Skip to main content

Advertisement

SpringerLink
Log in

Stability analysis and optimal control of a fractional HIV-AIDS epidemic model with memory and general incidence rate

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

We investigate the celebrated mathematical SICA model but using fractional differential equations in order to better describe the dynamics of HIV-AIDS infection. The infection process is modelled by a general functional response, and the memory effect is described by the Caputo fractional derivative. Stability and instability of equilibrium points are determined in terms of the basic reproduction number. Furthermore, a fractional optimal control system is formulated and the best strategy for minimizing the spread of the disease into the population is determined through numerical simulations based on the derived necessary optimality conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. UNAIDS: Fast-track-ending the AIDS epidemic by 2030. http://www.unaids.org

  2. M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, G.R. Jafari, M. Ausloos, Memory effects on epidemic evolution: the susceptible-infected-recovered epidemic model. Phys. Rev. E 95(2), 022–409 (2017). https://doi.org/10.1103/physreve.95.022409. 9

    Article  MathSciNet  Google Scholar 

  3. Y.A. Rossikhin, M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50(1), 15–67 (1997). https://doi.org/10.1115/1.3101682

    Article  ADS  Google Scholar 

  4. R.J. Marks, M.W. Hall, Differintegral interpolation from a bandlimited signal’s samples. IEEE Trans. Acoust. Speech Signal Process. 29(4), 872–877 (1981). https://doi.org/10.1109/TASSP.1981.1163636

    Article  MATH  Google Scholar 

  5. J.G. Liu, M.Y. Xu, Study on the viscoelasticity of cancellous bone based on higher-order fractional models. In: 2008 2nd International Conference on Bioinformatics and Biomedical Engineering, pp. 1733–1736. IEEE (2008). https://doi.org/10.1109/ICBBE.2008.761

  6. R.L. Magin, Fractional calculus in bioengineering, part 1. Critical Rev. Biomed. Eng. 32(1), 1–104 (2004). https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10

    Article  Google Scholar 

  7. E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance. Phys. A 284(1–4), 376–384 (2000). https://doi.org/10.1016/S0378-4371(00)00255-7

    Article  MathSciNet  Google Scholar 

  8. L. Song, S. Xu, J. Yang, Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simul. 15(3), 616–628 (2010). https://doi.org/10.1016/j.cnsns.2009.04.029

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. K.S. Cole, Electric conductance of biological systems. In: Cold Spring Harbor symposia on quantitative biology, vol. 1, pp. 107–116. Cold Spring Harbor Laboratory Press (1933). https://doi.org/10.1101/SQB.1933.001.01.014

  10. P. Agarwal, D. Baleanu, Y. Chen, S. Momani, J.A.T. Machado, Fractional Calculus, Springer Proceedings in Mathematics & Statistics, vol. 303. Springer Singapore (2019)

  11. P. Agarwal, S. Deniz, S. Jain, A.A. Alderremy, S. Aly, A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques. Phys. A 542, 122–769 (2020). https://doi.org/10.1016/j.physa.2019.122769

    Article  MathSciNet  Google Scholar 

  12. F.A. Rihan, Numerical modeling of fractional-order biological systems. Abstr. Appl. Anal. pp. Art. ID 816,803, 11 (2013). https://doi.org/10.1155/2013/816803

  13. H.G. Sun, W. Chen, H. Wei, Y.Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Special Topics 193(1), 185–192 (2011). https://doi.org/10.1140/epjst/e2011-01390-6

    Article  ADS  Google Scholar 

  14. F.A. Rihan, M. Sheek-Hussein, A. Tridane, R. Yafia, Dynamics of hepatitis C virus infection: mathematical modeling and parameter estimation. Math. Model. Nat. Phenom. 12(5), 33–47 (2017). https://doi.org/10.1051/mmnp/201712503

    Article  MathSciNet  MATH  Google Scholar 

  15. A.A.M. Arafa, S.Z. Rida, M. Khalil, A fractional-order model of HIV infection: numerical solution and comparisons with data of patients. Int. J. Biomath 7(4), 1450–036 (2014). https://doi.org/10.1142/S1793524514500363. 11

