[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Asymptotic Behavior Analysis and Threshold Sharpening of a Staged Progression AIDS/HIV Epidemic Model with L'{e}vy Jumps

Discontinuity, Nonlinearity, and Complexity 12(3) (2023) 583--613 | DOI:10.5890/DNC.2023.09.008

Driss Kiouach, Salim El Azami El-idrissi, Yassine Sabbar

Mathematics Department, Faculty of Sciences Dhar Al Mahraz, Sidi Mohammed Ben Abdellah University, P.O. Box 1796, Fez-Atlas, 30003 Fez, Morocco

Download Full Text PDF

 

Abstract

In this paper, we present and investigate a generalized stochastic AIDS/HIV epidemic model that includes both Brownian motions and L\'{e}vy jumps. Our proposed model is a staged progression compartmental one that takes the form of an It\^{o}-L\'{e}vy stochastic differential equations system. First, we demonstrate its well-posedness in the sense that it admits one and only one solution which is positive and global in time. Then, and based on some assumptions and nonstandard analytical techniques, we prove two principal asymptotic properties: extinction and persistence in the mean. The theoretical results show that the dynamical behavior of our AIDS/HIV model is mainly determined by the parameters that are closely related to the perturbations intensities. In the end, we provide some numerical simulation examples to corroborate our theoretical results and exhibit the effect of the new adopted mathematical techniques on the findings.

