A SIMPLE MATHEMATICAL MODEL THROUGH FRACTIONAL-ORDER DIFFERENTIAL EQUATION FOR PATHOGENIC INFECTION

Yıl 2019, Cilt: 3 Sayı: 1, 29 - 40, 31.01.2019

İlhan Öztürk Bahatdin Daşbaşı Gizem Cebe

https://doi.org/10.26900/jsp.3.004

Öz

The model in this study, examined the time-dependent changes in the
population sizes of pathogen-immune system, is presented mathematically by
fractional-order differential equations (FODEs) system. Qualitative analysis of
the model was examined according to the parameters used in the model. The
proposed system has always namely free-infection equilibrium point and the
positive equilibrium point exists when specific conditions dependent on
parameters are met, According to the threshold parameter R_0, it is founded the
stability conditions of these equilibrium points. Also, the qualitative
analysis was supported by numerical simulations.

Anahtar Kelimeler

Fractional-Order Differential Equation, Numerical Simulation, Stability Analysis

Kaynakça

• L. J. S. Allen, An Introduction to Mathematical Biology., 2007, ISBN 10: 0-13-035216-0.
• H. El-Saka and A. El-Sayed, Fractional Order Equations and Dynamical Systems. Germany: Lambrt Academic Publishing, 2013.
• A. M. A. El-Sayed and F.M. Gaafar, "Fractional order differential equations with memory and fractional-order relaxation oscillation model," (PU.M.A) Pure Math. Appl., vol. 12, 2001.
• F. A. Rihan, "Numerical Modeling of Fractional-Order Biological Systems," Abstract and Applied Analysis, pp. 1-11, 2013.
• W. Deng and C. Li, "Analysis of Fractional Differential Equations with Multi-Orders," Fractals, vol. 15, no. 2, pp. 173-182, 2007.
• M. Axtell and E. M. Bise, "Fractional calculus applications in control systems," in Proc. of the IEEE, New York, 1990, pp. 563-566.
• A. M. A. El-Sayed, "Fractional differential-difference equations," J. Fract. Calc., vol. 10, pp. 101–106, 1996.
• Xue-Zhi Li, Chun-Lei Tang, and Xin-Hua Ji, "The Criteria for Globally Stable Equilibrium in n-Dimensional Lotka-Volterra Systems," Journal of Mathematical Analysis and Applications, vol. 240, pp. 600-606, 1999.
• I. Podlubny and A.M.A. El-Sayed, On Two Definitions of Fractional Calculus.: Slovak Academy of Science, Institute of Experimental Phys., 1996.
• A. M. A. El-Sayed, E.M. El-Mesiry, and H.A.A. El-Saka, "Numerical solution for multi-term fractional (arbitrary) orders differential equations," Comput. Appl. Math., vol. 23, no. 1, pp. 33-54, 2004.
• B. Daşbaşı, "The Fractional-Order mathematical modeling of bacterial resistance against multiple antibiotics in case of local bacterial infection," Sakarya University Journal of Science, vol. 251, no. 3, pp. 1-13, 2017.
• B. Daşbaşı, "Dynamics between Immune System-Bacterial Loads," Imperial Journal of Interdisciplinary Research, vol. 2, no. 8, pp. 526-536, 2016.
• B. Daşbaşı and İ. Öztürk, "Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response," SpringerPlus, vol. 5, no. 408, pp. 1-17, April 2016.
• M. Mohtashemi and R. Levins, "Transient dynamics and early diagnosis in infectious disease," J. Math. Biol., vol. 43, pp. 446-470, 2001.
• B. Daşbaşı and İ. Öztürk, "The dynamics between pathogen and host with Holling type 2 response of immune system," Journal Of Graduate School of Natural and Applied Sciences, vol. 32, pp. 1-10, 2016.
• A. Pugliese and A. Gandolfi, "A simple model of pathogen–immune dynamics including specific and non-specific immunity," Math. Biosci., vol. 214, pp. 73–80, 2008.   T. Kostova, "Persistence of viral infections on the population level explained by an immunoepidemiological model," Math. Biosci., vol. 206, no. 2, pp. 309-319, 2007.
• M. Gilchrist and A. Sasaki, "Modeling host–parasite coevolution: A nested approach based on mechanistic models," J. Theor. Biol., vol. 218, pp. 289-308, 2002.
• M. Merdan, Z. Bekiryazici, T. Kesemen, and T. Khaniyev, "Comparison of stochastic and random models for bacterial resistance," Advances in Difference Equations, vol. 133, pp. 1-19.
• R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics.: Springer, Wien, 1997.
• E. M. El-Mesiry, A.M.A. El-Sayed, and H.A.A. El-Saka, "Numerical methods for multi-term fractional (arbitrary) orders differential equations," Appl. Math. Comput., vol. 160, no. 3, pp. 683–699, 2005.
• D. Matignon, "Stability results for fractional differential equations with applications to control processing," Comput. Eng. Sys. Appl. 2, vol. 963, 1996.
• I. Podlubny, Fractional Differential Equations. New York: Academic Press, 1999.
• B. Daşbaşı and T. Daşbaşı, "Mathematical Analysis of Lengyel-Epstein Chemical Reaction Model by Fractional-Order Differential Equation’s System with Multi-Orders," International Journal of Science and Engineering Investigations, vol. 6, no. 70, pp. 78-83, 2017.
• A. M. A. El-Sayed, E. M. El-Mesiry, and H. A. A. El-Saka, "On the fractional-order logistic equation," AML, vol. 20, pp. 817-823, 2007.