### Analysis of COVID-19 Disease Using Fractional Order SEIR Model

Received: 15 July 2021    Accepted: 9 August 2021    Published: 31 December 2021
Abstract

In this study, the spread of COVID-19 pandemic disease model is analyzed using fractional order SEIR model. Fractional order is concept of calculus such as derivative and integral that is measured by using non-natural order that recently being used in various applications in real-world such as engineering, physics, chemistry, biology etc. Starting from this concept, the mathematical model is used in this study is in the form of a dynamical system consisting of nonlinear fractional differential equations with order one. These equations represent four compartments with certain health conditions. Those four compartments are susceptible, exposed, infected and recovered that are considered to have a significant influence in the development of COVID-19 infectious diseases. Having described the model, some analysis in terms of region of the solutions, the equilibrium points and reproduction number are measured in which is useful in describing qualitatively the stability of the system described. This dynamical system is solved numerically and simultaneously by using a modification of the Euler method. The results obtained are the graphs that describe the behaviour of four compartments in which giving predictive results about how the disease behaves with various orders. This approach has the advantage in terms of giving flexibility in approaching the real case that is happening in the world.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 4) DOI 10.11648/j.ijamtp.20210704.15 Page(s) 126-130 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2021. Published by Science Publishing Group
Keywords

SEIR Model, Fractional Calculus, Dynamical System

References
 [1] R. Almeida, A. M. C. B. Cruz, N. Martins, M. Teresa T. Monteiro, An epidemiological MSEIR model described by the Caputo fractional derivative, preprint. [2] S. Annas, M. I. Pratama, M. Rifanda, W. Sanusi, S. Side, Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia, Chaos, Solitons and Fractals, 139 (2020) 110072. [3] M. S. Bartlett Some Evolutionary Stochastic Processes. J. R. Stat. Soc. Ser. B 1949, 11, 211–229. [4] Y. C. Chen, P. E. Lu, Graduate Student Member, IEEE, C. S. Chang, Fellow, dan T. H. Liu, A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons, preprint, arXiv: 2003.00122v5. [5] E. Demirci, A. Unal, N. Ozalp, A Fractional Order SEIR Model with Density Dependent ¨ Death Rate. [6] M. El-Shahed and A. Alsaedi, The fractional SIRC model and inuenza A. Math. Probl. Eng, 3 (2011) 378-387. [7] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71 (3) (2011), 876-902. [8] C. M. Ionescu, A. Lopes, D. Copot, J. A. T. Machado and J. H. T. Bates, The role of fractional calculus in modelling biological phenomena: A review. Commun. Nonlinear Sci. Numer. Simul. 51 (2017) 141-159. [9] R. Khalil, M. Alhorani A. Yoused dan M. Sababheh, A definition of fractional derivative, Journal of Computational Applied Mathematics, 264 (2014), 65-70. [10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006. [11] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol. 68 No 3 (2006), 615-626. [12] V. P. Latha, F. A. Rihan, R. Rakkiyappan and G. Velmurugan, A fractional-order delay differential model for Ebola infection and CD8 T-cells response: stability analysis and Hopf bifurcation, Int. J. Biomath, 10 (2017), 1750111. [13] K. S. Miller dan B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, INC 1993. [14] C. M. A. Pinto and A. R. N. Carvalho, A latency fractional order model for HIV dynamics. J. Comput. Appl. Math. 312 (2017) 240256. [15] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives, translated from the 1987 Russian original. Gordon and Breach, Yverdon, 1993. [16] A. G. M. Selvam dan B. Jacob, Analysis of fractional order SIR model, International Journal of Engineering Research and Technolgy, 5 (2017). [17] M. O. Zaid. And S. Momani, An Algorithm for the Numerical Solution of Differential Equations of Fractional Order, J. Appl. Math. and Informatics Vol. 26, 2008. [18] Anonim, https://www.worldometers.info/coronavirus/country/indonesia/, 2020.
• APA Style

Leli Deswita, Ponco Hidayah, Ali Mohamed Ali Hassan Ali, Syamsudhuha Syamdhuha. (2021). Analysis of COVID-19 Disease Using Fractional Order SEIR Model. International Journal of Applied Mathematics and Theoretical Physics, 7(4), 126-130. https://doi.org/10.11648/j.ijamtp.20210704.15

