Stability characterization of a fractional-order viral system with the non-cytolytic immune assumption

Year 2022, Volume: 2 Issue: 3, 164 - 176, 30.09.2022

Mouhcine Naim Yassine Sabbar Anwar Zeb

https://doi.org/10.53391/mmnsa.2022.013

Cited By: 13

Abstract

This article deals with a Caputo fractional-order viral model that incorporates the non-cytolytic immune hypothesis and the mechanism of viral replication inhibition. Firstly, we establish the existence, uniqueness, non-negativity, and boundedness of the solutions of the proposed viral model. Then, we point out that our model has the following three equilibrium points: equilibrium point without virus, equilibrium state without immune system, and equilibrium point activated by immunity with humoral feedback. By presenting two critical quantities, the asymptotic stability of all said steady points is examined. Finally, we examine the finesse of our results by highlighting the impact of fractional derivatives on the stability of the corresponding steady points.

Keywords

Viral model, non-cytolytic, immunity, fractional-order formulation, stability

References

• Wang, S., & Zou, D. Global stability of in-host viral models with humoral immunity and intracellular delays. Applied Mathematical Modelling, 36(3), 1313-1322, (2012).
• Wang, T., Hu, Z., Liao, F., & Ma, W. Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity. Mathematics and Computers in Simulation, 89, 13-22, (2013).
• Elaiw, A. M. Global stability analysis of humoral immunity virus dynamics model including latently infected cells. Journal of biological dynamics, 9(1), 215-228, (2015).
• Dhar, M., Samaddar, S., & Bhattacharya, P. Modeling the effect of non-cytolytic immune response on viral infection dynamics in the presence of humoral immunity. Nonlinear Dynamics, 98(1), 637-655, (2019).
• Hattaf, K., & Yousfi, N. Dynamics of SARS-CoV-2 infection model with two modes of transmission and immune response. Math. Biosci. Eng, 17(5), 5326-5340, (2020).
• Wodarz, D., Christensen, J.P., & Thomsen, A.R. The importance of lytic and nonlytic immune responses in viral infections. Trends in Immunology, 23(4), 194-200, (2002).
• Hollenberg, M.D., & Epstein, M. The innate immune response, microenvironment proteinases, and the COVID-19 pandemic: pathophysiologic mechanisms and emerging therapeutic targets. Kidney International Supplements, 12(1), 48-62, (2022).
• Prakasha, D.G., Malagi, N.S., Veeresha, P., & Prasannakumara, B.C. An efficient computational technique for time-fractional Kaup-Kupershmidt equation. Numerical Methods for Partial Differential Equations, 37(2), 1299-1316, (2021).
• Prakasha, D.G., Malagi, N.S., & Veeresha, P. New approach for fractional Schrödinger-Boussinesq equations with MittagLeffler kernel. Mathematical Methods in the Applied Sciences, 43(17), 9654-9670, (2020).
• Baishya, C., & Veeresha, P. Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel. Proceedings of the Royal Society A, 477(2253), 20210438, (2021).
• Fan, Y., Huang, X., Wang, Z., & Li, Y. Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function. Nonlinear Dynamics, 93(2), 611-627, (2018).
• Fan, Y., Huang, X., Wang, Z., & Li, Y. Global dissipativity and quasi-synchronization of asynchronous updating fractionalorder memristor-based neural networks via interval matrix method. Journal of the Franklin Institute, 355(13), 5998-6025, (2018).
• Agarwal, P., Baleanu, D., Chen, Y., Momani, S., & Machado, J. A. T. (Eds.). Fractional Calculus: ICFDA 2018, Amman, Jordan, July 16-18. Berlin, Germany: Springer, vol.303, (2019).
• Agarwal, P., Deniz, S., Jain, S., Alderremy, A.A., & Aly, S. A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques. Physica A: Statistical Mechanics and its Applications, 542, 122769, (2020).
• Song, L., Xu, S., & Yang, J. Dynamical models of happiness with fractional order. Communications in Nonlinear Science and Numerical Simulation, 15(3), 616-628, (2010).
• Magin, L.R. Fractional calculus in Bioengineering, Part 1. Critical Reviews in Biomedical Engineering, 32(1), 1–104, (2004).
• Sadek, O., Sadek, L., Touhtouh, S., & Hajjaji, A. The mathematical fractional modeling of TiO-2 nanopowder synthesis by sol–gel method at low temperature, Mathematical Modeling and Computing, 9(3), 616–626, (2022).
• Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).
• Özköse, F., Şenel, M.T., & Habbireeh, R. Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 67-83, (2021).
• Özköse, F., Yavuz, M., Şenel, M.T., & Habbireeh, R. Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom. Chaos, Solitons & Fractals, 157, 111954, (2022).
• Din, A., & Abidin, M.Z. Analysis of fractional-order vaccinated Hepatitis-B epidemic model with Mittag-Leffler kernels. Mathematical Modelling and Numerical Simulation with Applications, 2(2), 59-72, (2022).
