An Extensive Statistical Study for the Leukemia Mathematical Model using the RVT Technique

Abstract

This paper provides a probabilistic study for the four compartmental leukemia mathematical model. Our study focuses on the randomized endemic equilibrium state considering the proliferation rate of the immune cells due to the cancer relapse is a continuous random variable. This treatment makes the presented model to be more realistic and efficient. Depending on the Random Variable Transformation (RVT) technique, the first probability density functions (1-PDFs) for the solution processes of susceptible blood cells, infected blood cells, cancer cells and immune cells are explicitly derived at the equilibrium state. These PDFs are general and valid for any probabilistic distribution of the input random variable. Relying on the obtained PDFs, the main statistical properties, specifically, the mean and the variance functions for the solution processes are conducted. To test the validity of the theoretical findings associated to the proposed randomized leukemia model, some numerical results are presented through an illustrative example.

Keywords

• Leukemia mathematical model
• Random Variable Transformation (RVT) technique
• first probability density function (1-PDF)
• randomized endemic equilibrium state
• random proliferation rate

References

1. 1 I. Gihman, A. Skorohod, One-dimensional stochastic differential equations of first order, Springer-Verlag, 1972.
2. 2 E. Abdel Gawad, M. El-Tawil, General stochastic oscillatory systems, Appl. Math. Model. (AMM) 17(6) (1993), 329-335.
3. 3 A. Jahedi, G. Ahmadi, Application of Wiener-Hermite expansion to non stationary random vibrations of a Duffing oscillator, J. Appl. Mech. 50 (1983), 436-442.
4. 4 A. Nayfeh, Problems in perturbation, John Wiley, New York, 1993.
5. 5 M. El-Tawil, The application of WHEP technique on stochastic partial differential equations, Int. J. Differ. Equ. 7(3) (2003), 325-337.
6. 6 R. Walpole, R. Myers, S. Myers, Probability and statistics for engineers and scientists, sixth ed., Prentice-Hall, New Jersey, 1998.
7. 7 A. Papoulis, S. U. Pillai, Probability, random variables and stochastic processes, 4th Edition, McGraw-Hill, New York, 2002.
8. 8 M.-C. Casabán, J.-C. Cortés, J.-V. Romero, M.-D. Roselló, Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique, Commun. Nonlinear Sci. Numer. Simul. 24(1) (2015), 86-97.

9. 9 M.-C. Casabán, J.-C. Cortés, A. Navarro-Quiles, J.-V. Romero, M.-D. Roselló, R.-J. Villanueva, A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique, Commun. Nonlinear Sci. Numer. Simul. 32 (2016), 199-210.
10. 10 H. Slama, A. Hussein, N.A. El-Bedwhey, M.M. Selim, An approximate probabilistic solution of a random SIR-type epidemiological model using RVT technique, Appl. Math. Comp. 361 (2019), 144-156.
11. 11 J.-C. Cortés, S.K. El-Labany, A. Navarro-Quiles, H. Slama, A comprehensive probabilistic analysis of approximate SIR-type epidemiological models via full randomized discrete-time Markov chain formulation with applications, Math. Meth. Appl. Sci. 43(14) (2020), 8204-8222.
12. 12 A. Hussein, M.M. Selim, A general probabilistic solution of randomized radioactive decay chain (RDC) model using RVT technique, Eur. Phys. J. Plus 135(418) (2020), 1-16.
13. 13 A. Hussein, Mustafa M. Selim, A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique, J. Quant. Spect. Spectrosc. Radiat. Transf. 232 (2019), 54-65.
14. 14 A.S. Forkas, J.B Keller,B.D. Clarkson, Mathematical model of granulocytopoiesis and chronic myelogenous leukemia, Cancer Res. 51(8) (1991), 2084-2091.
15. 15 J.M. Goldman, J.V. Melo, Chronic myeloid leukemia-advances in biology and approaches to treatment, New Engl. J. Medic. 349(15) (2003), 1451-1464.
16. 16 M. Mamat, Subiyanto, A. Kartono, Mathematical model of cancer treatments using immunotherapy, Chemotherapy and bio chemotherapy, Appl. Math. Sci. 7(5) (2013), 247-261.
17. 17 M.C. Mackey, L. Pujo-Menjouet, Contribution to the study of periodic chronic myelogenous leukemia, Comptes Rendus Biologies 327(3) (2004), 235-244.
18. 18 M. Agarwal, A. S. Bhadauria, Mathematical modeling and analysis of leukemia: effect of external engineered T cells infusion, Appl. Appl. Math. 10(1) (2015), 249-266.
19. 19 S. Khatun, H.A. Biswas, Modeling infectious disease in healthcare problems for the medical systems improvement in Bangladesh, Proceedings of the 2nd European Conference on Industrial Engineering and Operations Management (IEOM) Paris, France, (2018), 3024-3031.