Analysis of an Epidemic Spreading Model with Exponential Decay Law

Mehmet Yavuz; Necati Özdemir

doi:10.36753/mathenot.691638

EN

Analysis of an Epidemic Spreading Model with Exponential Decay Law

Abstract

Mathematical modeling of infectious diseases has shown that combinations of isolation, quarantine, vaccine, and treatment are often necessary in order to eliminate most infectious diseases. Continuous mathematical models have been used to study the dynamics of infectious diseases within a human host and in the population. We have used in this study a SIR model that categorizes individuals in a population as susceptible (S), infected (I) and recovered (R). It also simulates the transmission dynamics of diseases where individuals acquire permanent immunity. We have considered the SIR model using the Caputo-Fabrizio and we have obtained special solutions and numerical simulations using an iterative scheme with Laplace transform. Moreover, we have studied the uniqueness and existence of the solutions

Keywords

epidemic model,exponential law,numerical simulation,fractional derivative,non-singularity

References

\bibitem{CVD} De Le\'{o}n, C. V.: \textit{Constructions of Lyapunov Functions for Classics SIS, SIR and SIRS Epidemic Model with Variable Population Size.} Foro-Red-Mat: Revista Electr\'{o}nica de Contenido Matem\'{a}tico. \textbf{26,} 1-12 (2009).
\bibitem{MKAM} Kermark, M., Mckendrick, A.: \textit{Contributions to the Mathematical Theory of Epidemics.} Part I. Proc. R. Soc. A. \textbf{115,} 700-721 (1927).
\bibitem{OAA} Arqub, O. A., El-Ajou, A.: \textit{Solution of the Fractional Epidemic Model by Homotopy Analysis Method.} Journal of King Saud University-Science. \textbf{25} (1), 73-81 (2013).
\bibitem{HES} El-Saka, H.: \textit{The Fractional-Order SIR and SIRS Epidemic Models with Variable Population Size.} Mathematical Sciences Letters. \textbf{2} (3), 195 (2013).
\bibitem{CNBI} Angstmann, C. N., Henry, B. I., McGann, A. V.: \textit{A Fractional-Order Infectivity SIR Model.} Physica A: Statistical Mechanics and its Applications. \textbf{452,} 86-93 (2016).
\bibitem{AJG} Arenas, A. J., González-Parra, G., Chen-Charpentier, B. M.: \textit{Construction of Nonstandard Finite Difference Schemes for the SI and SIR Epidemic Models of Fractional Order.} Mathematics and Computers in Simulation. \textbf{121,} 48-63 (2016).
\bibitem{CBAA} Angstmann, C., Henry, B., McGann, A.: \textit{A Fractional Order Recovery SIR Model from a Stochastic Process.} Bulletin of Mathematical Biology. \textbf{78} (3), 468-499 (2016).
\bibitem{EAN} Demirci, E., Unal, A., Özalp, N.: \textit{A Fractional Order SEIR Model with Density Dependent Death Rate.} Hacettepe Journal of Mathematics and Statistics. \textbf{40} (2), 287-295 (2011).
\bibitem{EAMA} Bonyah, E., Atangana, A., Khan, M. A.: \textit{Modeling the Spread of Computer Virus Via Caputo Fractional Derivative and the Beta-Derivative.} Asia Pacific Journal on Computational Engineering. \textbf{4} (1), 1-15 (2017).
\bibitem{EJA} Bonyah, E., G\'{o}mez-Aguilar, J., Adu, A.: \textit{Stability Analysis and Optimal Control of a Fractional Human African Trypanosomiasis Model.} Chaos, Solitons and Fractals. \textbf{117,} 150-160 (2018).