ACS Style

Leli Deswita; Ponco Hidayah; Ali Mohamed Ali Hassan Ali; Syamsudhuha Syamdhuha. Analysis of COVID-19 Disease Using Fractional Order SEIR Model. Int. J. Appl. Math. Theor. Phys. 2021, 7(4), 126-130. doi: 10.11648/j.ijamtp.20210704.15

AMA Style

Leli Deswita, Ponco Hidayah, Ali Mohamed Ali Hassan Ali, Syamsudhuha Syamdhuha. Analysis of COVID-19 Disease Using Fractional Order SEIR Model. Int J Appl Math Theor Phys. 2021;7(4):126-130. doi: 10.11648/j.ijamtp.20210704.15

• ```@article{10.11648/j.ijamtp.20210704.15,
author = {Leli Deswita and Ponco Hidayah and Ali Mohamed Ali Hassan Ali and Syamsudhuha Syamdhuha},
title = {Analysis of COVID-19 Disease Using Fractional Order SEIR Model},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {7},
number = {4},
pages = {126-130},
doi = {10.11648/j.ijamtp.20210704.15},
url = {https://doi.org/10.11648/j.ijamtp.20210704.15},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210704.15},
abstract = {In this study, the spread of COVID-19 pandemic disease model is analyzed using fractional order SEIR model. Fractional order is concept of calculus such as derivative and integral that is measured by using non-natural order that recently being used in various applications in real-world such as engineering, physics, chemistry, biology etc. Starting from this concept, the mathematical model is used in this study is in the form of a dynamical system consisting of nonlinear fractional differential equations with order one. These equations represent four compartments with certain health conditions. Those four compartments are susceptible, exposed, infected and recovered that are considered to have a significant influence in the development of COVID-19 infectious diseases. Having described the model, some analysis in terms of region of the solutions, the equilibrium points and reproduction number are measured in which is useful in describing qualitatively the stability of the system described. This dynamical system is solved numerically and simultaneously by using a modification of the Euler method. The results obtained are the graphs that describe the behaviour of four compartments in which giving predictive results about how the disease behaves with various orders. This approach has the advantage in terms of giving flexibility in approaching the real case that is happening in the world.},
year = {2021}
}
```
• ```TY  - JOUR
T1  - Analysis of COVID-19 Disease Using Fractional Order SEIR Model
AU  - Leli Deswita
AU  - Ponco Hidayah
AU  - Ali Mohamed Ali Hassan Ali
AU  - Syamsudhuha Syamdhuha
Y1  - 2021/12/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ijamtp.20210704.15
DO  - 10.11648/j.ijamtp.20210704.15
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 126
EP  - 130
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20210704.15
AB  - In this study, the spread of COVID-19 pandemic disease model is analyzed using fractional order SEIR model. Fractional order is concept of calculus such as derivative and integral that is measured by using non-natural order that recently being used in various applications in real-world such as engineering, physics, chemistry, biology etc. Starting from this concept, the mathematical model is used in this study is in the form of a dynamical system consisting of nonlinear fractional differential equations with order one. These equations represent four compartments with certain health conditions. Those four compartments are susceptible, exposed, infected and recovered that are considered to have a significant influence in the development of COVID-19 infectious diseases. Having described the model, some analysis in terms of region of the solutions, the equilibrium points and reproduction number are measured in which is useful in describing qualitatively the stability of the system described. This dynamical system is solved numerically and simultaneously by using a modification of the Euler method. The results obtained are the graphs that describe the behaviour of four compartments in which giving predictive results about how the disease behaves with various orders. This approach has the advantage in terms of giving flexibility in approaching the real case that is happening in the world.
VL  - 7
IS  - 4
ER  - ```
Author Information
• Department of Mathematics, Faculty of Mathematics and Natural Sciences, Riau University, Pekanbaru, Indonesia

• Department of Mathematics, Faculty of Mathematics and Natural Sciences, Riau University, Pekanbaru, Indonesia

• Department of Computer, Control and Management Engineering, Sapienza University of Rome, Rome, Italy

• Department of Mathematics, Faculty of Mathematics and Natural Sciences, Riau University, Pekanbaru, Indonesia

• Sections