spaces are perception schemes that reflect the levels of “subtlety” of perception. The same Volume at different levels of perception will manifest itself in its own way, retaining its main tasks. So, for example, the Umbilical Volume in the Event Space manifests itself through a series of situations in which a person connects something with something, arranges, controls, etc., in the Name Space - the same Volume will manifest itself through schematization. modeling, putting in order thoughts and views on the World, building plans, etc., in the Space of Reflections the entire emotional spectrum will also be colored by the tasks corresponding to this Volume.
The Volumetric-Spatial Model of the human body can be conditionally represented in the form of a diagram (Fig. 3.)
Fig.3. Volumetric-Spatial Model.
The scheme (Fig. 3.) clearly shows that each Space covers the entire spectrum of energy at a certain level of “subtlety”, where each Volume is a “sector” highlighting a certain energy range.
So - the Volumetric-Spatial Model allows in Man and in the World, which are perceived as dynamic energy structures, to highlight various qualities of energy. In perception, these qualities of energy are manifested through a certain combination of a wide variety of factors:
physiological processes (mechanical, thermal, chemical, electrodynamic), the dynamics of nerve impulses, the activation of certain modalities, the coloring of emotions and thinking, the combination of events, the interweaving of destinies; falling into the appropriate "external" conditions: geographical, climatic, social, political, historical, cultural...
Energy flows.
The scheme shown in Fig.3. gives us the energy model of the human body. From this point of view, the whole life of a person, as a manifestation, formation of this energy or as the dynamics of self-perception, can be represented as a movement-pulsation of a certain “pattern” on the diagram, where certain areas of the energy spectrum are activated at each moment of time (Fig. .4.).
However, the dynamics of self-perception and the movement of energy are not so arbitrary and diverse for an ordinary person. There are areas in which perception is, so to speak, fixed and fairly stable, some areas of the spectrum are available only occasionally and under special circumstances. There are areas that are practically inaccessible to awareness throughout life (for each person they are different: for one person, the experience of meaning is inaccessible, the other has not truly experienced his body in his entire life, the third is not able to experience a certain quality of emotions, events, thoughts and so on.).
The most probable trajectory of movement and fixations of perception and awareness is determined by the Dominant. It becomes clear that in order to break away from this most probable trajectory and stable positions of perception, some additional energy is needed and, most importantly, the ability to direct this energy in the right direction, so that it does not fall into the accumulated stereotypical channel.
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Fig.4. Dynamics of perception in time.
This explains the presence of ranges that are hard to reach and inaccessible to perception and awareness - usually a person does not have this additional energy; only sometimes it can be released as a result of some extraordinary, most often stressful, circumstances, which will allow perception to shift to a previously inaccessible range (such a sudden shift in perception can lead to the appearance in a person of some new abilities that are inaccessible in the usual state).
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For a more detailed description of this situation, we need to turn to the concept Energy flow. Energy flow - movement, development of a point impulse of perception in the Volumetric-Spatial energy system. You can also say this: Energy flow is a dynamic connection various areas in the Individual Sphere according to the general energy range (for example, according to one modality).
“Being in a continuous dialogue with the World, a person (IS) responds to almost all signals coming “from outside” by the movement of Energy Flows. Moreover, the sensitivity of I.S. well above the threshold of perception of the senses. Accordingly, there are many unconscious reactions.
Features of personal deformation of I.S. create permanent characteristic individual energy flows. What we are aware of as sensations, emotions, thoughts, body movements and vicissitudes of fate, memory, projections of the future, illnesses, features of culture and worldview - all this (and much more) is the movement of Energy Flows.”
It is possible to conditionally single out constructive and destructive energy flows. Constructive E. - the dynamics of perception, contributing to the elimination of deformations from I.S. - rigid, dominant structures. Destructive E. - the dynamics of perception, contributing to the emergence of new or reinforcement of existing deformations of I.S.
In turn, we will call the dynamics of Energy flows a multifactorial dynamic process that transfers a person's perception from one state to another (an example of the dynamics of Energy flows is shown in Fig. 5.).
Any Energy Flows are possible in the Whole organism, for which it (the organism) is absolutely transparent and permeable. Dynamics of Energy flows can, in such cases, transfer perception to any position. (This is equivalent to what we called through awareness in Chapter 1.).
Dynamics of Energy Flows is a multifactorial process, because any state manifests itself in the form of a combination of a large number of factors (for example, certain sensations, the nature of movements, facial expressions, voice parameters, certain emotions, etc.). Dynamics of Energy Flows transforms one state into another (more precisely, it is a process – a continuous change of states) and, accordingly, some factors and parameters through which Energy Flows manifest themselves can change.
