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A mass congestion of people always represents a potential danger and threat for their lives. In addition, every year in the world a very large number of people die because of the crush, the main cause of which is mass panic. Therefore, the study of the phenomenon of mass panic in view of her extreme social danger is an important scientific task. Available information, about the processes of her occurrence and spread refers to the category inaccurate. Therefore, the theory of fuzzy sets has been chosen as a tool for developing a mathematical model of the mechanism of transmitting panic state among people with various temperament species.

When developing an fuzzy model, it was assumed that panic, from the epicenter of the shocking stimulus, spreads among people according to the wave principle, passing at different frequencies through different environments (types of human temperament), and is determined by the speed and intensity of the circular reaction of the mechanism of transmitting panic state among people. Therefore, the developed fuzzy model, along with two inputs, has two outputs — the speed and intensity of the circular reaction. In the block «Fuzzyfication», the degrees of membership of the numerical values of the input parameters to fuzzy sets are calculated. The «Inference» block at the input receives degrees of belonging for each input parameter and at the output determines the resulting function of belonging the speed of the circular reaction and her derivative, which is a function of belonging for the intensity of the circular reaction. In the «Defuzzyfication» block, using the center of gravity method, a quantitative value is determined for each output parameter. The quality assessment of the developed fuzzy model, carried out by calculating of the determination coefficient, showed that the developed mathematical model belongs to the category of good quality models.

The result obtained in the form of quantitative assessments of the circular reaction makes it possible to improve the quality of understanding of the mental processes occurring during the transmission of the panic state among people. In addition, this makes it possible to improve existing and develop new models of chaotic humans behaviors. Which are designed to develop effective solutions in crisis situations, aimed at full or partial prevention of the spread of mass panic, leading to the emergence of panic flight and the appearance of human casualties.

Many properties of ordinary differential equations systems solutions are determined by the properties of the equations in variations. An ODE system, which includes both the original nonlinear system and the equations in variations, will be called an extended system further. When studying the properties of the Cauchy problem for the systems of ordinary differential equations, the transition to extended systems allows one to study many subtle properties of solutions. For example, the transition to the extended system allows one to increase the order of approximation for numerical methods, gives the approaches to constructing a sensitivity function without using numerical differentiation procedures, allows to use methods of increased convergence order for the inverse problem solution. Authors used the Broyden method belonging to the class of quasi-Newtonian methods. The Rosenbroke method with complex coefficients was used to solve the stiff systems of the ordinary differential equations. In our case, it is equivalent to the second order approximation method for the extended system.

As an example of the proposed approach, several related mathematical models of the blood coagulation process were considered. Based on the analysis of the numerical calculations results, the conclusion was drawn that it is necessary to include a description of the factor XI positive feedback loop in the model equations system. Estimates of some reaction constants based on the numerical inverse problem solution were given.

Effect of factor V release on platelet activation was considered. The modification of the mathematical model allowed to achieve quantitative correspondence in the dynamics of the thrombin production with experimental data for an artificial system. Based on the sensitivity analysis, the hypothesis tested that there is no influence of the lipid membrane composition (the number of sites for various factors of the clotting system, except for thrombin sites) on the dynamics of the process.