\bibitem{YavuzBonyah} Yavuz, M., Bonyah, E.: \textit{New approaches to the fractional dynamics of schistosomiasis disease model.} Physica A: Statistical Mechanics and its Applications, \textbf{525,} 373-393 (2019).
\bibitem{ARHAC} Aguirre-Ramos, H., Avina-Cervantes, J. G., Cruz-Aceves, I., Ruiz-Pinales, J., Ledesma, S.: \textit{Blood Vessel Segmentation in Retinal Fundus Images Using Gabor Filters, Fractional Derivatives, and Expectation Maximization.} Applied Mathematics and Computation. \textbf{339,} 568-587 (2018)
\bibitem{APMK} Prakash, A., Kumar, M.: \textit{Numerical Solution of Two Dimensional Time Fractional-order Biological Population Model.} Open Physics. \textbf{14,} 177-186 (2016)
\bibitem{Elazzouzi1} Elazzouzi, A., Alaoui, A. L., Tilioua, M., Torres, D. F.: \textit{Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse.} Statistics, Optimization \& Information Computing. \textbf{7} (3), 545-557 (2019).
\bibitem{MCMF} Caputo, M., Fabrizio, M.: \textit{A New Definition of Fractional Derivative without Singular Kernel.} Progress in Fractional Differentiation and Applications. \textbf{1} (2), 1-13 (2015).
\bibitem{AADB1} Atangana, A., Baleanu, D.: \textit{New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model.} Thermal Science. \textbf{20} (2), 763-769 (2016).
\bibitem{AAIK} Atangana, A., Koca, I.: \textit{Chaos in a Simple Nonlinear System with Atangana–Baleanu Derivatives with Fractional Order.} Chaos, Solitons and Fractals. \textbf{89,} 447-454 (2016).
\bibitem{XJYF} Yang, X. J., Gao, F., Machado, J. A., Baleanu, D.: \textit{A New Fractional Derivative Involving the Normalized Sinc Function without Singular Kernel.} The European Physical Journal Special Topics. \textbf{226,} 3567-3575 (2017).
\bibitem{OJJA} Algahtani, O. J. J.: \textit{Comparing the Atangana–Baleanu and Caputo–Fabrizio Derivative with Fractional Order: Allen Cahn Model.} Chaos, Solitons and Fractals. \textbf{89,} 552-559 (2016).
\bibitem{AADB2} Atangana, A., Baleanu, D.: \textit{Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer.} Journal of Engineering Mechanics. \textbf{143,} D4016005-1-5 (2017).
\bibitem{DBBA} Baleanu, D., Agheli, B., Al Qurashi, M. M.: \textit{Fractional Advection Differential Equation within Caputo and Caputo–Fabrizio Derivatives.} Advances in Mechanical Engineering. \textbf{8} (12), 1-8 (2016).
\bibitem{JFVF} G\'{o}mez-Aguilar, J. F., Morales-Delgado, V. F., Taneco-Hern\'{a}ndez, M. A., Baleanu, D., Escobar-Jim\'{e}nez, R. F., Al Qurashi, M. M.: \textit{Analytical Solutions of the Electrical Rlc Circuit Via Liouville–Caputo Operators with Local and Non-Local Kernels.} Entropy. \textbf{18,} 402 (2016).
\bibitem{JH1} Hristov, J.: \textit{Transient Heat Diffusion with a Non-Singular Fading Memory: From the Cattaneo Constitutive Equation with Jeffrey’s Kernel to the Caputo-Fabrizio Time-Fractional Derivative.} Thermal Science. \textbf{20} (2), 757-762 (2016).
\bibitem{JH2} Hristov, J.: \textit{Steady-State Heat Conduction in a Medium with Spatial Non-Singular Fading Memory: Derivation of Caputo-Fabrizio Space-Fractional Derivative from Cattaneo Concept with Jeffrey's Kernel and Analytical Solutions.} Thermal science. \textbf{21} (2), 827-839 (2017).
\bibitem{NAFA} Sheikh, N. A., Ali, F., Saqib, M., Khan, I., Jan, S. A. A., Alshomrani, A. S., Alghamdi, M. S.: \textit{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. \textbf{7,} 789-800 (2017).
\bibitem{MYNO1} Yavuz, M., Özdemir, N.: \textit{European Vanilla Option Pricing Model of Fractional Order without Singular Kernel.} Fractal and Fractional. \textbf{2} (3), 1-11 (2018).
\bibitem{FEMY} Evirgen, F., Yavuz, M.: \textit{An Alternative Approach for Nonlinear Optimization Problem with Caputo-Fabrizio Derivative,} In: Proceedings, ITM Web of Conferences: EDP Sciences, 01009, (2018).
\bibitem{JSDK} Singh, J., Kumar, D., Hammouch, Z., Atangana, A.: \textit{A Fractional Epidemiological Model for Computer Viruses Pertaining to a New Fractional Derivative.} Applied Mathematics and Computation. \textbf{316,} 504-515 (2018).
\bibitem{YavuzChar} Yavuz, M. \textit{Characterizations of two different fractional operators without singular kernel.} Mathematical Modelling of Natural Phenomena, \textbf{14} (3), 302 (2019).
\bibitem{DAAY} Avcı, D., Yetim, A. \textit{Analytical Solutions to the Advection-Diffusion Equation with the Atangana-Baleanu Derivative over a Finite Domain.} Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, \textbf{20} (2), 382-395 (2018).
\bibitem{DAYVZ} Avcı, D., Yavuz, M., Özdemir, N.: Fundamental solutions to the Cauchy and Dirich-let problems for a heat conduction equation equipped with the Caputo-Fabrizio differentiation. In: Heat Conduction: Methods, Applications and Research. Nova Science Publishers, 95-107 (2019).
