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12The main content of the bank of the NMPUD system is elementary densities (that is, those that are effec-
tively simulated by the inverse distribution function method; see Chapter 2.6 of the book [3]). These densities
can be obtained in sufficient quantities using the technology of sequential (inserted) integral substitutions; see
Chapter 14.2 of the book [3].
In this paper, the expediency of creating blocks within the NMPUD system for simulating distributions
with polynomial and piecewise polynomial densities using the modified superposition method (see Chapters
11.2, 11.3 of the book [3]) is substantiated.
This work was carried out under state contract with ICMMG SB RAS (0251-2021-0002).
References
1. Vasiliev T. V., Postovalov Ya. S., Cherkashin D. A. Project of a computer system for the selection and study of
simulated probability distributions // Proceedings of the 58th International Student Conference "Student and Scientific
and Technological Progress". Mathematics. Novosibirsk: NPTs NSU, 2020. P. 112 [in Russian].
2. Voytishek A. V., Postovalov Ya. S., Cherkashin D. A. The system of numerical modeling of one-dimensional random
variables NMPUD: formation of a bank of densities, automation of mathematical calculations and applications //
Proceedings of the XIX International Conference named after A. F. Terpugov �Information Technologies and Mathematical
Modelling� (Tomsk, December 2-5, 2020). Tomsk: Publishing house NTL, 2021. P. 363�368 [in Russian].
3. Voytishek A. V. Lectures on Computational Monte Carlo Methods. Novosibirsk: NPTs NSU, 2018 [in Russian].
Numerical modeling of boundary value problems for differential equations with random coefficients
B. S. Dobronets, O. A. Popova, A. M. Merko
Siberian Federal University, Krasnoyarsk
Email: BDobronets@yandex.ru
DOI 10.24412/cl-35065-2021-1-00-75
The article is devoted to the numerical modeling of differential equations with coefficients in the form of
random fields. Using the Karunen-Loeve expansion, the coefficients are approximated by the sum of inde-
pendent random variables and real functions. This allows us to use computational probabilistic analysis and, in
particular, we apply the technique of probabilistic extensions to construct the probability density functions of
the processes under study. As a result, we present a comparison of our approach with the Monte Carlo meth-
od in terms of the number of operations and demonstrate the results of numerical experiments for boundary
value problems for differential equations of elliptic type.
References
1. Dobronets B.S. Popova O.A. Computational Probabilistic Analysis: Models and Methods. Krasnoyarsk: Siberian
Federal University, 2020.
2. Soong T. Random Differential Equations in Science and Engineering. New York and London: Academic Press, 1973.
3. Dobronets B., Popova O. Computational aspects of probabilistic extensions. // Tomsk State University J. of Control
and Computer Science, 2019 pp 41�48.
Markov chain based stochastic modelling of HIV-1 and SARS-CoV-2 intracellular replication cycles
D. S. Grebennikov1,2, I. A. Sazonov3, G. A. Bocharov1,2
1Marchuk Institute of Numerical Mathematics RAS, Moscow, Russia
2Sechenov First Moscow State Medical University, Moscow, Russia
3Swansea University, Swansea, UK
Email: dmitry.ew@gmail.com
DOI 10.24412/cl-35065-2021-1-00-76
Understanding the dynamics of the intracellular virus replication is crucial for antiviral drug development.
We present the detailed deterministic models of the life cycles of HIV-1 [1] and SARS-CoV-2 [2] in target cells.
The Markov chain based models are formulated to study stochastic aspects prominent at low variable num-
bers. The hybrid simulation algorithm based on direct Gillespie method [3] with automatic switching between
stochastic and deterministic methods when variables exceed a specified threshold is implemented in C++. The
stochastic models are used to predict (1) the probability of target cell infection as function of multiplicity of
infection, (2) the heterogeneous structure in the evolution of the viral progeny number distribution, (3) the
processes having the biggest impact on the total progeny number, (4) the integrated provirus number distribu-
tion in HIV-infected cells [4]. The model extensions to describe the innate IFN-I response and the evasion
mechanisms by which HIV-1 and SARS-CoV-2 antagonize it are discussed.
This work was supported by the Russian Foundation for Basic Research (grants 20-01-00352, 20-04-60157) and the
Russian Science Foundation (grant 18-11-00171).
References
1. Shcherbatova et al. Modeling of the HIV-1 life cycle in productively infected cells to predict novel therapeutic
targets // Pathogens. 2020. V. 9. N. 4. P. 255.
2. Grebennikov et al. Intracellular life cycle kinetics of SARS-CoV-2 predicted using mathematical modelling //
Viruses. 2021. V. 13. N. 9. P. 1735.
3. Sazonov et al. Viral infection dynamics model based on a Markov process with time delay between cell infection
and progeny production // Mathematics. 2020. V. 8. N. 8. P. 1207.
4. Sazonov et al. Markov Chain-Based Stochastic Modelling of HIV-1 Life Cycle in a CD4 T Cell // Mathematics. 2021.
V. 9. No 17. P. 2025.
Solution of stochastic optimal control problems
S. A. Gusev
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Novosibirsk State Technical University
Email: sag@osmf.sscc.ru
DOI 10.24412/cl-35065-2021-1-00-77
The report is devoted to solution of the problem of stochastic optimal control of dynamical systems, which
are described by stochastic differential equations (SDE�s) of the Ito type [1]. The coefficients of the SDE of the
problem being solved depend on the random process, which is the control at each time item. In the problem, it is
required to define a control such that optimizes the mathematical expectation of the functional of the controlled
process. The determination of the optimal control is carried out on the basis of the principle of dynamic pro-
gramming and solving the HJB equation. Numerical solution of problems of this type is considered.
This work was supported by the budget project 0315-2019-0002 for ICMMG SB RAS.
References
1. Krylov N. V. Controlled Processes of Diffusion Type. M.: Nauka, 1977.
Numerical simulation of a two-dimensional electron gas transfer in a quantum well heterostructure
E. G. Kablukova1, K. K. Sabelfeld1, D. Yu. Protasov2 and K. S. Zhuravlev2
1Institute of Computational Mathematics and Mathematical Geophysics, SB RAS
2Rzhanov Institute of Semiconductor Physics SB RAS
Email: KablukovaE@sscc.ru
DOI 10.24412/cl-35065-2021-1-00-78
In this work, a question of an influence of quantum well subbands number on a drift velocity of a two-
dimensional electron gas in strong and weak electric fields is investigated by numerical modeling. A semicon-