• Fatmawati, M.A.K., Bonyah, E., Hammouch, Z., & Shaiful, E.M. A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model. AIMS Mathematics, 5(4), 2813-2842, (2020).
• M. Naim, F. Lahmidi, and A. Namir. Global stability of a fractional order SIR epidemic model with double epidemic hypothesis and nonlinear incidence rate. Communications in Mathematical Biology and Neuroscience, vol. 2020, Art. ID 38, 2020.
• Naim, M., Lahmidi, F., Namir, A., & Kouidere, A. Dynamics of an fractional SEIR epidemic model with infectivity in latent period and general nonlinear incidence rate. Chaos, Solitons & Fractals, 152, 111456, (2021).
• Gholami, M., Ghaziani, R.K., & Eskandari, Z. Three-dimensional fractional system with the stability condition and chaos control. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 41-47, (2022).
• Zahid, A., Masood, S., Mubarik, S., & Din, A. An efficient application of scrambled response approach to estimate the population mean of the sensitive variables. Mathematical Modelling and Numerical Simulation with Applications, 2(3), 127-146, (2022).
• Din, A., & Abidin, M.Z. Analysis of fractional-order vaccinated Hepatitis-B epidemic model with Mittag-Leffler kernels. Mathematical Modelling and Numerical Simulation with Applications, 2(2), 59-72, (2022).
• Sene, N. Second-grade fluid with Newtonian heating under Caputo fractional derivative: Analytical investigations via Laplace transforms. Mathematical Modelling and Numerical Simulation with Applications, 2(1), 13-25, (2022).
• Kumar, P., & Erturk, V.S. Dynamics of cholera disease by using two recent fractional numerical methods. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 102-111, (2021).
• Hammouch, Z., Yavuz, M., & Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).
• Daşbaşı, B. Stability analysis of an incommensurate fractional-order SIR model. Mathematical Modelling and Numerical Simulation with Applications, 1(1), (2021).
• Carvalho, A.R., Pinto, C., & Baleanu, D. HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load. Advances in Difference Equations, 2018(1), 1-22, (2018).
• Naik, P.A., Zu, J., & Owolabi, K.M. Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order. Physica A: Statistical Mechanics and its Applications, 545, 123816, (2020).
• Oustaloup, A., Levron, F., Victor, S., & Dugard, L. Non-integer (or fractional) power model to represent the complexity of a viral spreading: Application to the COVID-19. Annual Reviews in Control, 52, 523-542, (2021).
• Podlubny. I. Fractional Differential Equations. Academic Press, San Diego, 1999.
• Odibat, Z.M., & Shawagfeh, N.T. Generalized Taylor’s formula. Applied Mathematics and Computation, 186(1), 286-293, (2007).
• Lin, W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications, 332(1), 709-726, (2007).
• Li, H.L., Zhang, L., Hu, C., Jiang, Y.L., & Teng, Z. Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing, 54(1), 435-449, (2017).
• K. Diethelm. Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, (2010, January).
• Ali, A., Ullah, S., & Khan, M.A. The impact of vaccination on the modeling of COVID-19 dynamics: a fractional order model. Nonlinear Dynamics, 1-20, (2022).
• I. Petras. Fractional-order Nonlinear Systems: Modeling Anlysis and Simulation. Higher Education Press, Beijing, 2011.
• Vargas-De-León, C. Volterra-type Lyapunov functions for fractional-order epidemic systems. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 75-85, (2015).
• Aguila-Camacho, N., Duarte-Mermoud, M.A., & Gallegos, J.A. Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 19(9), 2951-2957, (2014).
• Huo, J., Zhao, H., & Zhu, L. The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Analysis: Real World Applications, 26, 289-305, (2015).
• Veeresha, P. A numerical approach to the coupled atmospheric ocean model using a fractional operator. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 1-10, (2021).
• Veeresha, P., Baskonus, H.M., & Gao, W. Strong interacting internal waves in rotating ocean: novel fractional approach. Axioms, 10(2), 123, (2021).
• Kiouach, D., & Sabbar, Y. The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps. International Journal of Biomathematics, 15(03), 2250004, (2022).
• Kiouach, D., & Sabbar, Y. Threshold analysis of the stochastic Hepatitis B epidemic model with successful vaccination and Levy jumps. In 2019 4th World Conference on Complex Systems (WCCS) (pp. 1-6), IEEE, (2019, April).
• Sabbar, Y., Kiouach, D., Rajasekar, S.P., & El-Idrissi, S.E.A. The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case. Chaos, Solitons & Fractals, 159, 112110, (2022).
• Kiouach, D., & Sabbar, Y. Modeling the impact of media intervention on controlling the diseases with stochastic perturbations. AIP Conference Proceedings (Vol. 2074, No. 1, p. 020026), AIP Publishing LLC, (2019, February).