Dynamical Analysis of  a Fractional Order HIV/AIDS Model

Septiangga Van Nyek Perdana Putra; Agus Suryanto; Nur Shofianah

doi:10.31764/jtam.v5i1.3224

Dynamical Analysis of a Fractional Order HIV/AIDS Model

Septiangga Van Nyek Perdana Putra, Agus Suryanto, Nur Shofianah

Abstract

This article discusses a dynamical analysis of the fractional-order model of HIV/AIDS. Biologically, the rate of subpopulation growth also depends on all previous conditions/memory effects. The dependency of the growth of subpopulations on the past conditions is considered by applying fractional derivatives. The model is assumed to consist of susceptible, HIV infected, HIV infected with treatment, resistance, and AIDS. The fractional-order model of HIV/AIDS with Caputo fractional-order derivative operators is constructed and then, the dynamical analysis is performed to determine the equilibrium points, local stability and global stability of the equilibrium points. The dynamical analysis results show that the model has two equilibrium points, namely the disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable when the basic reproduction number is less than one. The endemic equilibrium point exists if the basic reproduction number is more than one and is globally asymptotically stable unconditionally. To illustrate the dynamical analysis, we perform some numerical simulation using the Predictor-Corrector method. Numerical simulation results support the analytical results.

Keywords

Fractional-Order; PECE Method; HIV/AIDS; Epidemic Model;

Full Text:

DOWNLOAD [PDF]

References

Ahmed, E., El-Sayed, A. M. A., & El-Saka, H. A. A. (2006). On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Physics Letters, Section A: General, Atomic and Solid State Physics, 358(1), 1–4. https://doi.org/10.1016/j.physleta.2006.04.087

Das, S., & Gupta, P. K. (2011). A mathematical model on fractional Lotka-Volterra equations. Journal of Theoretical Biology, 277(1), 1–6. https://doi.org/10.1016/j.jtbi.2011.01.034

Diethelm, K. (2010). The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, 2004, 1–30.

Hassouna, M., Ouhadan, A., & El Kinani, E. H. (2018). On the solution of fractional order SIS epidemic model. Chaos, Solitons and Fractals, 117, 168–174. https://doi.org/10.1016/j.chaos.2018.10.023

Huo, H. F., Chen, R., & Wang, X. Y. (2016). Modelling and stability of HIV/AIDS epidemic model with treatment. Applied Mathematical Modelling, 40(13–14), 6550–6559. https://doi.org/10.1016/j.apm.2016.01.054

Kemenkes RI. (2017). Stop HIV AIDS. Kementerian Kesehatan Republik Indonesia, 1–3. http://www.kemkes.go.id/development/site/depkes/pdf.php?id=1-17042500008

Li, H. L., Zhang, L., Hu, C., Jiang, Y. L., & Teng, Z. (2017). Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. Journal of Applied Mathematics and Computing, 54(1–2), 435–449. https://doi.org/10.1007/s12190-016-1017-8

Moore, E. J., Sirisubtawee, S., & Koonprasert, S. (2019). A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment. Advances in Difference Equations, 2019(1). https://doi.org/10.1186/s13662-019-2138-9

Nyabadza, F., Mukandavire, Z., & Hove-Musekwa, S. D. (2011). Modelling the HIV/AIDS epidemic trends in South Africa: Insights from a simple mathematical model. Nonlinear Analysis: Real World Applications, 12(4), 2091–2104. https://doi.org/10.1016/j.nonrwa.2010.12.024

Petras, I. (2011). Fractional-Orde Nonlinear System. Springer-Verlag.

Pinto, C. M. A., & Carvalho, A. R. M. (2015). Effect of drug-resistance in a fractional complex-order model for HIV infection. IFAC-PapersOnLine, 28(1), 188–189. https://doi.org/10.1016/j.ifacol.2015.05.162

Rihan, F. A. (2013). Numerical modeling of fractional-order biological systems. Abstract and Applied Analysis, 2013. https://doi.org/10.1155/2013/816803

Rihan, F. A., Baleanu, D., Lakshmanan, S., & Rakkiyappan, R. (2014). On fractional SIRC model with salmonella bacterial infection. Abstract and Applied Analysis, 2014. https://doi.org/10.1155/2014/136263

S.M., S., & A.M., Y. (2017). On a fractional-order model for HBV infection with cure of infected cells. Journal of the Egyptian Mathematical Society, 25(4), 445–451. https://doi.org/10.1016/j.joems.2017.06.003

Shaikh, A. S., & Sooppy Nisar, K. (2019). Transmission dynamics of fractional order Typhoid fever model using Caputo–Fabrizio operator. Chaos, Solitons and Fractals, 128, 355–365. https://doi.org/10.1016/j.chaos.2019.08.012

Silva, C. J., & Torres, D. F. M. (2017). Global Stability For a HIV/AIDS Model. Communications Faculty of Sciences Universitiy of Ankara Series A1 Mathematics and Statistics, 1–8.

Silva, C. J., & Torres, D. F. M. (2019). Stability of a fractional HIV/AIDS model. Mathematics and Computers in Simulation, 164, 180–190. https://doi.org/10.1016/j.matcom.2019.03.016

Solís-Pérez, J. E., Gómez-Aguilar, J. F., & Atangana, A. (2019). A fractional mathematical model of breast cancer competition model. Chaos, Solitons and Fractals, 127, 38–54. https://doi.org/10.1016/j.chaos.2019.06.027

Sweilam, N. H., AL-Mekhlafi, S. M., Mohammed, Z. N., & Baleanu, D. (2020). Optimal control for variable order fractional HIV/AIDS and malaria mathematical models with multi-time delay. Alexandria Engineering Journal, 59(5), 3149–3162. https://doi.org/10.1016/j.aej.2020.07.021

Vargas-De-León, C. (2015). Volterra-type Lyapunov functions for fractional-order epidemic systems. Communications in Nonlinear Science and Numerical Simulation, 24(1–3), 75–85. https://doi.org/10.1016/j.cnsns.2014.12.013

DOI: https://doi.org/10.31764/jtam.v5i1.3224