Mathematical Model of Spread of the Epidemic Inside a Railway Compartment Coach
https://doi.org/10.30932/1992-3252-2020-18-6-06-29
Abstract
This article discusses an aspect of the most pressing problem of 2020, that of the spread of infectious diseases. The work considers a railway compartment coach as a particular object of spread of infectious diseases. The objective is to describe spread of the epidemic in a railway coach using a stochastic model. The model of the coach is represented as a network. The processes occurring on the network are considered to be Markov processes. In this paper, two methods of stochastic modelling are applied: modelling based on Kolmogorov equations and Gillespie algorithm. Kolmogorov equations are used to test applicability of Gillespie algorithm, which, in turn, is used to simulate the model of the coach. The obtained data were analysed, and based on that analysis it is possible to make a conclusion about applicability of the model to the case of a typical passenger train.
About the Authors
A. S. Bratus
Russian University of Transport
Russian Federation
Russian Federation
Bratus, Alexander S. – D.Sc. (Physics and Mathematics), Professor at the Department of Mathematical Modelling and System Analysis of the Institute of Management, Control and Digital Technologies
Moscow
A. S. Ocheretyanaya
Russian University of Transport
Russian Federation
Russian Federation
Ocheretyanaya, Alexandra S. – Ph.D. student at the Department of Mathematical Modelling and System Analysis of the Institute of Management, Control and Digital Technologies
Moscow
References
1. Choisy, M., Guégan, J.-F., Rohanil, P. Mathematical Modeling of Infectious Diseases Dynamics. Encyclopedia of Infectious Diseases: Modern Methodologies by Tibayrenc. M. John Wiley & Sons, Inc., 2007, pp. 379–403.
2. Barrat, A., Barthélemy, M., Colizza, V., Vespignani, V. The role of the airline transportation network in the prediction and predictability of global epidemics. PNAS, 14 February, 2006, Vol. 103 (7), pp. 2015–2020. DOI: www.pnas.org/cgi/doi/10.1073/pnas.051052103.
3. Balcan, D., Colizza, V., Gonçalves, B., Hu, H., Ramasco, J. J., Vespignani, A. Multiscale mobility networks and the spatial spreading of infectious diseases. PNAS, 22 December, 2009, Vol. 106 (51), pp. 21484–21489. DOI: https://doi.org/10.1073/pnas.0906910106.
4. Mari, L., Bertuzzo, E., Righetto L. [et al]. Modelling cholera epidemics: the role of waterways, human mobility and sanitation. J. R. Soc. Interface, 2012, Vol. 9 (67), pp. 376–388. DOI: 10.1098/rsif.2011.0304.
5. Gatto, M., Mari, L., Bertuzzo, E., Casagrandi, R., Righetto, L., Rodriguez-Iturbe, I., Rinaldo, A. Generalized reproduction numbers and the prediction of patterns in waterborne disease. PNAS, 27 November, 2012, Vol. 109 (48), pp. 19703–19708. DOI: 1073/pnas.1217567109.
6. Wu, J. T., Leung, K., Leung, G. M. Now casting and forecasting the potential domestic and international spread of the 2019‑ncov outbreak originating in Wuhan, China: A modelling study. The Lancet, 29 February, 2020, Vol. 395 (10225), pp. 689–697. DOI: 10.1016/S0140-6736(20)30260-9.
7. Hagberg, A. A., Schult, D. A., Swart, P. J. Exploring network structure, dynamics, and function using NetworkX. Proceedings of the 7th Python in Science Conference (SciPy 2008). Eds.: Gäel Varoquaux, Travis Vaught, Jarrod Millman, Pasadena, CA, USA, August 2008, pp. 11–15.
8. Gillespie, D. T. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics, December, 1976, Vol. 22 (4), pp. 403–434.
9. Bratus, A. S., Novozhiov, A. S., Platonov, A. P. Dynamic systems of biology models [Dinamicheskie sistemy i modeli biologii]. Moscow, Fizmatlit publ., 2010, 400 p.
10. Kiss, I., Miller, J., Simon, P. Mathematics of Epidemics on Networks: from Exact to Approximate Models. Springer Math, 2007, 413 p.
11. Deng, Xiaomin; Wang, Xiaomeng. The Application of Gillespie Algorithm in Spreading. 3rd International Conference on Mechatronics Engineering and Information Technology (ICMEIT 2019), April 2019, pp. 688–695. DOI: 10.2991/icmeit‑19.2019.110.
12. Official website of JSC Russian Railways. [Electronic resource]: https://www.rzd.ru/. Last accessed 15.06.2020.
13. Venttsel, E. S. Study of operations: tasks, principles, methodology: Study guide [Issledovanie operatsii: zadachi, printsipy, metodologiya: Ucheb. posobie]. 5th ed., ster. Moscow, Nauka publ.; Chief editorial board of physics-mathematical literature, 1980, 208 p.
14. EoN (Epidemics on Networks): a fast, flexible Python package for simulation, analytic approximation, and analysis of epidemics on networks. [Electronic resource]: https://joss.theoj.org/papers/10.21105/joss.01731. Last accessed 19.06.2020.
15. Gillespie, D. T. Exact stochastic simulation of coupled chemical reactions. Journal of Computational Physics, 01 December, 1977, Vol. 81 (25), pp. 2340–2361. DOI: 10.1021/j100540a008.
16. Kamina, K. M., Mwalili, S., Wanjoya, A. The Modeling of a Stochastic SIR Model for HIV/AIDS Epidemic Using Gillespie’s Algorithm. International Journal of Data Science and Analysis, 2019, Vol. 5, No. 6, pp. 117–122. DOI: 10.11648/j.ijdsa.20190506.12.
17. Mo, Baichuan; Feng, Kairui; Shen, Yu; Tam, Clarence; Li, Daqing; Yin, Yafeng; Zhao, Jinhua. Modeling Epidemic Spreading through Public Transit using Time-Varying Encounter Network. [Electronic resource]: https://arxiv.org/pdf/2004.04602.pdf. Last accessed 19.06.2020.
Review
For citations:
Bratus A.S., Ocheretyanaya A.S. Mathematical Model of Spread of the Epidemic Inside a Railway Compartment Coach. World of Transport and Transportation. 2020;18(6):6-29. https://doi.org/10.30932/1992-3252-2020-18-6-06-29
Views:
474
Email this article 