Analysis of Fractional-Order Model of COVID-19 Pandemics With a Nonlinear Incidence Rate

Vinod Varghese, Sonal Bhoyar, Kottakkaran Sooppy Nisar


Background. Several mathematical representations of contagious disease COVID-19 were evolved in order to capture the pragmatic aspect of unfurling of the disease. It is learned that individuals who became receptive were infected with a rate proportional to the fraction of the individuals affected by the infection, in the comprehensive population as well as the infected individuals recuperate at a sustained rate. It is also observed that in the SIR model, all contacts impart the disease with an identical probability.

Objective. We will estimate the dynamic epidemic behaviour of inflected population for India with the use of fractional-order SIR simulations and compare our results with the results obtained for extrapolated actual cases of the infected people.

Methods. We have obtained the approximate solutions of the fractional-order Susceptible-Infectious-Recovered model within the framework of the modified Riemann–Liouville fractional differential operator using a new iterative fractional complex transform technique.

Results. The optimal values of the fractional-order SIR model parameters were identified with the use of the New Iterative Method. The dynamic incident rate with high and low reproduction number is predicted as well as the illustrated graphical with actual data is provided. To sum, the fractional calculus model for a complex system proposed here is just an indication to show what might happen if we do not control the reproduction number in the community.

Conclusions. The control measures that have already been found like swift surveillance, quarantine and social distancing means, such as face masks and closures, assisted in curtailing coronavirus transmission – estimated by the average number of people each infected individual infects, or reproduction number, to close to the level of 1 in each month.


SIR-modeling; modified Riemann–Liouville fractional operator; numerical simulations; COVID-19 outbreak; basic reproduction number

Full Text:



Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc A Math Phys Eng Sci. 1927;115:700-21. DOI: 10.1098/rspa.1927.0118

Cooke KL. Stability analysis for a vector disease model. Rocky Mountain J Math. 1979;9(1):31-42. DOI: 10.1216/RMJ-1979-9-1-31

Brauer F, Castillo-Chavez C. Mathematical models in population biology and epidemiology. Texts in Applied Mathematics. New York: Springer; 2012. DOI: 10.1007/978-1-4614-1686-9

Grassly NC, Fraser C. Mathematical models of infectious disease transmission. Nat Rev Microbiol. 2008;6(6):477-87. DOI: 10.1038/nrmicro1845

Vitanov NV, Ausloos MR. Knowledge epidemics and population dynamics models for describing idea diffusion. arXiv [Preprint] 2012. Available from: arXiv:1201.0676v1

Caputo M. Linear models of dissipation whose Q is almost frequency independent—II. Geophys J Int. 1967;13(5):529-39.

Podlubny I. Fractional differential equations. New York: Academic Press; 1999.

Diethelm K. The analysis of fractional differential equations: an application-oriented exposition using operators of Caputo type. Springer; 2004.

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Vol. 204. Elsevier Science; 2006.

Angstmann CN, Henry BI, McGann AV. A fractional-order recovery SIR model from a stochastic process. Bull Math Biol. 2016;78(3):468-99. DOI: 10.1007/s11538-016-0151-7

Angstmann CN, Henry BI, McGann AV. A fractional-order infectivity SIR model. Phys A Stat Mech Appl. 2016;452:86-93. DOI: 10.1016/j.physa.2016.02.029

Hamdan NI, Kilicman A. A fractional-order SIR epidemic model for dengue transmission. Chaos Solitons Fractal. 2018;114:55-62. DOI: 10.1016/j.chaos.2018.06.031

Mouaouine A, Boukhouima A, Hattaf K, Yousfi N. A fractional-order SIR epidemic model with nonlinear incidence rate. Adv Differ Equ. 2018;1:160. DOI: 10.1186/s13662-018-1613-z

Wang X, Wang Z, Huang X, Li Y. Dynamic analysis of a delayed fractional-order SIR model with saturated incidence and treatment functions. Int J ifurcation Chaos. 2018;28(14):1850180. DOI: 10.1142/S0218127418501808

Sene N. SIR epidemic model with Mittag–Leffler fractional derivative. Chaos Solitons Fractals. 2020;137:109833. DOI: 10.1016/j.chaos.2020.109833

Shaikh AS, Jadhav VS, Timol MG, Nisar KS, I. Khan I. Analysis of the COVID-19 pandemic spreading in India by an epidemiological model and fractional differential operator. Preprints [Preprint] 2020. preprint. DOI: 10.20944/preprints202005.0266.v1

Oldham KB, Spanier J. The Fractional calculus. New York: Academic Press; 1974.

Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: Theory and applications. Gordon and Breach Science Publishers; 1993.

Miller KS, Ross B. An Introduction to the fractional calculus and fractional differential equation. New York: Wiley; 1993.

Gorenflo R, Mainardi F. Fractional calculus: integral and differential equations of fractional order. In: Carpineti A, Mainardi F, editors. Fractals and fractional calculus in coninum mechanics. Vienna: Springer; 1997.

Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput Math Appl. 2006;51(9-10):1367-76. DOI: 10.1016/j.camwa.2006.02.001

Li ZB, He JH. Fractional complex transform for fractional differential equations. Math Comput Appl. 2010; 15:970-973.

Daftardar-Gejji V, Jafari H. An iterative method for solving nonlinear functional equations. J Math Anal Appl. 2006;316:753-63.

Bhalekar S, Daftardar-Gejji V. Convergence of the new iterative method. Int J Diff Eq. 2011;2011:989065. DOI: 10.1155/2011/989065

Ministry of Health and Family Welfare, Govt. of India [Internet]. 2020 [cited 2020 May 31]. Available from:

Nesteruk I. Simulations and predictions of COVID-19 pandemic with the use of SIR model. Innov Biosyst Bioeng. 2020;4(2):110-21. DOI: 10.20535/ibb.2020.4.2.204274

GOST Style Citations

Copyright (c) 2020 The Author(s)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.