The Dynamics of a Discrete Fractional-Order Logistic Growth Model with Infectious Disease

Hasan S Panigoro, Emli Rahmi

= http://dx.doi.org/10.20473/conmatha.v3i1.26938
Abstract views = 576 times | downloads = 334 times

Abstract


In this paper, we study the dynamics of a discrete fractional-order logistic growth model with infectious disease. We obtain the discrete model by applying the piecewise constant arguments to the fractional-order model. This model contains three fixed points namely the origin point, the disease-free point, and the endemic point. We confirm that the origin point is always exists and unstable, the disease-free point is always exists and conditionally stable, and the endemic point is conditionally exists and stable. We also investigate the existence of forward, period-doubling, and Neimark-Sacker bifurcation. The numerical simulations are also presented to confirm the analytical results. We also show numerically the existence of period-3 solution which leads to the occurrence of chaotic behavior.

Keywords


logistic growth model; infectious disease; piecewise constant arguments; bifurcation

Full Text:

PDF

References


Zhang, F.W. & Nie, L.F., 2017, Dynamics of SIS epidemic model with varying total population and multivaccination control strategies, Stud. Appl. Math., 139(4) 533–550.

Kermack, W.O. & McKendrick, A.G., 1927, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London. Ser. A, Contain. Pap. a Math. Phys., 115(772), 700–721.

Dos Santos, J.P.C., Monteiro, E., & Vieira, G.B., 2017, Global stability of fractional SIR epidemic model, 5, 1–7.

Hassouna, M., Ouhadan, A., & El Kinani, E.H., 2018, On the solution of fractional order SIS epidemic model, Chaos Soliton Fract., 117, 168–174.

Hoang, M.T., Zafar, Z.U.A., & Ngo, T.K.Q., 2020, Dynamics and numerical approximations for a fractional-order SIS epidemic model with saturating contact rate, Comput. Appl. Math., 39(4), 277.

Panigoro, H.S. & Rahmi, E., 2020, Global stability of a fractional-order logistic growth model with infectious disease, Jambura J. Biomath., 1(2), 49–56.

Widya, E., Miswanto, M., and Alfiniyah, C., 2020, Analisis kestabilan model matematika penyebaran penyakit schistosomiasis dengan saturated incidence rate, Contemp. Math. Appl., 2(2), 71-88.

Ahaya, S. O. S. P., Rahmi, E., and Nurwan, N., 2020, Analisis dinamik model SVEIR pada penyebaran penyakit campak, Jambura J. Biomath., 1(2), 57–64.

Jajarmi, A., Yusuf, A., Baleanu, D., and Inc, M. , 2019, A new fractional HRSV model and its optimal control: A non-singular operator approach, Phys. A Stat. Mech. its Appl., 547, 123860.

Fatmawati, Khan, M. A., Alfiniyah, C., and Alzahrani, E., 2020, Analysis of dengue model with fractal-fractional Caputo–Fabrizio operator, Adv. Differ. Equations, 2020(1), 422.

Okyere, E., Ackora-Prah, J., and Oduro, F. T. , 2020, A Caputo based SIRS and SIS fractional order models with standard incidence rate and varying population, Commun. Math. Biol. Neurosci., 2020(60).

Abdelaziz, M. A. M., Ismail, A. I., Abdullah, F. A., and Mohd, M. H. , 2018, Bifurcations and chaos in a discrete SI epidemic model with fractional order, Adv. Differ. Equations, 2018(1), 44.

Shi, Y., Ma, Q., and Ding, X., 2018, Dynamical behaviors in a discrete fractional-order predator-prey system, Filomat, 32(17), 5857–5874.

Elettreby, M. F., Ahmed, E., and Alqahtani, A. S., 2020, A discrete fractional-order Prion model motivated by Parkinson’s disease,” Math. Probl. Eng., 2020, 1–12.

Agarwal, R. P., El-Sayed, A. M. A., and Salman, S. M. , 2013, Fractional-order Chua’s system: discretization, bifurcation and chaos, Adv. Differ. Equations, 2013(1), 320.

El-Sayed, A. M. A. and Salman, S. M., 2013, On a discretization process of fractional order Riccati differential equation, J. Fract. Calc. Appl., 4(2), 251–259.

Podlubny, I., 1999, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego CA: Academic Press.

Diethelm, K., 2010, The analysis of fractional differential equations: an application-oriented exposition using differential operators of caputo type, Berlin, Heidelberg: Springer.

El Raheem, Z. F. & Salman, S. M., 2014, On a discretization process of fractional-order Logistic differential equation, J. Egypt. Math. Soc., 22(3), 407–412.

Elsadany, A. A. & Matouk, A. E., 2015, Dynamical behaviors of fractional-order Lotka–Volterra predator–prey model and its discretization, J. Appl. Math. Comput., 49(1–2), 269–283.

Din, Q., Elsadany, A. A., & Khalil, H., 2017, Neimark-sacker bifurcation and chaos control in a fractional-order plant-herbivore model, Discret. Dyn. Nat. Soc., 2017(3), Article ID 6312964.

Mokodompit, R., Nurwan, and Rahmi, E., 2020, Bifurkasi periode ganda dan Neimark-Sacker pada model diskret Leslie-Gower dengan fungsi respon ratio-dependent,” Limits J. Math. Its Appl., 17(1), 19.

Elaydi S., Discrete Chaos with Applications in Science and Engineering, 2nd ed, Boca Raton: Chapman and Hall/CRC; 2008.

Singh, A., Elsadany, A. A., and Elsonbaty, A., 2019, Complex dynamics of a discrete fractional-order Leslie-Gower predator-prey model, Math. Methods Appl. Sci., 42(11), 3992–4007.


Refbacks

  • There are currently no refbacks.


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

View My Stats

Contemporary Mathematics and Applications is licensed under a  Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Indexed by: