Research ArticleNo Access

On stochastic models in biology and medicine

Iveta Nikolova

Faculty of Mathematics and Natural Sciences, South-West University “Neofit Rilski”, 66 “Ivan Mihaylov” Str., Blagoevgrad 2700, Bulgaria

E-mail Address: iveta.nikolova@abv.bg

Search for more papers by this author

https://doi.org/10.1142/S1793557120501685Cited by:4 (Source: Crossref)

Abstract

Stochastic models along with deterministic models are successfully used for mathematical description of biological processes. They apply knowledge from probability theory and mathematical statistics to analyze specific characteristics of living systems. The paper is devoted to some stochastic models of various phenomena in biology and medicine. Basic concepts and definitions used in classical probability models are considered and illustrated by several examples with solutions. The stochastic kinetic modeling approach is described. A new kinetic model of autoimmune disease is presented. It is a system of nonlinear partial integro-differential equations supplemented by corresponding initial conditions. The modeling problem is solved computationally.

Communicated by J. Koppitz

Keywords:

AMSC: 05A05, 93B40, 92B05

Remember to check out the Most Cited Articles!
Check out our Mathematics books for inspirations & up-to-date information in your research area today!

References

1. R. Aris , Mahematical Modelling Techniques (Dover, 1994). Google Scholar
2. L. Arkeryd , On the Boltzmann equation, Arch. Rational Mech. Anal. 45 (1972) 1–34. Crossref, Web of Science, Google Scholar
3. L. Arlotti and N. Bellomo , Population dynamics with stochastic interaction, Transp. Theory Stat. Phys. 24(1–3) (1995) 431–443. Crossref, Google Scholar
4. L. Arlotti, N. Bellomo and M. Lachowicz , Kinetic equations modeling population dynamics, Transp. Theory Stat. Phys. 29 (2000) 125–139. Crossref, Web of Science, Google Scholar
5. L. Arlotti and M. Lachowicz , Qualitative analysis of an equation modeling tumor-host dynamics, Math. Comput. Model. 23 (1996) 11–29. Crossref, Web of Science, Google Scholar
6. A. Belleni-Morante , Applied Semigroups and Evolution Equations (Oxford University Press, 1979). Google Scholar
7. N. Bellomo, A. Palczewski and G. Toscani , Mathematical Topics in Nonlinear Kinetic Theory (World Scientific, 1988). Google Scholar
8. H. Benaroya, S. M. Han and M. Nagurka , Probabilistic Models for Dynamical Systems (CRC Press, 2013). Crossref, Google Scholar
9. W.-Y. Chen and S. Bokka , Stochastic modeling of nonlinear epidemiology, J. Theor. Biol. 234 (2005) 455–470. Crossref, Web of Science, Google Scholar
10. R. Durbin, S. R. Eddy, A. Krogh and G. Mitchison , Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids (Cambridge University Press, 1998). Crossref, Google Scholar
11. B. Gnedenko , The Theory of Probability and the Elements of Statistics (CRC Press, 2005). Google Scholar
12. K. Heberger and J. M. Andrade , Procrustes rotation and pair-wise correlation: A parametric and a non-parametric method for variable selection, Croatica Chemica Acta 77 (2004) 117–125. Web of Science, Google Scholar
13. S. Iwami, Y. Takeuchi, Y. Miura, T. Sasaki and T. Kajiwara , Dynamical properties of autoimmune disease models: Tolerance, flare-up, dormancy, J. Theor. Biol. 246 (2007) 646–659. Crossref, Web of Science, Google Scholar
14. Z. Jackiewicz, C. L. Jorcyk, M. Kolev and B. Zubik-Kowal , Correlation between animal and mathematical models for prostate cancer progression, Comput. Math. Methods Med. 10 (2009) 241–252. Crossref, Web of Science, Google Scholar
15. I. Kalchev, K. Yordzhev and V. Vachkov , Stochastic Measurement Systems (Technical University, 2007) (in Bulgarian). Google Scholar
16. E. Karashtranova and I. Atanasova , A model for implementation of the dependence between random events in knowledge bases, Econ. Manag. Faculty Econ. South-West Univ. Neofit Rilski 6 (2010) 17–24. Google Scholar
17. M. Kolev , Mathematical analysis of an autoimmune diseases model: Kinetic approach, Mathematics 7 (2019) 1024. Crossref, Web of Science, Google Scholar
18. M. Kolev, A. Korpusik and A. Markovska , Adaptive immunity and CTL differentiation — A kinetic modeling approach, Math. Eng. Sci. Aerospace (MESA) 3 (2012) 285–293. Google Scholar
19. M. Kolev, A. Markovska and A. Korpusik , On a mathematical model of adaptive immune response to viral infection, Lect. Notes Comput. Sci. Numer. Anal. Appl. 8236 (2013) 355–362. Crossref, Google Scholar
20. M. Kolev, S. Nawrocki and B. Zubik-Kowal , Numerical simulations for tumor and cellular immune system interactions in lung cancer treatment, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 1473–1480. Crossref, Web of Science, Google Scholar
21. M. Kolev and I. Nikolova , On a kinetic type model of autoimmune disease, in Proc. 23rd National Conf. Applicatons of Mathematics in Biology and Medicine, eds. U. Forys and J. Smieja (Silesian University of Technology, University of Warsaw, 2017), pp. 101–107. Google Scholar
22. A. Kolmogorov and S. Fomin , Introductory Real Analysis (Dover, 1975). Google Scholar
23. A. Korpusik and M. Kolev , Single injection of $CD 8 + T$ $CD 8 + T$ lymphocytes derived from hematopoietic stem cells — Mathematical and numerical insights, BioSystems 144 (2016) 46–54. Crossref, Web of Science, Google Scholar
24. M. Lachowicz , Modele matematyczne $w$ $w$ biologii: Wprowadzenie (Mathematical models in biology: An introduction.), Math. Appl. (Mat. Stosow.) 1 (2000) 3–34. Google Scholar
25. I. Peneva , Nonparametric Statistical Methods (Deylikont, Sofia, 2015). Google Scholar
26. R. P. N. Rao, B. A. Olshausen and M. S. Lewicki1 , eds. Probabilistic Models of the Brain (The MIT Press, 2002). Crossref, Google Scholar
27. M. Regenwetter, B. Grofman, A. A. J. Marley and I. M. Tsetlin , Behavioral Social Choice. Probabilistic Models, Statistical Inference, and Applications (Cambridge University Press, 2006). Google Scholar
28. S. Rothman and S. A. Walker , Acute disseminated lupus erythematosus with remission followed by relapse, with discoid lesions, Arch. Derm. Syphilol. 61 (1950) 1043–1044. Google Scholar
29. J. Stewart and T. Day , Biocalculus: Calculus, Probability, and Statistics for the Life Sciences (Cengage Learning, 2016). Google Scholar
30. C. J. Thompson , Mathematical Statistical Mechanics (Princeton University Press, 1972). Google Scholar
31. J. C. Wooley and H. S. Lin , eds., Catalyzing Inquiry at the Interface of Computing and Biology (National Academies Press, 2005). Google Scholar
32. K. Yordzhev , Parametric and nonparametric statistical methods — Essence and conditions for application, J. Basic Appl. Res. Int. 11 (2015) 131–134. Google Scholar
33. K. Yordzhev , Random permutations, random sudoku matrices and randomized algorithms, Int. J. Math. Sci. Eng. Appl. 6 (2012) 291–302. Google Scholar