Discrete-Time Fractional Order SIR Epidemic Model with Saturated Treatment Function

References

[1] A. J. Lotka, Elements of mathematical biology, Dover, New York, 1956.Search in Google Scholar

[2] V. Volterra, Opere matematiche, Memorie e Note, vol. V, Acc. Naz. dei Lincei, Rome, Cremona, 1962.Search in Google Scholar

[3] H. Poincaré, Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation, Acta mathematica. 7 (1885), 259–380.10.1007/BF02402204Search in Google Scholar

[4] S. Khajanchi and S. Banerjee, Quantifying the role of immunotherapeutic drug T11 target structure in progression of malignant gliomas: Mathematical modeling and dynamical perspective, Math. Biosci. 289 (2017), 69–77.10.1016/j.mbs.2017.04.006Search in Google Scholar PubMed

[5] A. A. Elsadany and A. E. Matouk, Dynamical behaviors of fractional-order Lotka–Volterra predator–prey model and its discretization, J. Appl. Math. Comput. 49 (2015), 269–283.10.1007/s12190-014-0838-6Search in Google Scholar

[6] S. M. Salman and A. A. Elsadany, On the bifurcation of Marotto’s map and its application in image encryption, J. Comput. Appl. Math. 328 (2018), 177–196.10.1016/j.cam.2017.07.010Search in Google Scholar

[7] Q. Chen, Z. Teng and Z. Hu, Bifurcation and control for a discrete-time Prey-Predator model with Holling-IV functional response, Int. J. Appl. Math. Comput. Sci. 23(2) (2013), 247–261.10.2478/amcs-2013-0019Search in Google Scholar

[8] A. A. Elsadany, Amr Elsonbaty and H. N. Agiza, Qualitative dynamical analysis of chaotic plasma perturbations model, Commun. Nonlin. Sci. Numer. Simul. 59 (2018), 409–423.10.1016/j.cnsns.2017.11.020Search in Google Scholar

[9] Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model, Math. Comput. Simul. 97 (2014), 80–93.10.1016/j.matcom.2013.08.008Search in Google Scholar

[10] J. Li and N. Cui, Bifurcation and chaotic behavior of a discrete-time SIS model, Discrete dynamics in nature and society volume 2013, Article ID 705601, 8 pages.10.1155/2013/705601Search in Google Scholar

[11] M. A. M. Abdelaziz, A. I. Ismail, F. A. Abdullah and M. H. Mohd, Bifurcations and chaos in a discrete SI epidemic model with fractional order, Adv. Diff. Eq. Springer, 2018 (2018), 44.10.1186/s13662-018-1481-6Search in Google Scholar

[12] S. Khajanchi, Dynamic behavior of a Beddington-DeAngelis type stage structured predator-prey model, Appl. Math. Comput. 244 (2014), 344–360.10.1016/j.amc.2014.06.109Search in Google Scholar

[13] S. Khajanchi, Bifurcation and oscillatory dynamics in a tumor immune interaction model, BIOMAT 2015: International Symposium on Mathematical and Computational Biology, (2016), 241–259.10.1142/9789813141919_0016Search in Google Scholar

[14] S. Khajanchi, Bifurcation analysis of a delayed mathematical model for tumor growth, Chaos Solitons Fractals. 77 (2015), 264–276.10.1016/j.chaos.2015.06.001Search in Google Scholar

[15] D. Ghosh, S. Khajanchi, S. Mangiarotti, F. Denis, S. K. Dana and C. Letellier, How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment? Biosystems. 158 (2017), 17–30.10.1016/j.biosystems.2017.05.001Search in Google Scholar PubMed

[16] B. Buonomo and D. Lacitignola, On the backward bifurcation of a vaccination model with nonlinear incidence, Nonlinear Anal. Modell. Control. 16 (2011), 30–46.10.15388/NA.16.1.14113Search in Google Scholar

[17] X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl. 348 (2008), 433–443.10.1016/j.jmaa.2008.07.042Search in Google Scholar

[18] L. Zhoua and M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl. 13 (2012), 312–324.10.1016/j.nonrwa.2011.07.036Search in Google Scholar

[19] L. Li, Y. Bai and Z. Jin, Periodic solutions of an epidemic model with saturated treatment, Nonlinear Dyn. 76 (2014), 1099–1108.10.1007/s11071-013-1193-0Search in Google Scholar

[20] J. Li, Z. Teng, G. Wang, L. Zhang and C. Hu, Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment, Chaos Solitons Fractals. 99 (2017), 63–71.10.1016/j.chaos.2017.03.047Search in Google Scholar

[21] I. M. Wangari and L. Stone, Analysis of a heroin epidemic model with saturated treatment function, J. Appl. Math. 2017, Article ID 1953036, 21pages.10.1155/2017/1953036Search in Google Scholar

[22] S. Khajanchi, D. K. Das and T. K. Kar, Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation, Physica A: Stat. Mech. Appl. 497 (2018), 52–71.10.1016/j.physa.2018.01.014Search in Google Scholar

[23] L. Zhou and M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl. 13 (2012), 312–324.10.1016/j.nonrwa.2011.07.036Search in Google Scholar

