The Fractional-Order mathematical modeling of bacterial resistance against multiple antibiotics in case of local bacterial infection

Yıl 2017, , 442 - 453, 01.06.2017

Bahatdin Daşbaşı

https://doi.org/10.16984/saufenbilder.298934

Cited By: 1

Öz

In this study, it is described
the general forms of fractional-order differential equations and asymtotic
stability of their system’s equilibria. In addition that, the stability
analysis of equilibrium points of the local bacterial infection model which is
fractional-order differential equation system, is made. Results of this
analysis are supported via numerical simulations drawn by datas obtained from
literature for mycobacterium tuberculosis and the antibiotics isoniazid (INH),
rifampicin (RIF), streptomycin (SRT) and pyrazinamide (PRZ) used against this
bacterial infection.

Anahtar Kelimeler

Fractional-order differential equation system, Mathematical model, Stability analysis, Equilibrium points, Multiple antibiotics

Lokal Bakteriyel enfeksiyon durumunda çoklu antibiyotik tedavisine karşı bakteriyel direncin kesirsel mertebeden matematiksel modellemesi

Öz

Bu çalışmada kesirsel mertebeden diferansiyel denklemlerin genel biçimi ve bu denklemlerin sistemlerinin
dengelerinin asimptotik kararlılıkları tanımlandı. Ayrıca kesirsel mertebeden diferansiyel denklem sistemi şeklinde
ifade edilen lokal bir bakteriyel enfeksiyon modelinin denge noktalarının kararlılık analizi yapıldı. Bu analizin
sonuçları mycobacterium tuberculosis bakterisi ve bu bakterinin neden olduğu enfeksiyona karşı kullanılan isoniazid
(INH), rifampicin (RIF), streptomycin (SRT) ve pyrazinamide (PRZ) antibiyotikleri için literatürden elde edilen
veriler kullanılarak çizilen nümerik simülasyonlar vasıtasıyla desteklendiler.  

Anahtar Kelimeler

Kesirsel mertebeden diferansiyel denklem sistemi, Matematiksel model, Kararlılık analizi, Denge noktaları, Çoklu antibiyotik tedavisi

Kaynakça

• [1] E I Mondragón et al., "Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations," BioSystems, vol. 117, pp. 60–67, 2014.
• [2] 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.
• [3] A G Mahmoud and L B Rice, "Antifungal agents: mode of action, mechanisms of resistance, and correlation of these mechanisms with bacterial resistance, and correlation," Clin. Microbiol. Rev., vol. 12, no. 4, pp. 501–517, 1999.
• [4] L Ternent, R J Dyson, A M Krachler, and S Jabbari, "Bacterial fitness shapes the population dynamics of antibiotic resistant and susceptible bacteria in a model," J. Theor. Biol., vol. 372, pp. 1-11, 2014.
• [5] D P Arya, Aminoglycoside Antibiotics: From Chemical Biology to Drug Discovery. New Jersey: Wiley, 2007.
• [6] M S Butler and A D Buss, "Natural products - The future scaffolds for novel antibiotics?," Biochem. Pharmacol., vol. 71, no. 7, pp. 919-929, 2006.
• [7] A E Clatworthy, E P Pierson, and D T Hung, "Targeting virulence: a new paradigm for antimicrobial therapy," Nature Chem. Biol., vol. 3, pp. 541-548, 2007.
• [8] K Lewis, "Platforms for antibiotic discovery," Nat. Rev. Drug Discov., vol. 12, pp. 371-387, 2013.
• [9] A J McMichael, "La “epidemiología molecular”: nueva ruta de investigación o compañero de viaje?," Bol. Oficina. Sanit. Panam., vol. 119, no. 3, pp. 243–254, 1995.
• [10] Y Zhang, "Mechanisms of drug resistance in Mycobacterium tuberculosis," Int. J. Tuberc. Lung Dis., vol. 13, no. 11, pp. 1320–1330, 2009.
• [1] Y. Xue and J. Wang, "Backward bifurcation of an epidemic model with infectious force in infected and immune period and treatment," vol. 14, 2012.
• [2] H W Hethcote, "The mathematics of infectious diseases," SIAM Rev., vol. 42, pp. 599-653, 2000.
• [3] B Singer, "Mathematical Models of infectious diseases: seeking new tools for planning and evaluating control programs," Supplement to Popul. Dev. Rev., vol. 10, pp. 347–365, 1984.
• [4] M Mohtashemi and R Levins, "Transient dynamics and early diagnosis in infectious disease," J. Math. Biol., vol. 43, pp. 446-470, 2001.
• [5] 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.
• [6] A Whitman and H Ashrafiuon, "Asymptotic theory of an infectious disease model," J. Math. Biol., vol. 53, no. 2, pp. 287-304, 2006.
• [7] I. Podlubny, Fractional Differential Equations.: Academic Press, 1999.
• [8] 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.
• [9] A.M.A. El-Sayed, F.M. Gaafar, and H.H. Hashem, "On the maximal and minimal solutions of arbitrary orders nonlinear functional integral and differential equations," Math. Sci. Res. J., vol. 8, no. 11, pp. 336–348, 2004.
• [10] D. Matignon, "Stability results for fractional differential equations with applications to control processing," Comput. Eng. Sys. Appl. 2, vol. 963, 1996.
• [1] I. Podlubny and A.M.A. El-Sayed, On Two Definitions of Fractional Calculus.: Slovak Academy of Science, Institute of Experimental Phys., 1996.
• [2] E. Ahmed, A.M.A. El-Sayed, and H.A.A. El-Saka, "On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems," Phys. Lett. A, vol. 358, 2006.
• [3] E. Ahmed, A.M.A. El-Sayed, and H.A.A. El-Saka, "Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models," J. Math. Anal. Appl., vol. 325, pp. 542-553, 2007.
• [4] H.A. El-Saka, E. Ahmed, M.I. Shehata, and A.M.A. El-Sayed, "On stability, persistence and Hopf Bifurcation in fractional order dynamical systems," Nonlinear Dyn., vol. 56, pp. 121-126, 2009.
• [5] H. El-Saka and A. El-Sayed, Fractional Order Equations and Dynamical Systems. Germany: Lambrt Academic Publishing, 2013.
• [6] Bahatdin Daşbaşı and İlhan 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, no. 1, pp. 1-10, 2016.
• [7] K. Diethelm and N. J. Ford, "Analysis of fractional differential equations," Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229-248, 2002.
• [8] Health Organization World, "The Evolving Threat of Antimicrobial Resistance," in Options for Action, 2012, pp. 1503-1518 ISBN: 978 924.
• [9] J Alavez et al., "Within-host population dynamics of antibiotic-resistant M. tuberculosis," Math. Med. Biol., vol. 24, pp. 35-56, 2006.
• [10] P. Coll, "Fármacos con actividad frente a Mycobacterium tuberculosis," Enfer-medades Infecciosas y Microbiologa Clnica, vol. 27, no. 8, pp. 474–480, 2009.
• Y Zhang, Y Dhandayuthapani, S V Deretic,Molecular basis for the exquisite sensitivity of Mycobacterium Tuberculosis to isoniazid,13212-13216,PNAS,vol. 93,1996
• [32] J. Romero, E. Ibargüen, L. Esteva,Un modelo matemático sobre bacteriassensibles y resistentes a antibióticos,55-73,Matemáticas: Ense˜nanza Universitaria,2011,vol. 20