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


Abstract views: 449 / PDF downloads: 340

Authors

DOI:

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

Keywords:

Fractional-Order Differential Equation, Numerical Simulation, Stability Analysis.

Abstract

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 R0 , it is founded the stability conditions of these equilibrium points. Also, the qualitative analysis was supported by numerical simulations.

Downloads

Download data is not yet available.

References

[1] L. J. S. Allen, An Introduction to Mathematical Biology., 2007, ISBN 10: 0-13-035216-0.
[2] H. El-Saka and A. El-Sayed, Fractional Order Equations and Dynamical Systems. Germany: Lambrt Academic Publishing, 2013.
[3] 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.
[4] F. A. Rihan, "Numerical Modeling of Fractional-Order Biological Systems," Abstract and Applied Analysis, pp. 1-11, 2013.
[5] W. Deng and C. Li, "Analysis of Fractional Differential Equations with Multi-Orders," Fractals, vol. 15, no. 2, pp. 173-182, 2007.
[6] M. Axtell and E. M. Bise, "Fractional calculus applications in control systems," in Proc. of the IEEE, New York, 1990, pp. 563-566.
[7] A. M. A. El-Sayed, "Fractional differential-difference equations," J. Fract. Calc., vol. 10, pp. 101–106, 1996.
[8] 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.
[9] I. Podlubny and A.M.A. El-Sayed, On Two Definitions of Fractional Calculus.: Slovak Academy of Science, Institute of Experimental Phys., 1996.
[10] 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.
[11] 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.
[12] B. Daşbaşı, "Dynamics between Immune System-Bacterial Loads," Imperial Journal of Interdisciplinary Research, vol. 2, no. 8, pp. 526-536, 2016.
[13] 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.
[14] M. Mohtashemi and R. Levins, "Transient dynamics and early diagnosis in infectious disease," J. Math. Biol., vol. 43, pp. 446-470, 2001.
[15] 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.
[16] 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.
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] 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.
[22] D. Matignon, "Stability results for fractional differential equations with applications to control processing," Comput. Eng. Sys. Appl. 2, vol. 963, 1996.
[23] I. Podlubny, Fractional Differential Equations. New York: Academic Press, 1999.
[24] 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.
[25] 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.

Downloads

Published

2019-01-31

How to Cite

ÖZTÜRK, İlhan, DAŞBAŞI, B., & CEBE, G. (2019). A SIMPLE MATHEMATICAL MODEL THROUGH FRACTIONAL-ORDER DIFFERENTIAL EQUATION FOR PATHOGENIC INFECTION. HEALTH SCIENCES QUARTERLY, 3(1), 29–40. https://doi.org/10.26900/jsp.3.004

Issue

Section

Letter to the Editor