Analysis through the FDE Mathematical Model with Multiple Orders the Effects of the Specific Immune System Cells and the Multiple Antibiotic Treatment against Infection

Year 2018, Volume: 10 Issue: 3, 207 - 236, 04.11.2018

Bahatdin Daşbaşı , İlhan Öztürk , Nurcan Menekşe

https://doi.org/10.24107/ijeas.458642

Abstract

In this study, the infection process in infectious individual is
mathematically modeled by using a system of multiple fractional order
differential equations. Qualitative analysis of the model was done. To
mathematically examine the effect of Pseudomonas Aeruginosa and Mycobacterium
tuberculosis and their treatment methods, the results of the proposed model are
compared with numerical simulations with the help of datas obtained from the
literature.

Keywords

Fractional-Order Differential Equation, Mathematical model, Qualitative Analysis, Numerical Simulation

References

• Wang, J., Xu, T.-Z., Wei, Y.-Q. Wei, and Xie, J.-Q., Numerical solutions for systems of fractional order differential equations with Bernoulli wavelets. International Journal Of Computer Mathematics, DOI: 10.1080/00207160.2018.1438604, 1-20, 2018.
• Stankovic, T.M., Atanackovis, B., On a numerical scheme for solving differential equations of fractional order. Mechanics Research Communications, 35, 429 – 438, 2008.
• Daher Okiye, M.A., Aziz-Alaoui, M., Boundedness and global stability for a predator – prey model with modified Leslie – Gower and Holling-type II schemes. Applied Math. Lett., 16, 1069-1075, 2003.
• Caputo, M., Linear models of dissipation whose Q is almost frequency independent. Geophys. J. Int., 13, 5, 529–539, 1967.
• Clark, C.W., Mathematical models in the economics of renewable resources. SIAM Rev., 21, 81 – 89, 1979.
• Mukherjee, T., Chaudhari, R.N., Das, K.S., Bioeconomic harvesting of a prey – predator fishery. J. Biol. Dyn., 3, 447-462, 2009.
• Davis, H.D., The Theory of Linear Operators. Indiana: Principia Press, 1936.
• Peng, Y., Wang, R., Du, M., Effect of protection zone in the diffusive Leslie predator – prey model.. J. Differ. Equ, 3932 – 3956, 2009.
• El-Sayed, A.M.A., Multivalued fractional differential equations. Aplied Math and Comput., 80, 1-11, 1994.
• El-Sayed, A.M.A., Fractional order evolution equations. Journal Of Fractional Calculus, 7, 89-100, 1995.
• Alidousti, K., Eshkaftaki, J., Ghaziani, B., Stability and dynamics of a fractional order Leslie-Gower prey-predator model. Applied Math. Modelling, 1-12, 2013.
• Podlubny, I., Fractional Differential Equations. New York: Academic Press, 1999.
• Jesus, I.S., Machado, J.A.T., Fractional control of heat diffusion systems. Nonlinear Dynamics, 54, 3, 2008.
• Vinagre, B.M., Petras, I., Merchan, P., Dorcak, L., Two Digital Realizations of Fractional Controllers: Application to Temperature Control of a Solid. in Proceedings of the European Control Conference 2001 , Porto, Portugal, 2001, 1765-1767.
• Parada, F.J.V., Tapia, J.A.O., Ramirez, J.A., Effective medium equations for fractional Ficks law in porous media. Physica A, 373, 339-353, 2007.
• Torvik, P.J., Bagley, R.L., On the Appearance of the Fractional Derivative in the Behavior of Real Materials. Transactions of the ASME, 51, 294-298, 1984.
• Gaul, L., Klein, P., Kempfle, S., Damping description involving fractional operators. Mech. Syst. Signal Process, 5, 81-88, 1991.
• Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993.
• Samko, G., Kilbas, A., Marichev, O., Fractional Integrals and Derivatives: Theory and Applications. Amsterdam: Gordon and Breach, 1993.
• Zaslavsky, G.M., Hamiltonian Chaos and Fractional Dynamics. Oxford: Oxford University Press, 2005.
• Matsuzaki, T., Nakagawa, M., A chaos neuron model with fractional differential equation. J. Phys. Soc. Japan, 72, 2678-2684, 2003.
• Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A., Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. J. Math. Anal. Appl, 325, 542-553, 2007.
• Turchin, P., Does population ecology have general laws?. Oikos, 94, 17-26, 2001.
• Malthus, T.R., An Essay On The Principle Of Population as it Affects The Future Improvement of Society. London: J. Johnson, 1798.
• Pearl, R., Reed, L.J., On the rate of growth of the population of the united states since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences, 6, 6, 275 – 288, 1920.
• Verhuslt, P.F., Notice sur la loi que la population suit dans son accroissement. Correpondance mathematique et Physique, 10, 112–121, 1838.
