OPTIMAL CONTROL OF A POPULATION-VARYING
HETEROSEXUAL HIV/AIDS MATHEMATICAL MODEL
Calisto Munongi1, Justin M.W. Munganga2
1,2 Department of Mathematical Sciences
University of South Africa Science Campus
Florida, Unisa 0003, SOUTH AFRICA

Abstract

We propose a heterosexual HIV transmission model of variable population size. The model is proved to be well-posed both mathematically and epidemiologically. A threshold for the existence of the disease is established, together with the existence and asymptotic stability of equilibria. An optimal control problem is formulated in which four controls (sex education, screening, HIV prevention methods, and treatment of infectives) are administered. It is shown that there exist an optimal control set and the optimal controls are characterised by applying Pontryagin's maximum principle. We numerically solve the optimal control problem using the forward-backward sweep method and ODE45 solver. The control strategy causes an important decrease in susceptibles, unaware infectives and AIDS individuals. The strategy also causes an outstanding transfer of infectives to the HIV treatment class.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 5
Year: 2021

DOI: 10.12732/ijam.v34i5.4

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] J.M. Baeten, J. Overbaugh, Measuring the infectiousness of persons with HIV-1: opportunities for preventing sexual HIV-1 transmission, Current HIV Research, 1, No 1 (2003), 69-86.
  2. [2] S. Busenberg, P. van den Driessche, Analysis of a Disease Transmission model in a population with varying size, Journal of Mathematical Biology, 28, No 3 (1990), 257-270.
  3. [3] M. Chopra, L. Townsend, L. Johnston, et al., Estimating HIV prevalence and risk behaviors among high-risk heterosexual men with multiple sex partners: Use of respondent-driven sampling, Journal of Acquired Immune Deficiency Syndrome, 51 (2009), 72-77.
  4. [4] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, TMH Edition, Tata McGraw-Hill Publ. Co. Ltd, New Delhi (1972).
  5. [5] O. Diekmann, J. A. Heesterbeek, J.A. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28, No 4 (1990), 365-382.
  6. [6] L. Esteva, C. Vargas, A model for dengue disease with variable human population, Journal of Mathematical Biology, 38, No 3 (1999), 220-240.
  7. [7] W.H. Fleming, R.W. Rishel, Deterministic and stochastic optimal control, 1st Ed., Springer-Verlag, Berlin-Heidelberg-New York (1975).
  8. [8] H. Gaff, E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Mathematical Biosciences and Engineering, 6, No 3 (2009), 469-492.
  9. [9] H.W. Hethcote, The mathematics of infectious diseases, SIAM Review 42, No 4 (2000), 599-653.
  10. [10] J.L. Juusola, M.L. Brandeau, HIV Treatment and Prevention: A simple model to determine optimal investment, Medical Decision Making, 36, No 3 (2016), 391-409.
  11. [11] W. Kermack, A. McKendrick, A contribution to the mathematical theory of epidemics, Proc. of the Royal Society of London. Ser. A, Containing Papers of a Mathematical and Physical Character, 115, No 772 (1927), 700-721.
  12. [12] J.H. Kim, HIV transmissions by stage and sex role in long-term concurrent sexual partnerships, Acta Biotheoretica, 63 (2015), 33-54.
  13. [13] J. Lenhart, S. andWorkman, Optimal Control Applied to Biological Models, Chapman Hall/CRC, Boca Raton-London-New York (2007).
  14. [14] M.Y. Li, J.R. Graef, L. Wang, Global dynamics of a SEIR model with a varying total population size, Mathematical Biosciences, 160, No 2 (1999), 191-213.
  15. [15] A. Mallela, S. Lenhart, N.K. Vaidya, HIV/TB co-infection treatment: Modeling and optimal control theory perspectives, Journal of Computational and Applied Mathematics, 307 (2015), 143-161.
  16. [16] C.C. Mccluskey, A model of HIV / AIDS with staged progression and amelioration, Mathematical Biosciences, 181 (2003), 1-16.
  17. [17] Ministry of Health and Child Care Zimbabwe AIDS and TB Unit, Zimbabwe HIV Drug Resistance EarlyWarning Indicators (EWI) Survey: 2013 Report, Technical Report, Harare, (2013).
  18. [18] R.E. Miron, R.J. Smith, Resistance to protease inhibitors in a model of HIV-1 INFECTION WITH IMPULSIVE DRUG EFFECTS, Bull. of Mathematical Biology, 76 No 1 (2014), 59-97.
  19. [19] G.A. Ngwa, W.S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32, No 7-8 (2000), 747-763.
  20. [20] C. Orrell, Antiretroviral adherence in a resource-poor setting, Current HIV/AIDS Reports, 2, No 4 (2005), 171-176.
  21. [21] L. Pontryagin, V. Boltyanskii, R. Gamkrelidze, E. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, New YorkLondon (1962).
  22. [22] Z. Shuai, P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM Journal on Applied Mathematics, 73, No 4 (2013), 1513-1532.
  23. [23] R.F. Stengel, Mutation and control of the human immunodeficiency virus, Mathematical Biosciences 213, No 2 (2008), 93-102.
  24. [24] T. Sterling, D. Vlahov, et al., Initial PLASMA HIV-1 RNA levels and progression to AIDS in women and men, New England Journal of Medicine, 344 (2001), 720-726.
  25. [25] UNAIDS, Global AIDS Update, Techn. Report, Geneva (2016).
  26. [26] UNAIDS, Fact Sheet, N. Hiv, 2017 Global HIV Statistics, Techn. Report, July (2018).
  27. [27] UNAIDS, GLOBAL REPORT: UNAIDS Report on the global AIDS epidemic (2013).
  28. [28] P. van den Driessche, J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180, No 1-2 (2002), 29-48.
  29. [29] B. Varghese, J.E. Maher, etal, Reducing the risk of sexual HIV transmission: quantifying the per-act risk for HIV on the basis of choice of partner, sex act, and condom use, Sexually Transmitted Diseases, 29 (2002), 38-43.
  30. [30] WHO, Number of deaths due to HIV/AIDS by country, Technical Report (2017).
  31. [31] K.H. Wong, K.C. Chan, S.S. Lee, Delayed progression to death and to AIDS in a Hong Kong cohort of patients with advanced HIV type 1 disease during the era of highly active antiretroviral therapy, Clinical Infectious Diseases, 39 (2004), 853-860.
  32. [32] Y. Zhou, Y. Liang, J. Wu, An optimal strategy for HIV multitherapy, Journal of Computational and Applied Mathematics, 263 (2014), 326-337.
  33. [33] ZIMSTAT, ICF International, Demographic and Health Survey Key Findings, Technical Report (2016).