    Article  MathSciNet  MATH  Google Scholar 

  16. W. Wojtak, C.J. Silva, D.F.M. Torres, Uniform asymptotic stability of a fractional tuberculosis model. Math. Model. Nat. Phenom. 13(1), 10 (2018). https://doi.org/10.1051/mmnp/2018015

    Article  MathSciNet  MATH  Google Scholar 

  17. G. González-Parra, A.J. Arenas, B.M. Chen-Charpentier, A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1). Math. Methods Appl. Sci. 37(15), 2218–2226 (2014). https://doi.org/10.1002/mma.2968

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. S. Pooseh, H.S. Rodrigues, D.F.M. Torres, Fractional derivatives in dengue epidemics. AIP Conf. Proc. 1389(1), 739–742 (2011). https://doi.org/10.1063/1.3636838

    Article  ADS  Google Scholar 

  19. N.H. Sweilam, S.M. Al-Mekhlafi, D. Baleanu, Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains. J. Adv. Res. 17, 125–137 (2019). https://doi.org/10.1016/j.jare.2019.01.007

    Article  Google Scholar 

  20. S. Rosa, D.F.M. Torres, Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection. Chaos Solitons Fractals 117, 142–149 (2018). https://doi.org/10.1016/j.chaos.2018.10.021

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. H. Kheiri, M. Jafari, Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing. J. Appl. Math. Comput. 60(1–2), 387–411 (2019). https://doi.org/10.1007/s12190-018-01219-w

    Article  MathSciNet  MATH  Google Scholar 

  22. C.M.A. Pinto, Tenreiro J.A. Machado, Fractional model for malaria transmission under control strategies. Comput. Math. Appl 66(5), 908–916 (2013). https://doi.org/10.1016/j.camwa.2012.11.017

    Article  MathSciNet  Google Scholar 

  23. A. Boukhouima, K. Hattaf, N. Yousfi, Dynamics of a fractional order hiv infection model with specific functional response and cure rate. Int. J. Differ. Equ. 2017(8), 140–8372 (2017). https://doi.org/10.1155/2017/8372140

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Mouaouine, A. Boukhouima, K. Hattaf, N. Yousfi, A fractional order sir epidemic model with nonlinear incidence rate. Adv. Differ. Equ. (2018). https://doi.org/10.1186/s13662-018-1613-z

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Boukhouima, K. Hattaf, N. Yousfi, Modeling the Memory and Adaptive Immunity in Viral Infection, pp. 271–297. Springer International Publishing, Cham (2019). https://doi.org/10.1007/978-3-030-23433-1_18

  26. P. Agarwal, R. Singh, Modelling of transmission dynamics of nipah virus (Niv): A fractional order approach. Physica A: Statistical Mechanics and its Applications p. 124243 (2020). https://doi.org/10.1016/j.physa.2020.124243. http://www.sciencedirect.com/science/article/pii/S0378437120300625

  27. I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198 (Academic Press Inc, San Diego, CA, 1999)

  28. M. Du, Z. Wang, H. Hu, Measuring memory with the order of fractional derivative. Sci. Rep. 3(1), 3431 (2013). https://doi.org/10.1038/srep03431

    Article  ADS  Google Scholar 

  29. K. Hattaf, A.A. Lashari, Y. Louartassi, N. Yousfi, A delayed SIR epidemic model with general incidence rate (Electron. J. Qual. Theory Differ, Equ, 2013). https://doi.org/10.14232/ejqtde.2013.1.3

    Book  Google Scholar 

  30. W. Lin, Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332(1), 709–726 (2007). https://doi.org/10.1016/j.jmaa.2006.10.040

    Article  MathSciNet  MATH  Google Scholar 

  31. Z.M. Odibat, N.T. Shawagfeh, Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007). https://doi.org/10.1016/j.amc.2006.07.102