• Winsor, C.P., The Gompertz curve as a growth curve. Proceedings of the national academy of sciences, 18, 1, 1-8, 1932.
• Gompertz, B., On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, 513-585, 1825.
• Durbin, P.W., Jeung, N., Williams, M.H., Arnold, J.S., Construction of a growth curve for mammary tumors of the rat. Cancer Research, 27, 1341-1347, 1967.
• Chow, G.C., Technological change and the demand for computers. The American Economic Review, 57, 5, 1117–1130, 1967.
• Stranndberg, P.E., The chemostat. Tech. rep.. Univeristy of Linköping, 2003.
• Gard, T.C., Hallam, T.G., Persistence in food webs: I Lotka-Volterra food chains. Bull. Math. Biol., 41, 877-891, 1979.
• Lotka, A.J., Elements of physical biology. Baltimore: Williams and Wilkins, 1925.
• Volterra, V., Fluctuations in the abundance of a species considered mathematically. Nature, 118, 558–560, 1926.
• Kolmogorov, A.N., Sulla teoria di Volterra della lotta per l' esistenza. Giornale Istituto Ital. Attuari, 7, 74-80, 1936.
• May, R.M., Limit cycles in predator-prey communities. Science, 177, 900-902, 1972.
• Sterman, J.D., Business dynamics: Systems Thinking and Modeling for a Complex World., 2000.
• Daşbaşı, B., The Fractional-Order mathematical modeling of bacterial resistance against multiple antibiotics in case of local bacterial infection. Sakarya University Journal of Science, 251, 3, 1-13, 2017.
• Mondragón E.I., et al., Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations. BioSystems, 117, 60–67, 2014.
• Ternent, L., Dyson, R.J., Krachler, A.M., Jabbari, S., Bacterial fitness shapes the population dynamics of antibiotic resistant and susceptible bacteria in a model. J. Theor. Biol., 372, 1-11, 2014.
• Daşbaşı, B., Öztürk, İ., Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response. SpringerPlus, 5, 408, 1-17, April 2016.
• Daşbaşı, B., Öztürk, İ., The dynamics between pathogen and host with Holling type 2 response of immune system. Journal Of Graduate School of Natural and Applied Sciences, 32, 1-10, 2016.
• Kostova, T., Persistence of viral infections on the population level explained by an immunoepidemiological model. Math. Biosci., 206, 2, 309-319, 2007.
• Daşbaşı, B., Dynamics between Immune System-Bacterial Loads. Imperial Journal of Interdisciplinary Research, 2, 8, 526-536, 2016.
• Pugliese, A., Gandolfi, A., A simple model of pathogen–immune dynamics including specific and non-specific immunity. Math. Biosci., 214, 73–80, 2008.
• Daşbaşı B., Öztürk, İ., On The Stability Analysis of The General Mathematical Modeling of Bacterial Infection. International Journal Of Engineering & Applied Sciences, 10, 2, 93-117, 2018.
• Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A., Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl., 325, 542-553, 2007.
• Handel, A., Margolis, E., Levin, B., Exploring the role of the immune response in preventing antibiotic resistance. J.Theor.Biol., 256, 655–662, 2009.
• Smith, A., McCullers, J., Adler, F., Mathematical model of a three-stage innate immune response to a pneumococcal lung infection. J. Theor. Biol., 276, 106–116, 2011.
• Campion, J.J., McNamara, P.J., Evans, M.E., Pharmacodynamic modeling of ciprofloxacin resistance in Staphylococcus aureus. Antimicrob. Agents Chemother, 49, 1, 209-219, 2005.
• Chung, P., McNamara, P.J., Campion, J.J., Evans, M.E., Mechanism-based pharmacodynamic models of fluoroquinolone resistance in Staphylococcus aureus. Antimicrob. Agents Chemother, 50, 9, 2957–2965, 2006.
• Health Organization World, The Evolving Threat of Antimicrobial Resistance. in Options for Action, 2012, 1503-1518 ISBN: 978 924.
• Alavez, J., et al., Within-host population dynamics of antibiotic-resistant M. tuberculosis. Math. Med. Biol., 24, 35-56, 2006.
• Mohtashemi, M., Levins, R., Transient dynamics and early diagnosis in infectious disease. J. Math. Biol., 43, 446-470, 2001.
• Coll, P., Fármacos con actividad frente a Mycobacterium tuberculosis. Enfer-medades Infecciosas y Microbiologa Clnica, 27, 8, 474–480, 2009.
• Zhang, Y., Mechanisms of drug resistance in Mycobacterium tuberculosis. Int. J. Tuberc. Lung Dis., 13, 11, 1320–1330, 2009.
• Romero, J., Ibargüen, E., Esteva, L., Un modelo matemático sobre bacteriassensibles y resistentes a antibióticos. Matemáticas: Ensĕnanza Universitaria, 20, 1, 55-73, 2011.