[24] S. R. J. JANG, Backward bifurcation in a discrete SIS model with vaccination, J. Biol. Syst. 16(4) (2008), 479–494.10.1142/S0218339008002630Search in Google Scholar

[25] J. Wang, S. Liu, B. Zheng and Y. Takeuchi, Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Math. Comput. Modell. 55 (2012), 710–722.10.1016/j.mcm.2011.08.045Search in Google Scholar

[26] R. P. Agarwal, A. M. A. El-Sayed and S. M. Salman, Fractional-order Chua’s system: discretization, bifurcation and chaos, Adv. Diff. Eq. 2013 (2013), 320.10.1186/1687-1847-2013-320Search in Google Scholar

[27] A. E. Matouk, A. A. Elsadany, E. Ahmed and H. N. Agiza, Dynamical behavior of fractional-order Hastings-Powell food chain model and its discretization, Commun. Nonlinear Sci. Numer. Simulat. 27 (2015), 153–167.10.1016/j.cnsns.2015.03.004Search in Google Scholar

[28] Q. Din, A. A. Elsadany and H. Khalil, Neimark-Sacker bifurcation and chaos control in a fractional-order plant-herbivore model, Discrete Dynamics in Nature and Society Volume 2017, Article ID 6312964, 15pages.10.1155/2017/6312964Search in Google Scholar

[29] J. S. Duan, The periodic solution of fractional oscillation equation with periodic input, Adv. Math. Phys. 2013 (2013), Article ID 869484, 6pages.10.1155/2013/869484Search in Google Scholar

[30] W. M. Ahmad and J. C. Sprott, Chaos in fractional-order autonomous nonlinear systems, Chaos, Solitons Fractals. 16 (2003), 339–351.10.1016/S0960-0779(02)00438-1Search in Google Scholar

[31] I. Podlubny, Fractional differential equations, Academic Press, New York, NY, 1999.Search in Google Scholar

[32] K. Diethelm, The analysis of fractional differential equations, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[33] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in: North Holland mathematics studies, vol. 204, Elsevier Science, Publishers BV, Amsterdam, 2006.Search in Google Scholar

[34] M. Caputo, Linear models of dissipation qhose Q is almost frequency independent II, Geophys. J. R. Astron. Soc. 13 (1967), 529–539.10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[35] S. Rana, S. Bhattacharya, J. Pal, G. M. N’Guérékata and J. Chattopadhyay, Paradox of enrichment: A fractional differential approach with memory. Phys. A. 392 (2013), 610–3621.10.1016/j.physa.2013.03.061Search in Google Scholar PubMed PubMed Central

[36] R. Toledo-Hernandez, V. Rico-Ramirez, G. A. Iglesias-Silva and U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions, Chem. Eng. Sci. 117 (2014), 217–228.10.1016/j.ces.2014.06.034Search in Google Scholar

[37] C. Xianbing, D. Abhirup, F. AL BASIR and R. P. Kumar, Fractional-order model of the disease Psoriasis: A control based mathematical approach, J. Sys. Sci. Complexity. 29 (2016), 1565–1584.10.1007/s11424-016-5198-xSearch in Google Scholar

[38] Q. Chen, Z. Teng and L. H. Jiang, The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence, Nonlinear Dyn. 71 (2013), 55–73.10.1007/s11071-012-0641-6Search in Google Scholar

[39] A. M. A. El-Sayed and S. M. Salman, On a discretization process of fractional order Riccati’s differential equation, J. Fractional Calculus Appl. 4 (2013), 251–259.Search in Google Scholar

[40] H. N. Agiza, E. M. ELabbasy, H. EL-Metwally and A. A. Elsadany, Chaotic dynamics of a discrete prey–predator model with Holling type II, Nonlinear Anal. Real World Appl. 10 (2009), 116–129.10.1016/j.nonrwa.2007.08.029Search in Google Scholar

[41] M. Martcheva, An introduction to mathematical epidemiology, Texts in applied mathematics, vol. 61, Springer, New York, 2015.10.1007/978-1-4899-7612-3Search in Google Scholar

[42] A. A. Elsadany, Dynamics of a Cournot duopoly game with bounded rationality based on relative profit maximization, Appl. Math. Comput. 294 (2017), 253–263.10.1016/j.amc.2016.09.018Search in Google Scholar

[43] Li-G. Yuan and Qi-G. Yang, Bifurcation, invariant curve and hybrid control in a discrete-time predator–prey system, Appl. Math. Modell. 39 (2015), 2345–2362.10.1016/j.apm.2014.10.040Search in Google Scholar

[44] S. Khajanchi and S. Banerjee, Stability and bifurcation analysis of delay induced tumor immune interaction model, Appl. Math. Comput. 248 (2014), 652–671.10.1016/j.amc.2014.10.009Search in Google Scholar

[45] S. Khajanchi, Uniform persistence and global stability for a brain tumor and immune system interaction, Biophys. Rev. Lett. 12 (2017), 1–22.10.1142/S1793048017500114Search in Google Scholar

[46] S. Khajanchi, M. Perc and D. Ghosh, The influence of time delay in a chaotic cancer model, CHAOS. 28 (2018), 103101.10.1063/1.5052496Search in Google Scholar PubMed