Hopf Bifurcation Analysis of a Zika Virus Transmission Model with Two Time Delays
Abstract
This study focuses on a mathematical model of Zika virus transmission that incorporates multiple time delays. The inclusion of time delays in the model takes into account the incubation period in humans and the latency of disease transmission from mosquitoes. The qualitative behavior of the model was examined in four different conditions by analyzing the characteristic equation corresponding to the endemic equilibrium point. Furthermore, the two distinct time lags were selected as the bifurcation parameter, while the existence of a Hopf bifurcation at the endemic equilibrium point for threshold parameters was confirmed. Subsequently, numerical simulations were used to validate the theoretical analysis for each case using MATLAB.
Keywords
Bifurcation, Hopf bifurcation, Stability
Ethical Statement
It is declared that during the preparation process of this study, scientific and ethical principles were followed and all the studies benefited from are stated in the bibliography.
References
- [1] P. Baydemir, H. Merdan, E. Karaoglu, et al., Complex dynamics of a discrete-time prey–predator system with Leslie type, International Journal of Bifurcation and Chaos, 30(10) (2020), Article ID 2050149, https://doi.org/10.1142/S0218127420501497
- [2] S. Chomcheon, Y. Lenbury, W. Sarika, Stability, Hopf bifurcation and effects of impulsive antibiotic treatments in a model of drug resistance with conversion delay, Adv. Differ. Equ., 274 (2019), 1-18.
- [3] M. E. Hajji, M. F. S. Aloufi, M. H. Alharbi, Influence of seasonality on Zika virus transmission[J], AIMS Mathematics, 9(7) (2024), 19361-19384.
- [4] A. F. Nindjin, M. A. Aziz-Alaoui, M. Cadivel, Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay, Nonlinear Analysis: Real World Applications, 7(5) (2006), 1104-1118.
- [5] S. Olaniyi, Dynamics of Zika virus model with nonlinear incidence and optimal control strategies, Appl. Math. Inf. Sci., 12 (5) (2018), 969-982.
- [6] R. P. Singh, Bifurcation and stability analysis of delayed SIR model, J. Phys.: Conf. Ser., 2267 (2022), https://doi.org/10.1088/1742-6596/2267/1/012011
- [7] L. Wang, X. Wu, Stability and Hopf bifurcation for an SEIR epidemic model with delay, Advances in the Theory of Nonl. Anal. and its Appl., 2(3) (2018), 113-127.
- [8] C. Çelik, H. Merdan, Hopf bifurcation analysis of a system of coupled delayed-differential equations, Appl. Math. Comput., 219(12) (2013), 6605-6617.
- [9] E. Karaoğlu, H. Merdan, Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays, Chaos Solitons Fract., 68 (2014), 159-168.
- [10] J. Li, Y. Kuang, Analysis of a model of the glucose-insulin regulatory system with two delays, SIAM J. Appl. Math., 67(3) (2007), 757-776.