Fig.5. An example of the dynamics of Energy Flows, which transfers perception from a state with a rigidly localized structure (A) to a more holistic one (D), within the same space
If we now turn to psychotherapy, we find the following:
The patient is in a certain state of perception (determined by his Dominant), which, obviously, is not Holistic, there are rigidly localized structures in his energy, which makes it impossible to shift perception to other positions. To get out of this situation, it is necessary to set the Energy Flows, allowing you to shift to another state, which the patient will perceive as more positive. This is where psychotherapy usually ends.
If you look at it from a more general perspective, it turns out that a non-patient or a cured patient is, by and large, not much different from a "sick" patient. The only difference is that the "sick" perceives his condition as uncomfortable, and the "healthy" - as more or less comfortable and, perhaps, having more degrees of freedom. However, this has nothing to do with Integrity, because. and the state of “sick” and “healthy” is, as a rule, all the same limited, localized and set by the Dominant fixation of perception.
Integrity means the ability independent tasks of any Energy flows and experience of the World totally, simultaneously by the whole organism.
In the previous chapter, we considered models that are a static reflection of systems at certain points in time. In this sense, the considered versions of the "black box" model, the composition model and the structural model are called static models, which emphasizes their immobility.
The next step in system research is to understand and describe how the system "works" to fulfill its intended purpose. Such models should describe the behavior of the system, fix changes that occur over time, capture cause-and-effect relationships, and adequately reflect the sequence of processes occurring in the system and the stages of its development. Such models are called dynamic. When studying a particular system, it is necessary to determine the direction of possible changes in the situation. If such a list is exhaustive, then it characterizes the number of degrees of freedom, and therefore is sufficient to describe the state of the system. As it turned out, dynamic models are divided into the same types as static ones (“black box”, composition and “white box”), only the elements of these models are temporary.
2.4.1. Dynamic black box model
In mathematical modeling of a dynamic system, its specific implementation is described in the form of a correspondence between the possible values of some integral characteristic of the system c and time points t. If we denote by C the set of possible values c, and by T the ordered set of time points t, then building a model of a dynamic system is equivalent to building a mapping
Г->С:с(t)ϵСͭͭ,
where Cͭ is the value of the integral characteristic at the point t ϵ .
In the dynamic model of the "black box" it is assumed that the input stream x is divided into two components: and - controlled inputs, y - uncontrolled inputs (Fig. 2.9).
Thus, it is expressed by a combination of two processes:
Xͭ = (u(t), y(t)); u(t)eU; y(f)eK;
Rice. 2.9. Dynamic black box model
this transformation is assumed to be unknown.
From of this type models, the so-called inertialess systems have been studied to the greatest extent. They do not take into account the time factor and work according to the "if-then" scheme. For example, if water is heated to
100 ° C, then it will boil. Or: if you correctly authorized your credit card, then the ATM will immediately give you the requested amount of money. That is, the effect comes into force immediately after the cause.
Definition 1. A dynamical system is called inertialess if it instantly converts input into output, i.e. if y(t)
is a function of only x(t) at the same time.
The search for an unknown function y(/) = Ф(x(t)) is carried out by observing the inputs and outputs of the system under study. In essence, this problem is about the transition from the "black box" model to the "white box" model based on observations of inputs and outputs in the presence of information about the system's inertialessness.
However, the class of inertialess systems is very narrow. In economics, such systems are very rare. Unless individual stock exchange transactions can be classified as non-inertial with some stretch.
When modeling economic systems, it must be remembered that there is always a delay in them and, moreover, the consequence (result) may appear in a completely different place where it was expected. Thus, when dealing with economic systems, one must be prepared for the fact that the consequences may be separated from the cause that caused them in time and space.
For example, if a company's sales department allows pre-sales service to take its course and concentrates all its efforts on sales, the after-sales service department will suffer. But this will not appear immediately, but later certain time. On the face of the manifestation of the investigation "not there and not at that time." Or: it may take several weeks of an advertising campaign to change buying habits, and not necessarily tangible changes will begin immediately after it ends.
Feedback acts along a chain of cause-and-effect relationships that form a closed loop, and it takes time to bypass it. The more dynamic complexity a system has, the more time it takes for the feedback signal to run through its structure (network of interconnections). One delay is enough to provide a strong signal delay.
Definition 2. The time required for the feedback signal to pass through all the links of the system and return to the starting point is called the system memory.
