Stochastization of classical models with dynamical invariants

  • Karachanskaya Elena V., Far-Earstern State Transport University, 47 Serysheva Street, Khabarovsk 680027, Russia
Keywords: stochastic model, invariant of dynamical model, construction of Itô equations


The article considers the possibility of construction of stochastic analogues for classical models described by a differential equations system and having an invariant function. The suggested method of stochastization is based on both the concept of the first integral for a stochastic differential Ito equations system (SDE) and the theorem for construction of the SDE system using its first integral. We consider three models. The first is a simple mathematical model of epidemics or the SIR (susceptible-infectedrecovered) model, the second is the predator-prey model, and the third model is the multistage multi-enzymatic chemical reaction. The process of stochastization includes three constituents of the classical model. The first constituent is the diffuse component and described by the Wiener process violence, the second describes the intermittent transition connected with the Poisson jumps realization, and the third is the complementary function for the drift coefficient. We construct systems of stochastic differential Ito equations with Poisson jumps for the SIR model and the predator-prey model. The stochastic analogue for the third model is described with the use of a stochastic differential Ito equations system without Poisson jumps. We use a collection of functions, including an invariant function, several complementary functions, and arbitrary functions to construct coefficients for the differential equations system. It is shown that the choice of complementary functions allows us to obtain such coefficients that ensure that the solution of the differential equations system satisfies some reasonable limitations. Examples of differential equations system construction are obtained using the author’s MathCad computer software for analytical simulation. The obtained numerical solutions of the systems verify the theoretical propositions and preserve the constant value for the dynamic invariant function for any realizations of the SDE solutions.


Dubko V. A., “Integral of a System of Stochastic Differential Equations [in Russian],” Preprint No. 78–27, Inst. Mat. Ukr. Akad. Nauk, Kiev (1978).

Dubko V. A. , Problems of the Theory of Stochastic Differential Equations and Their Applications [in Russian], Dal’nevost. Nauch. Tsentr, Vladivostok (1989).

Dubko V. A., “Open evolving systems and their modelling [in Russian],” Vestn. DVO RAN, No. 4–5, 55–64 (1993).

Karachanskaya E. V., Stochastic Processes with Invariants [in Russian], Izdat. Tikhookeansk. Gos. Univ., Khabarovsk (2014).

Gikhman I. I. and Skorokhod A. V., Stochastic Differential Equations, Springer, New York (1972).

Karachanskaya E., “The generalized Itˆo–Venttsel’s formula in the case of a noncentered Poisson measure, a stochastic first integral, and a first integral [in Russian],” Sib. Adv. Math., 25, No. 3, 191–205 (2015).

Dubko V. A. and Karachanskaya E. V., “Stochastic first integrals, kernels of integral invariants, and the Kolmogorov equations [in Russian],” Dal’nevost. Mat. Zh., 14, No. 2, 200–216 (2014).

Erugin N. P., “Construction of the whole set of differential equations having a given integral curve [in Russian],” Prikl. Mat. Mekh., 16, 658–670 (1952).

Mukhametzyanov I. A. and Mukharlyamov R. G., Equations of Programmed Motions [in Russian], Izdat. RUDN, Moscow (1986).

Chalykh E., "Constructing the set of program controls with probability 1 for one class of stochastic systems,” Autom. Remote Control, 70, No. 8, 1364–1375 (2009).

Karachanskaya E. V., “Construction of programmed controls for a dynamic system based on the set of its first integrals,” J. Math. Sci., 199, No. 5, 547–555 (2014).

Karachanskaya E. V., “Construction of program controls with probability 1 for a dynamical system with Poisson perturbations [in Russian],” Vestn. Tikhookeansk. Gos. Univ., No. 2, 51–60 (2011).

Karachanskaya E. V., “Construction of a set of differential equations with a given set of first integrals [in Russian],” Vestn. Tikhookeansk. Gos. Univ., No. 3, 47–56 (2011).

Karachanskaya E. V., Integral Invariants of Stochastic Systems and Program Control with Probability 1 [in Russian], Tikhookeansk. Gos. Univ., Khabarovsk (2015).

Averina T., Karachanskaya E., and Rybakov K., “Statistical analysis of diffusion systems with invariants,” Russ. J. Numer. Anal. Math. Model., No. 33, 1–13 (2018).

Karachanskaya E. V. and Petrova A. P., “Modeling of the programmed control with probability 1 for some financial tasks [in Russian],” Mat. Zametki SVFU, 25, No. 1, 25–37 (2018).

Karachanskaya E. V., “Programmed controls with probability 1 for systems with random disturbances [in Russian],” in: Anal. Mekh., Ustoych., Upravl., Proc. XI Int. Chetaev Conf., 3, pp. 242–248 Kazan (2017).

Kermack W. O. and McKendrick A. G., “Contributions to the mathematical theory of epidemics,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 115, No. 772, 700–721 (1927).

Romanyukha A. A., Mathematical Models in Immunology and Epidemiology of Infectious Diseases [in Russian], BINOM, Moscow (2015).

Varfolomeev S. D. and Lukovenkov A. V., “Stability in chemical and biological systems. Multistage polyenzyme reactions [in Russian],” J. Phys. Chem., 84, No. 8, 1448–1457 (2010).

Trubetskov D. I., “Phenomenon of the Lotka–Volterra mathematical model and similar ones [in Russian],” Iz. Vuzov, No. 2, 69–88 (2011).

Romanov V. P. and Akhmadeev B. A., “Modeling the innovation ecosystem based on the "Predator-victim" model [in Russian],” Bizness-Infor., No. 1, 7–17 (2015).

Dormidontov A. V., Mironova L. V., and Mironov V. S., “On the possibility of applying the mathematical model of counteraction to assessing the level of security of transport infrastructure facilities [in Russian],” Nauch. Vestn. MGTU GA, 21, No. 3, 67–77 (2018).
How to Cite
Karachanskaya, E. (2020) “Stochastization of classical models with dynamical invariants”, Mathematical notes of NEFU, 27(1), pp. 69-87. doi:
Mathematical Modeling