On the Stability Analysis of the Generalized Mathematical Model with Fractional-Order for Mycobacterium Tuberculosis

Yıl 2019, Cilt: 9 Sayı: 1, 272 - 287, 01.03.2019

Bahatdin Daşbaşı

https://doi.org/10.21597/jist.450193

Öz

In this study, the mathematical model, based on the system of fractional-order differential equations and examined the dynamics among concentrations of multiple antibiotic, immune system cells of host and sensitive and resistant bacterial populations to antibiotic in case of bacterial infection, was proposed. The existence and uniqueness of this model solutions were showed. In addition, according to the specific conditions of the parameters used in the model, the existence of disease-free equilibrium point and the stability of this point was examined. The proposed model with numerical simulations by using the parameter values obtained from the literature for Mycobacterium Tuberculosis (Mtb) was supported, which is fully compatible with the recommended treatment method.

Anahtar Kelimeler

Fractional-order differential equation, equilibrium point, stability, numerical simulation, mycobacterium tuberculosis

Mycobacterium Tuberculosis için Genelleştirilmiş Kesirsel Mertebeden Matematiksel Modelin Kararlılık Analizi Üzerine

Öz

Bu çalışmada kesirsel mertebeden diferansiyel denklem sistemi temel alınarak bakteriyel bir enfeksiyon durumunda çoklu antibiyotik konsantrasyonu, bu antibiyotiklere hassas ve dirençli bakteri popülasyonları ve konakçının bağışıklık sistemi hücrelerinin aralarındaki dinamikleri inceleyen bir matematiksel model önerildi. Modelin çözümünün varlığı ve tekliği gösterildi. Ayrıca modelde kullanılan parametrelerin özel durumlarına göre, enfeksiyondan bağımsız denge noktasının varlığı ve bu denge noktasının kararlılığı bulundu. Bunlara ek olarak Mycobacterium Tuberculosis (Mtb) için literatürden elde edilen parametre değerleri kullanılarak önerilen tedavi yöntemiyle bire bir uyumlu Nümerik simülasyonlarla önerilen model desteklendi.

Anahtar Kelimeler

Kesirli mertebeden diferansiyel denklem, denge noktası, kararlılık, nümerik simülasyon, mycobacterium tuberculosis

Kaynakça

• Alavez J, Avenda R, Esteva L, Fuentes J, Garcia G, Gómez G, 2006. Within-host population dynamics of antibiotic-resistant M. tuberculosis. Math. Med. Biol., 24: 35-56.
• Allen LJ, 2007. An Introduction to Mathematical Biology. London: Pearson Education.
• Coll P, 2009. Fármacos con actividad frente a Mycobacterium tuberculosis. Enfer-medades Infecciosas y Microbiologa Clnica, 27: 474–480.
• D’Agata E, Magal P, Olivier D, Ruan S, Webb GF, 2007. Modeling antibiotic resistance in hospitals: the impact of minimizing treatment duration. J. Theor. Biol., 249: 487-499.
• Daşbaşı B, 2017. The Fractional-Order mathematical modeling of bacterial resistance against multiple antibiotics in case of local bacterial infection. Sakarya University Journal of Science, 251: 1-13.
• Daşbaşı B, 2018. Çoklu Kesirli Mertebeden Diferansiyel Denklem Sistemlerinin Kalitatif Analizi, Analizdeki Bazi Özel Durumlar ve Uygulamasi: Av-Avci Modeli. Fen Bilimleri ve Matematik'te Akademik Araştırmalar (1. b., s. 127-157). içinde Ankara: Gece Kitaplığı.
• Daşbaşı B, Öztürk İ, 2016. Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response. SpringerPlus, 5: 1-17.
• Dokuyucu MA, Çelik E, 2016. Nonlinear diffusion for chemotaxis and birth-death process for Keller-Segel model. New Trends in Mathematical Sciences, 4: 204-211.
• El-Saka H, El-Sayed A, 2013. Fractional Order Equations and Dynamical Systems. Germany: Lambrt Academic Publishing.
• Fang C-Q, Sun H-Y, Gu J-P, 2015. Application of Fractional Calculus Methods to Viscoelastic Response of Amorphous Shape Memory Polymers. Journal of Mechanics, 4: 427-432.
• Gomez-Aguilar J, Razo-Hernandez R, Granados-Lieberman D, 2014. A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response. Revista Mexicana de Fisica, 60: 32–38.
• Ionescu C, Caponetto R, Chen Y-Q, 2013. Special Issue on "Fractional Order Modeling and Control in Mechatronics". Mechatronics, 23: 739-740.
• Meilanov RP, Magomedov RA, 2014. Thermodynamics in Fractional Calculus. Journal of Engineering Physics and Thermophysics, 87: 1521-1531.
• Miljković N, Popović N, Djordjević O, Konstantinović L, Šekara TB, 2017. ECG artifact cancellation in surface EMG signals by fractional order calculus application. Computer Methods and Programs in Biomedicine, 140: 259-264.
• Mondragón EI, Mosquera S, Cerón M, Burbano-Rosero EM, Hidalgo-Bonilla SP, Esteva L, 2014. Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations. BioSystems, 117: 60–67.
• Odibat Z, Corson N, Aziz-Alaoui M, Alsaedi A, 2017. Chaos in Fractional Order Cubic Chua System and Synchronization. International Journal of Bifurcation and Chaos, 27: 1-13.
• Owolabi KM, 2018. Riemann-Liouville Fractional Derivative and Application to Model Chaotic Differential Equations. Progr. Fract. Differ. Appl., 4: 99-110.
• Pugliese A, Gandolfi A, 2008. A simple model of pathogen–immune dynamics including specific and non-specific immunity. Math. Biosci., 214: 73–80.
• Rihan FA, Hashish A, Al-Maskari F, Sheek-Hussein M, Ahmed E, Riaz MB, 2016. Dynamics of Tumor-Immune System with Fractional-Order. Journal of Tumor Research, 2: 1-6.
• Romero J, Ibargüen E, Esteva L, 2011. Un modelo matemático sobre bacteriassensibles y resistentes a antibióticos. Matemáticas: Ense˜nanza Universitaria, 20: 55-73.
• Sheikh NA, Ali F, Saqib M, Khan I, Jan SA, Alshomrani AS, 2017. Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results in Physics, 7: 789-800.
• Sikora R, 2017. Fractional derivatives in electrical circuit theory – critical remarks. Archives of Electrical Engineering, 66: 155-163.
• Smith A, McCullers J, Adler F, 2011. Mathematical model of a three-stage innate immune response to a pneumococcal lung infection. J. Theor. Biol., 276: 106–116.
• Tarasova VV, Tarasov, VE, 2016. Elasticity For Economic Processes With Memory: Fractional Differential Calculus Approach. Fractional Differential Calculus, 6: 219-232.
• Ternent L, Dyson RJ, Krachler AM, Jabbari S, 2014. Bacterial fitness shapes the population dynamics of antibiotic resistant and susceptible bacteria in a model. J. Theor. Biol., 372: 1-11.
• Zhang Y, Dhandayuthapani Y, Deretic SV, 1996. Molecular basis for the exquisite sensitivity of Mycobacterium Tuberculosis to isoniazid. PNAS, 93: 13212-13216.