\bibitem{OUE} Özdemir, N., Uçar, S., Eroğlu, B. B. İ.: \textit{Dynamical Analysis of Fractional Order Model for Computer Virus Propagation with Kill Signals.} International Journal of Nonlinear Sciences and Numerical Simulation, (2020). https://doi.org/10.1515/ijnsns-2019-0063
\bibitem{FOUZ} Evirgen, F., Uçar, S., Özdemir, N., Hammouch, Z.: \textit{System Response of an Alcoholism Model Under the Effect of Immigration via Non-singular Kernel Kerivative.} Discrete \& Continuous Dynamical Systems-S, (2020). https://doi.org/10.3934/dcdss.2020145
\bibitem{UOA} Ucar, E., Özdemir, N., Altun, E.: \textit{Fractional Order Model of Immune Cells Influenced by Cancer Cells.} Mathematical Modelling of Natural Phenomena, \textbf{14} (3), 308 (2019).
\bibitem{MY111} Yavuz, M.: \textit{Fundamental Solution of Heat Problem with a New Fractional Derivative Operator Involving Normalized Sinc Function.} In: Proceedings of the Mathematical Studies and Applications, 4-6 October 2018, Karaman, TURKEY. Proceedings, 194-198 (2018).
\bibitem{MY112} Yavuz, M., Özdemir, N.: \textit{An Integral Transform Solution for Fractional Advection-Diffusion Problem.} In: Proceedings of the Mathematical Studies and Applications, 4-6 October 2018, Karaman, TURKEY. Proceedings, 442-446 (2018).
\bibitem{MYTHABET} Yavuz, M., Abdeljawad, T: \textit{On a Nem Integral Transformation Applied to Fractional Derivative with Mittag-Leffler Nonsingular Kernel.} Electronic Research Archive, \textbf{28} (1), 1-11 (2020). https://doi.org/10.3934/era.2020039 \bibitem{bahar1} Acay, B., Bas, E., Abdeljawad, T.: \textit{Non-local fractional calculus from different viewpoint generated by truncated M-derivative.} Journal of Computational and Applied Mathematics, \textbf{366,} 112410, (2020).
\bibitem{ramazan1} Bas, E., Ozarslan, R., Baleanu, D., Ercan, A.: \textit{Comparative simulations for solutions of fractional Sturm–Liouville problems with non-singular operators.} Advances in Difference Equations, \textbf{2018} (1), 350 (2018).
\bibitem{sene} Sene, N.: \textit{Stability Analysis of the Fractional Differential Equations with the Caputo-Fabrizio Fractional Derivative.} Journal of Fractional Calculus and Applications, \textbf{11} (2), 160-172 (2020).
\bibitem{parvaiz} Naik, P. A., Zu, J., Ghoreishi, M.: \textit{Estimating the approximate analytical solution of HIV viral dynamic model by using homotopy analysis method.} Chaos, Solitons \& Fractals, \textbf{131,} 109500 (2019).
\bibitem{MYNOB} Yavuz, M., Ozdemir, N., Baskonus, H. M.: \textit{Solutions of Partial Differential Equations Using the Fractional Operator Involving Mittag-Leffler Kernel.} The European Physical Journal Plus. \textbf{133,} 215 (2018).
\bibitem{USUE} Uçar, S., Uçar, E., Ozdemir, N., Hammouch, Z.: \textit{Mathematical Analysis and Numerical Simulation for a Smoking Model with Atangana–Baleanu Derivative.} Chaos, Solitons and Fractals. \textbf{118,} 300-306 (2019)
\bibitem{MY} Yavuz, M.: \textit{Novel Recursive Approximation for Fractional Nonlinear Equations within Caputo-Fabrizio Operator.} In: Proceedings, ITM Web of Conferences: EDP Sciences, 01008, (2018).
\bibitem{Aysegul} Keten, A., Yavuz, M., Baleanu, D.: \textit{Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces.} Fractal and Fractional, \textbf{3} (2), 27 (2019).
\bibitem{Tukur} Sulaiman, T. A., Yavuz, M., Bulut, H., Baskonus, H. M.: \textit{Investigation of the fractional coupled viscous Burgers’ equation involving Mittag-Leffler kernel.} Physica A: Statistical Mechanics and its Applications, \textbf{527}, 121126, (2019).
\bibitem{Zakia} Asif, N. A., Hammouch, Z., Riaz, M. B., Bulut, H.: \textit{Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative.} The European Physical Journal Plus, \textbf{133} (7), 272 (2018).
\bibitem{bahar2} Bas, E., Acay, B., Ozarslan, R.: \textit{Fractional models with singular and non-singular kernels for energy efficient buildings.} Chaos: An Interdisciplinary Journal of Nonlinear Science, \textbf{29} (2), 023110 (2019).
\bibitem{YLRG} Luchko, Y., Gorenflo, R.: \textit{An operational method for solving fractional differential equations with the Caputo derivatives.} Acta Mathematica Vietnamica, \textbf{24} (2), 207-233 (1999).
\bibitem{DBKD} Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J.: Fractional Calculus: Models and Numerical Methods, World Scientific, (2012)
\bibitem{KSMB} Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, John Wiley and Sons, (1993)
\bibitem{AABST} Atangana, A., Alkahtani, B. S. T.: \textit{New Model of Groundwater Flowing within a Confine Aquifer: Application of Caputo-Fabrizio Derivative.} Arabian Journal of Geosciences. \textbf{9} (8), 1-6 (2016).
\bibitem{JLJJ} Losada, J., Nieto, J. J.: \textit{Properties of a New Fractional Derivative without Singular Kernel.} Progr. Fract. Differ. Appl. \textbf{1} (2), 87-92 (2015).
\bibitem{BSLS} Shulgin, B., Stone, L., Agur, Z.: \textit{Pulse Vaccination Strategy in the Sir Epidemic Model.} Bulletin of Mathematical Biology. \textbf{60} (6), 1123-1148 (1998).

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Mehmet Yavuz
Türkiye

Necati Özdemir This is me
Türkiye

Publication Date

March 20, 2020

Submission Date

January 15, 2019

Acceptance Date

-

Published in Issue

Year 2020 Volume: 8 Number: 1

DOI

https://doi.org/10.36753/mathenot.691638

IZ

https://izlik.org/JA35FY25AA

APA

Yavuz, M., & Özdemir, N. (2020). Analysis of an Epidemic Spreading Model with Exponential Decay Law. Mathematical Sciences and Applications E-Notes, 8(1), 142-154. https://doi.org/10.36753/mathenot.691638

AMA

1.Yavuz M, Özdemir N. Analysis of an Epidemic Spreading Model with Exponential Decay Law. Math. Sci. Appl. E-Notes. 2020;8(1):142-154. doi:10.36753/mathenot.691638

Chicago

Yavuz, Mehmet, and Necati Özdemir. 2020. “Analysis of an Epidemic Spreading Model With Exponential Decay Law”. Mathematical Sciences and Applications E-Notes 8 (1): 142-54. https://doi.org/10.36753/mathenot.691638.

EndNote

Yavuz M, Özdemir N (March 1, 2020) Analysis of an Epidemic Spreading Model with Exponential Decay Law. Mathematical Sciences and Applications E-Notes 8 1 142–154.

IEEE

[1]M. Yavuz and N. Özdemir, “Analysis of an Epidemic Spreading Model with Exponential Decay Law”, Math. Sci. Appl. E-Notes, vol. 8, no. 1, pp. 142–154, Mar. 2020, doi: 10.36753/mathenot.691638.

ISNAD

Yavuz, Mehmet - Özdemir, Necati. “Analysis of an Epidemic Spreading Model With Exponential Decay Law”. Mathematical Sciences and Applications E-Notes 8/1 (March 1, 2020): 142-154. https://doi.org/10.36753/mathenot.691638.

JAMA

1.Yavuz M, Özdemir N. Analysis of an Epidemic Spreading Model with Exponential Decay Law. Math. Sci. Appl. E-Notes. 2020;8:142–154.

MLA

Yavuz, Mehmet, and Necati Özdemir. “Analysis of an Epidemic Spreading Model With Exponential Decay Law”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 1, Mar. 2020, pp. 142-54, doi:10.36753/mathenot.691638.

Vancouver

1.Mehmet Yavuz, Necati Özdemir. Analysis of an Epidemic Spreading Model with Exponential Decay Law. Math. Sci. Appl. E-Notes. 2020 Mar. 1;8(1):142-54. doi:10.36753/mathenot.691638