    Article  MathSciNet  MATH  Google Scholar 

  32. E.M. Lotfi, M. Mahrouf, M. Maziane, C.J. Silva, D.F.M. Torres, N. Yousfi, A minimal HIV-AIDS infection model with general incidence rate and application to Morocco data. Stat. Optim. Inf. Comput 7(3), 588–603 (2019). https://doi.org/10.19139/soic.v7i3.834

    Article  MathSciNet  Google Scholar 

  33. C. Vargas-De-León, Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24(1–3), 75–85 (2015). https://doi.org/10.1016/j.cnsns.2014.12.013

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal. Real World Appl. 26, 289–305 (2015). https://doi.org/10.1016/j.nonrwa.2015.05.014

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. S. Lenhart, J.T. Workman, Optimal control applied to biological models, (Chapman & Hall/CRC Mathematical and Computational Biology Series. Chapman & Hall/CRC, Boca Raton, FL, 2007)

  37. J.P. Mateus, P. Rebelo, S. Rosa, C.M. Silva, D.F.M. Torres, Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete Contin. Dyn. Syst. Ser. S 11(6), 1179–1199 (2018). https://doi.org/10.3934/dcdss.2018067

    Article  MathSciNet  MATH  Google Scholar 

  38. H. Kheiri, M. Jafari, Optimal control of a fractional-order model for the HIV/AIDS epidemic. Int. J. Biomath 11(7), 1850–086 (2018). https://doi.org/10.1142/S1793524518500869. 23

    Article  MathSciNet  MATH  Google Scholar 

  39. K. Diethelm, N.J. Ford, A.D. Freed, Y. Luchko, Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Engrg. 194(6–8), 743–773 (2005). https://doi.org/10.1016/j.cma.2004.06.006

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. S. Rosa, D.F.M. Torres, Optimal control and sensitivity analysis of a fractional order TB model. Stat. Optim. Inf. Comput 7(3), 617–625 (2019). https://doi.org/10.19139/soic.v7i3.836

    Article  MathSciNet  Google Scholar 

  41. S. Rosa, D.F.M. Torres, Parameter estimation, sensitivity analysis and optimal control of a periodic epidemic model with application to HRSV in Florida. Stat. Optim. Inf. Comput 6(1), 139–149 (2018). https://doi.org/10.19139/soic.v6i1.472

    Article  MathSciNet  Google Scholar 

  42. P. Rodrigues, C.J. Silva, D.F.M. Torres, Cost-effectiveness analysis of optimal control measures for tuberculosis. Bull. Math. Biol. 76(10), 2627–2645 (2014). https://doi.org/10.1007/s11538-014-0028-6

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Rosa was funded by The Portuguese Foundation for Science and Technology (FCT–Fundação para a Ciência e a Tecnologia) through national funds under the Project UIDB/EEA/50008/2020; Torres by FCT through CIDMA, reference UIDB/50008/2020. The authors would like to acknowledge the comments and suggestions from three anonymous reviewers, which helped them to enriched their work.

Author information

Authors and Affiliations

  1. Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O Box 7955, Sidi Othman, Casablanca, Morocco

    Adnane Boukhouima, El Mehdi Lotfi, Marouane Mahrouf & Noura Yousfi

  2. Instituto de Telecomunicações and Department of Mathematics, Universidade da Beira Interior, 6201-001, Covilhã, Portugal

    Silvério Rosa

  3. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Delfim F. M. Torres

Authors
  1. Adnane Boukhouima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. El Mehdi Lotfi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marouane Mahrouf
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Silvério Rosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Delfim F. M. Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Noura Yousfi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Delfim F. M. Torres.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boukhouima, A., Lotfi, E.M., Mahrouf, M. et al. Stability analysis and optimal control of a fractional HIV-AIDS epidemic model with memory and general incidence rate. Eur. Phys. J. Plus 136, 103 (2021). https://doi.org/10.1140/epjp/s13360-020-01013-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-01013-3

Search

Navigation