The existence of Neimark-Sacker bifurcation on a discrete-time SIS-Epidemic model incorporating logistic growth and allee effect

Amelia Tri Rahma Sidik, Hasan S. Panigoro, Resmawan Resmawan, Emli Rahmi


This article investigates the dynamical properties of a discrete time SIS-Epidemic model incorporating logistic growth rate and Allee effect. The forward Euler discretization method is employed to obtain the discrete-time model. All possible fixed points are identified including their local dynamics. Some numerical simulations by varying the step size parameter are explored to show the analytical findings, the existence of Neimark-Sacker bifurcation, and the occurrence of period-10 and 20 orbits


Discrete time model; Epidemic model; Neimark-Sacker bifurcation; Allee effect

Full Text:



D. Aldila, N. Nuraini, and E. Soewono, “Optimal control problem in preventing of swine flu disease transmission,” Applied Mathematical Sciences, vol. 8, no. 69-72, pp. 3501–3512, 2014. DOI: 10.12988/ams.2014.44275

K. M. Kabir, K. Kuga, and J. Tanimoto, “Analysis of SIR epidemic model with information spreading of awareness,” Chaos, Solitons and Fractals, vol. 119, pp. 118–125, 2019. DOI: 10.1016/j.chaos.2018.12.017

Windarto, M. A. Khan, and Fatmawati, “Parameter estimation and fractional derivatives of dengue transmission model,” AIMS Mathematics, vol. 5, no. 3, pp. 2758–2779, 2020. DOI: 10.3934/math.2020178

S. Pal, S. K. Sasmal, and N. Pal, “Chaos control in a discrete-time predatorprey model with weak Allee effect,” International Journal of Biomathematics, vol. 11, no. 7, pp. 1–26, 2018. DOI: 10.1142/S1793524518500894

Y. Chen and M. Li, “Mathematical Model of Allee Effect on Animal Single Population Dynamics and Behavior of Solutions,” Revista Cientifica-Facultad De Ciencias Veterinarias, vol. 29, no. 4, pp. 955–962, 2019.

L. Berec, V. Bernhauerová, and B. Boldin, “Evolution of mate-finding Allee effect in prey,” Journal of Theoretical Biology, vol. 441, pp. 9–18, 2018. DOI: 10.1016/j.jtbi.2017.12.024

X. Lin, H. Liu, X. Han, and Y. Wei, “Stability and Hopf bifurcation of an SIR epidemic model with density-dependent transmission and Allee effect,” Mathematical Biosciences and Engineering, vol. 20, no. 2, pp. 2750–2775, 2022. DOI: 10.3934/mbe.2023129

H. S. Panigoro, E. Rahmi, A. Suryanto, and I. Darti, “A fractional order predator–prey model with strong allee effect and Michaelis–Menten type of predator harvesting,” AIP Conference Proceedings, vol. 2498, no. 1, p. 20018, 2022. DOI: 10.1063/5.0082684

E. Rahmi, I. Darti, A. Suryanto, Trisilowati, and H. S. Panigoro, “Stability Analysis of a Fractional-Order Leslie-Gower Model with Allee Effect in Predator,” Journal of Physics: Conference Series, vol. 1821, no. 1, p. 012051, 2021. DOI: 10.1088/1742-6596/1821/1/012051

N. Hasan and A. Suryanto, “Dynamics of a fractional-order eco-epidemic model with Allee effect and refuge on prey,” Communications in Mathematical Biology and Neuroscience, vol. 2022, p. Article ID 117, 2022. DOI: 10.28919/cmbn/7742

Z. Hu, Z. Teng, and L. Zhang, “Stability and bifurcation analysis in a discrete SIR epidemic model,” Mathematics and Computers in Simulation, vol. 97, pp. 80–93, 2014. DOI: 10.1016/j.matcom.2013.08.008

W. J. Du, J. G. Zhang, S. Qin, and J. N. Yu, “Bifurcation analysis in a discrete sir epidemic model with the saturated contact rate and vertical transmission,” Journal of Nonlinear Science and Applications, vol. 9, no. 7, pp. 4976–4989, 2016. DOI: 10.22436/jnsa.009.07.02

Q. Cui, Z. Qiu, W. Liu, and Z. Hu, “Complex dynamics of an SIR epidemic model with nonlinear saturate incidence and recovery rate,” Entropy, vol. 19, no. 7, pp. 1–16, 2017. DOI: 10.3390/e19070305

M. A. Abdelaziz, A. I. Ismail, F. A. Abdullah, and M. H. Mohd, “Bifurcations and chaos in a discrete SI epidemic model with fractional order,” Advances in Difference Equations, vol. 2018, no. 1, p. 44, 2018. DOI: 10.1186/s13662-018-1481-6

P. K. Santra, H. S. Panigoro, and G. S. Mahapatra, “Complexity of a DiscreteTime Predator-Prey Model Involving Prey Refuge Proportional to Predator,” Jambura Journal of Mathematics, vol. 4, no. 1, pp. 50–63, 2022. DOI: 10.34312/jjom.v4i1.11918

H. S. Panigoro and E. Rahmi, “The Dynamics of a Discrete Fractional-Order Logistic Growth Model with Infectious Disease,” Contemporary Mathematics and Applications, vol. 3, no. 1, pp. 1–18, 2021. DOI: 10.20473/conmatha.v3i1.26938

H. S. Panigoro, E. Rahmi, N. Achmad, S. L. Mahmud, R. Resmawan, and A. R. Nuha, “A discrete-time fractional-order Rosenzweig-Macarthur predatorprey model involving prey refuge,” Communications in Mathematical Biology and Neuroscience, vol. 2021, p. Article ID 77, 2021. DOI: 10.28919/cmbn/6586

R. Mokodompit, N. Nurwan, and E. Rahmi, “Bifurkasi Periode Ganda dan Neimark-Sacker pada Model Diskret Leslie-Gower dengan Fungsi Respon Ratio-Dependent,” Limits: Journal of Mathematics and Its Applications, vol. 17, no. 1, p. 19, 2020. DOI: 10.12962/limits.v17i1.6809


Copyright (c) 2022 Amelia Tri Rahma Sidik, Hasan S. Panigoro, Resmawan Resmawan, Emli Rahmi

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

Jambura Journal of Biomathematics (JJBM) has been indexed by:


 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango 96554, Gorontalo, Indonesia
 +6281356190818 (Call/SMS/WA)
 Jambura Journal of Biomathematics (JJBM) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.  Powered by Public Knowledge Project OJS.