Not only living systems have memory. In economics, for example, this vividly demonstrates the process of introducing a new product to the market. As soon as a new product that is in demand appears on the market, there are immediately many who want to produce it. Many firms are launching the production of this product, and as long as there is demand, they are increasing its volumes. The market is gradually saturated, but manufacturers do not feel it yet. When the volume of production exceeds a certain critical value, demand will begin to fall. The production of goods by a certain inertia will continue for some time. Overstocking of warehouses with finished products will begin. Supply will greatly exceed demand. The price of a commodity will fall. Many firms will stop producing this product. And this situation will continue until the supply drops to such values that it cannot cover the existing demand. The market will immediately catch the emerging deficit and react by raising prices. After that, the revival of production and a new cycle of rise and fall of the market will begin. This will continue until several manufacturers remain on the market, who either agree among themselves or intuitively find quotas for the production of goods, the total volume of which will correspond to the required supply and demand ratio (Fig. 2.10).
The graphs of inflation and deflation of the money market, the rise and fall of the stock market, the replenishment and spending of the family budget look exactly the same. The thing is that cause and effect are separated by a delay in time. All this time, the system “remembers” how it should respond to the cause. At first, it seems that there is no consequence. But over time, the effect appears. Misled (in our example, entrepreneurs) react too late and too strongly to peaks in supply and demand. And the balancing feedback, which works with a time delay, is to blame for everything.
Rice. 2.11. commodity market fluctuation
In such a situation, there are two solutions. First, the measurement can be made more reliable by monitoring the market continuously or periodically. Secondly, you should take into account the time difference and strive to be where you need to be by the time the signal feedback will have time to go through all the links of the system. When you understand how the process is carried out, it becomes possible to change the situation in the desired direction.
In very complex systems consequences may appear after a very long time. By the time it makes itself felt, the critical threshold may have passed and it will be too late to fix anything. This danger is especially evident in the impact of industrial waste on the environment. What we do now will affect our future life when the consequences of our actions appear. By our actions today, we shape the future.
Essentially, nothing will change in the appearance of the dynamic black box model, except that the moment of appearance of the output y will need to be corrected for the delay time ∆, i.e. the output of the system will take the form y(t + ∆) (see Fig. 2.10). However, the main difficulty in modeling is to determine the value of D and the place where y will appear. This is best done within the framework of constructing the so-called lag models, which are studied by mathematical statistics.
2.4.2. Dynamic Composition Model
In systems theory, two types of dynamics are distinguished: functioning and development. Functioning refers to the processes that occur in a system that consistently implements a fixed goal (an enterprise is functioning, clocks are functioning, urban transport is functioning, etc.). Development is understood as a change in the state of the system due to external and internal causes. Development, as a rule, is associated with the movement of systems in the phase space.
The study of the functioning of economic systems is carried out by specialists in the field of economic analysis. The initial basis for this study is accounting data, statistical reporting and statistical observations. In most cases, the task of economic analysis is solved by analytical methods of accounting or is reduced to the construction and implementation of correlation and regression models. The richest toolkit of economic analysis is studied within a number of disciplines of the cycle "Accounting and statistics".
Development in most cases is due to a change in the external goals of the system. A characteristic feature of development is that the existing structure ceases to correspond to new goals, and in order to ensure the necessary compliance, it is necessary to change the structure of the system, i.e. carry out its reorganization. Economic systems (enterprises, organizations, corporate entities) in a market economy must constantly be in the development phase in order to survive in the competitive struggle. Only constant renewal of the range of products or services provided, improvement of production technology and management methods, advanced training and education of personnel can provide the economic system with certain competitive advantages and expanded reproduction.
In this section, without denying the importance of the system functioning phase, we will mostly talk about the phase of its development, although with an extended interpretation of the system functioning as movement towards the intended goal (plan), the arguments below are quite applicable to modeling the system functioning phase.
The dynamic version of the composition model corresponds to a list of development stages or system states in the simulated time interval. Under the state of the system we mean such a set of parameters characterizing the spatial position of the system, which exhaustively determines its current position.
State fixation is determined by introducing various variables, each of which reflects one essential aspect of the system under study. In this case, the exhaustiveness of the description is important for revealing the purpose of the system, which is being studied within the framework of this model.
The state of the system is most clearly defined through the degrees of freedom. This concept was introduced in mechanics and means the number of independent coordinates that uniquely describe the position of the system. Thus, a rigid body in mechanics is a system with six degrees of freedom: three linear coordinates fix the position of the center of mass, and three angular coordinates fix the position of the body relative to the center of mass.
In economic research, each coordinate (degree of freedom) is associated with a certain indicator (a quantitatively measured characteristic of the system). The key task here is to ensure the independence of the indicators selected to build the system model. Therefore, it is necessary to deeply understand the nature of economic phenomena and indicators reflecting them in order to correctly form the basis for building a model of the composition of the economic system.
The development of a system is not a habitual movement, but some abstraction that describes a change in its state. Thus, the dynamic properties of an object are characterized through the change in state parameters over time. On fig. 2.12 shows a graphical representation of the motion of the system in three-dimensional space (in systems theory, such a space is called the state space, or phase space).
Rice. 2.12. System Development Trajectory
Then the state of the system at time ts is described by the vector Cs = (C1s,C2s,C3s). Its initial SN and final CK states are described similarly, and changes in the system are displayed by a certain curve - the trajectory of development. Each point on this curve captures the state of the system at a certain point in time. Then the motion of the system is equivalent to moving the point along the trajectory C2.
Extrapolating this description to the case of and independent coordinates and remembering that each coordinate (parameter) depends on time t, the development of the system can be described by a set of functions с1= с1(t), с2=с2(t) ,..., сn =сn( t), or a vector (с1(t), с2 (t),...,сn =сn(t)) belonging to the state space С.
Thus, the dynamic model of the composition of the system is nothing but an ordered sequence of its states, the last of which is equivalent to the goal of the system, i.e.
Сн =С0 ->СJ ->Ct ->...->CT=Ск,
where Cn - initial;
Sk - final;
С, = (c1 (t), c2 (t),..., сn (t)), t ϵ - current state of the system.
The case when the boundary states of the system are strictly defined belongs to the category of the simplest, since it is far from always possible to describe the state with specific values. A more general situation is when certain conditions are imposed on the initial and final states of the system. Each of the conditions in the state space is represented by some surface or area, the dimension of which should not be greater than the number of degrees of freedom of the system. Then the state vector of the system at the boundary moments of time should be on a given surface or in a given area, which will mean the fulfillment of the conditions.
2.4.3. Dynamic structural model
In dynamic systems, elements can enter into a wide variety of relationships with each other. And since each of them is capable of being in many different states, even with a small number of elements they can be connected by many various ways. To build a model of such a system, providing for a change in the states of some elements of the system, depending on what happens to its other elements, is a very difficult task. Nevertheless, modern science has developed many approaches to modeling such systems. Let's take a closer look at two of them, which have become classics.
As in the case of the static structural model, the dynamic structural model is a symbiosis of the dynamic black box model and the dynamic composition model. In other words, the dynamic structural model must link into a single whole the input to the system X = (х(t)) = (u(t),v(t)), u(t)ϵu, v(t)ϵV, intermediate states
Ct = , t ϵ, and output y=(y(t)),
where, U - set of controlled inputs u(t);
U - set of uncontrolled inputs v(t);
X = U U X - set of all inputs to the system;
T is the system modeling horizon;
C, - intermediate state of the system at the time t ϵ .
Depending on whether the intermediate states of the system are displayed in a strictly defined ordered sequence
Ct (t = 0.1, 2, ..., T) or one indefinite function Ct = Ф(t, xt), as a result of modeling, either a dynamic structural model of a network type or a dynamic structural model of an analytical type is obtained.
Network dynamic models. In a dynamic structural model of a network type, for each pair of neighboring states of the system Сt-1 and Сt (t ϵ ), a control action u(t) is set, which transfers the system from the state Ct-l to the state Ct. In this case, it is obvious that u(t) at each step of the trajectory can take values from a certain set of admissible control actions at this step
Ut: u(t)ϵUt. (2.1)
Thus, the intermediate state of the system at some point t of the trajectory of its development is written as follows
Сt=F(Ct-i,u(t)), tϵ.
Denote by Ct the set of all system states to which it can be transferred from the initial state C0=CH in t steps using the control actions u(t) ϵ Ut (t = 0,1, 2,..., t). The reachability set Ct is defined using the following recursive relations:
Сt = (Ct: Сt = ƒ(Сt-1, u(t); u(t ϵUt; t = 0.1, 2,...,t).
In the task for further development or initial development of the system, a list of its admissible final states is indicated, which must belong to a certain area
СtϵС-Т. (2.2)
The control U =(u(1), u(2),..., u(t),..., u(T)) , consisting of step-by-step control actions, will be admissible if it transfers the system from the initial state Сн = С0 to the final state Ск =СТ satisfying condition (2.2).
Let us derive the conditions for the admissibility of the control. To do this, consider the last T-step. Due to the limitedness of the set UT, it is possible to transfer the system to the state ST ϵ ST not from any state CT-1, but only from-T-1,St-1 G с,
Where, C is a set that satisfies the condition
VCT=1 ϵ C-T-1zu(T)ϵUT: su =/(SU-1, u(T))&st.
In other words, in order to be able to enter the area of admissible states C after T-th step-r of control, it is necessary-r-1 to be in the area C after (T - 1) steps.
Similar sets of admissible states c" are formed for all other steps t = 1, T - 1.
To achieve the goal of building (development) of the system, it is necessary to fulfill the conditions
C "PS" * 0, / \u003d 1, T. (2.3)
Otherwise, the goal of the system cannot be achieved. To overcome this obstacle, it will be necessary either to change the goal of the system T, thereby changing C, or to expand the area of possible control actions ut = 1,T (primarily at those steps of the system trajectory at which condition 2.3 is not satisfied).
Let, as a result of overcoming (t -1) steps, the system passed into the state Ct-1. Then the set of admissible control actions on t-th step is defined as follows:
U(t) = (u(t): Сt =ƒ(Сt-1, u(t) ϵс-t) (2.4)
Combining (2.1) and (2.4), we can write down the conditions for the controlled purposeful development of the system:
U(t)ϵ(t)nU(f) = 1e. (2.5)
Conditions (2.5) mean that the control must be possible in terms of its realizability and admissible in terms of ensuring that the system reaches the given region of final states.
Thus, the construction of a dynamic structural model of a network type system consists in a formalized description of the trajectory of its development by specifying intermediate states of the system and control actions that sequentially transfer the system from the initial state to the final state corresponding to the goal of its development.
Since, as a rule, there are many paths from the "beginning" to the "end", the system development trajectory can be determined according to various criteria (minimum time, maximum effect, minimum costs, etc.). The choice of criterion is determined by the purpose of modeling the system.
This approach to modeling dynamic systems, as a rule, leads to the construction of network models different types(network graphs, technological networks, Petri nets, etc.). Regardless of the type of network model, their essence lies in the fact that they describe a certain set of logically linked works, the execution of which should ensure the construction of a certain system (enterprise, road, political party) or its transfer to another state corresponding to new goals and requirements of the time.
The concretization of dynamical systems, of course, does not end there. These models are most likely individual examples of real systems. In the class of models of dynamic systems, there are also stationary models, soft and hard models, which are used in the study of specific applied problems.
1. Give several definitions of the system and a meaningful description of each of them.
2. What is the difference between a philosophical category and a natural scientific concept?
3. List and interpret the main properties of the system.
4. What is system emergence?
5. How do the concepts of "integrity" and "emergence" relate?
6. What is the essence of reductionism? How is it different from the systems approach?
7. What is the difference between external and internal system links?
8. What property underlies the division of systems into open and closed (closed)?
9. Give examples of closed economic systems.
10. How is the stability of the system ensured?
11. What are the internal and external goals of the system?
12. How are the internal and external strategies of the system aligned?
13. How to set the boundaries of the economic system?
14. What is the reason for the unsatisfactory forecasts obtained as a result of econometric modeling.
15. Describe the transactional environment of the economic system.
16. Due to what open economic systems retain their individual characteristics?
17. How (on what scales) are the emergent properties of systems measured?
18. Name the necessary condition for the existence of the emergent property of the system.
19. What is the essence of the property of purposefulness. How does this property manifest itself in economic systems?
20. Give examples of reactive, responsive, self-adjusting and active economic systems.
21. What is the essence of the property of the hierarchy of economic systems?
22. Are the concepts of “level of hierarchy” and “stratum” equivalent?
23. What is the essence of the property of the multidimensionality of the economic system?
24. Give a systematic definition of the concept of "compromise".
25. Give practical examples of using the property of multidimensionality in the study of economic systems.
26. What is the essence of the multiplicity property of the economic system?
27. Give examples of the multiplicity of functions of the economic system.
28. How is the plurality of the structure of the economic system manifested?
29. Give examples of equifinality and multifinality of economic systems.
30. List the reasons for the counterintuitive behavior of economic systems.
31. What classification feature is the basis for the primary classification of systems?
32. What are the main characteristics of natural systems. Give examples.
33. What are the main characteristics of artificial systems. Give examples.
34. What is the specificity of socio-cultural systems?
35. What class of primary systems do economic systems belong to?
36. To what extent are the natural, technical and human sciences involved in the analysis of economic systems?
37. Place the factors in descending order of influence on the configuration of the system: external environment, internal connections of the system, connections of the system with the external environment, elements of the system.
38. Explain how the moral values of a decision maker materialize in a real economic system.
39. What is the environment in which economic systems exist and function?
40. Define the economic system.
41. What classification features form the basis of the spatio-temporal classification of economic systems?