References

  1. [1]  Klimas, N., Koneru, A.O., and Fletcher, M.A. (2008), Overview of HIV, Psychosomatic Medicine, 70, 523-530.
  2. [2]  Ruelas, D.S. and Greene, W.C. (2013), An integrated overview of HIV-1 latency, Cell, 155, 519-529.
  3. [3]  Joint United Nations Programme on HIV and AIDS (2020), HIV and AIDS - Basic facts, UNAIDS official website.
  4. [4]  Centers for Disease Control and Prevention (2020), HIV transmission, CDC official website.
  5. [5]  World Health Organization (2020), https://www.who.int/news-room/fact-sheets/detail/hiv-aids, WHO official website.
  6. [6]  Marfatia, Y.S., Jose, S.K., Baxi, R.R., and Shah, R.J. (2017), Pre-and post-sexual exposure prophylaxis of HIV: An update, Indian journal of sexually transmitted diseases and AIDS, 38, 1.
  7. [7]  Joint United Nations Programme on HIV and AIDS (2020), HIV prevention, UNAIDS official website.
  8. [8]  Boily, M.-C. and Brunham, R.C. (1993), The impact of HIV and other stds on human populations: Are predictions possible? Infectious disease clinics of North America, 7, 771-792.
  9. [9]  Huang, G., Ma, W., and Takeuchi, Y. (2011), Global analysis for delay virus dynamics model with beddington-deangelis functional response, Applied Mathematics Letters, 24, 1199-1203.
  10. [10]  Lv, C., Huang, L., and Yuan, Z. (2014), Global stability for an HIV-1 infection model with beddington-deangelis incidence rate and ctl immune response, Communications in Nonlinear Science and Numerical Simulation, 19, 121-127.
  11. [11]  Hyman, J.M., Li, J., and Stanley, E.A. (2003), Modeling the impact of random screening and contact tracing in reducing the spread of HIV, Mathematical biosciences, 181, 17-54.
  12. [12]  Otunuga, O.M. (2018), Global stability for a 2n+ 1 dimensional HIV/AIDS epidemic model with treatments, Mathematical Biosciences, 299, 138-152.
  13. [13]  Nelson, P.W. and Perelson, A.S. (2002), Mathematical analysis of delay differential equation models of HIV-1 infection, Mathematical Biosciences, 179, 73-94.
  14. [14]  Cai, L.-M., Guo, B.-Z., and Li, X.-Z. (2012), Global stability for a delayed HIV-1 infection model with nonlinear incidence of infection, Applied Mathematics and Computation, 219, 617-623.
  15. [15]  Lin, X., Hethcote, H.W., and Vanden Driessche, P. (1993), An epidemiological model for HIV/AIDS with proportional recruitment, Mathematical biosciences, 118, 181-195.
  16. [16]  Hyman, J.M., Li, J., and Stanley, E.A. (1999), The differential infectivity and staged progression models for the transmission of HIV, Mathematical Biosciences, 155, 77-109.
  17. [17]  McCluskey, C.C. (2003), A model of HIV/AIDS with staged progression and amelioration, Mathematical Biosciences, 181, 1-16.
  18. [18]  Cheng, Y., Zhang, F., and Zhao, M. (2019), A stochastic model of HIV infection incorporating combined therapy of haart driven by l{e}vy jumps, Advances in Difference Equations, 2019, 1-17.
  19. [19]  Cai, L., Fang, B., and Li, X. (2014), A note of a staged progression HIV model with imperfect vaccine, Applied Mathematics and Computation, 234, 412-416.
  20. [20]  Guo, H. and Li, M.Y. (2011), Global dynamics of a staged-progression model for HIV/AIDS with amelioration, Nonlinear Analysis: Real World Applications, 12, 2529-2540.
  21. [21]  Han, B., Jiang, D., Hayat, T., Alsaedi, A., and Ahmad, B. (2020), Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation, Chaos, Solitons $\&$ Fractals, 140, 110238.
  22. [22]  Cai, Y., Kang, Y., Banerjee, M., and Wang, W. (2016), A stochastic epidemic model incorporating media coverage, Communications in Mathematical Sciences, 14, 893-910.
  23. [23]  Ivorra, B., Ferr{a}ndez, M.R., Vela-P{e}rez, M., and Ramos, A. (2020), Mathematical modeling of the spread of the coronavirus disease 2019 (covid-19) taking into account the undetected infections. The case of China. Communications in Nonlinear Science and Numerical Simulation, 88, 105303.
  24. [24]  Wang, Y., Jiang, D., Hayat, T., and Alsaedi, A. (2019), Stationary distribution of an HIV model with general nonlinear incidence rate and stochastic perturbations, Journal of the Franklin Institute, 356, 6610-6637.
  25. [25]  Liu, Q., Jiang, D., Hayat, T., and Ahmad, B. (2017), Asymptotic behavior of a stochastic delayed HIV-1 infection model with nonlinear incidence, Physica A: Statistical Mechanics and Its Applications, 486, 867-882.
  26. [26]  Liu, Q. and Jiang, D. (2020), Dynamics of a stochastic multigroup s-di-a model for the transmission of HIV, Applicable Analysis, 101(2), 747-772.
  27. [27]  Naik, P.A., Owolabi, K.M., Yavuz, M., and Zu, J. (2020), Chaotic dynamics of a fractional order hiv-1 model involving aids-related cancer cells, Chaos, Solitons $\&$ Fractals, 140, 110272.
  28. [28]  Yavuz, M. and Sene, N. (2020), Stability analysis and numerical computation of the fractional predator-prey model with the harvesting rate, Fractal and Fractional, 4, 35.
  29. [29]  Naik, P.A., Yavuz, M., and Zu, J. (2020), The role of prostitution on hiv transmission with memory: A modeling approach, Alexandria Engineering Journal, 59, 2513-2531.
  30. [30]  Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., and Townley, S. (2020), Modeling and analysis of covid-19 epidemics with treatment in fractional derivatives using real data from pakistan, The European Physical Journal Plus, 135, 1-42.
  31. [31]  Yavuz, M. and {\"O}zdemir, N. (2020), Analysis of an epidemic spreading model with exponential decay law, Mathematical Sciences and Applications E-Notes, 8, 142-154.
  32. [32]  Yavuz, M. and Bonyah, E. (2019), New approaches to the fractional dynamics of schistosomiasis disease model, Physica A: Statistical Mechanics and its Applications, 525, 373-393.
  33. [33]  Kiouach, D. and Sabbar, Y. (2021), The long-time behaviour of a stochastic {SIR} epidemic model with distributed delay and multidimensional Levy jumps, arXiv preprint arXiv:2003.08219.
  34. [34]  Kiouach, D., El-idrissi, S.E.A., and Sabbar, Y. (2021), Advanced and comprehensive research on the dynamics of covid-19 under mass communication outlets intervention and quarantine strategy: a deterministic and probabilistic approach, arXiv preprint arXiv:2101.00517.
  35. [35]  Rajasekar, S. and Pitchaimani, M. (2019), Qualitative analysis of stochastically perturbed sirs epidemic model with two viruses, Chaos, Solitons $\&$ Fractals, 118, 207-221.
  36. [36]  Rajasekar, S., Pitchaimani, M., and Zhu, Q. (2019), Dynamic threshold probe of stochastic sir model with saturated incidence rate and saturated treatment function, Physica A: Statistical Mechanics and its Applications, 535, 122300.
  37. [37]  Rajasekar, S., Pitchaimani, M., Zhu, Q., and Shi, K. (2021), Exploring the stochastic host-pathogen tuberculosis model with adaptive immune response, Mathematical Problems in Engineering, 2021.
  38. [38]  Zhang, X. and Wang, K. (2013), Stochastic SIR model with jumps, Applied Mathematics Letters, 26, 867-874.
  39. [39]  Kiouach, D., Sabbar, Y., and El-idrissi ElAzami, S. (2021), New results on the asymptotic behavior of an {SIS} epidemiological model with quarantine strategy, stochastic transmission, and Levy disturbance, Mathematical Methods in the Applied Sciences.
  40. [40]  Kiouach, D. and Sabbar, Y. (2020), Ergodic stationary distribution of a stochastic hepatitis b epidemic model with interval-valued parameters and compensated poisson process, Computational and Mathematical Methods in Medicine, 2020.
  41. [41]  Kiouach, D. and Sabbar, Y. (2021), Dynamic characterization of a stochastic sir infectious disease model with dual perturbation, International Journal of Biomathematics, p. 2150016.
  42. [42]  Zhao, D., Yuan, S., and Liu, H. (2019), Stochastic dynamics of the delayed chemostat with Levy noises, International Journal of Biomathematics, 12, 1950056.
  43. [43]  Zhao, D. and Yuan, S. (2018), Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat, Applied Mathematics and Computation, 339, 199-205.
  44. [44]  Liu, Q., Jiang, D., Hayat, T., and Alsaedi, A. (2018), Stationary distribution and extinction of a stochastic dengue epidemic model, Journal of the Franklin Institute, 355, 8891-8914.
  45. [45]  \O ksendal, B.K. and Sulem, A. (2007), Applied Stochastic Control of Jump Diffusions, 498, Springer.
  46. [46]  Mao, X. (2007), Stochastic Differential Equations and Applications, Elsevier.
  47. [47]  Karatzas, I. and Shreve, S.E. (1998), Brownian Motion and Stochastic Calculus, Springer.
  48. [48]  Peng, S. and Zhu, X. (2006), Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Processes and Their Applications, 116, 370-380.
  49. [49]  Ji, C. and Jiang, D. (2014), Threshold behaviour of a stochastic SIR model, Applied Mathematical Modelling, 38, 5067-5079.
  50. [50]  Yin, S. (2017), A new generalization on cauchy-schwarz inequality, Journal of Function Spaces, 2017.
  51. [51]  Liu, Q., Jiang, D., Hayat, T., Alsaedi, A., and Ahmad, B. (2020), Dynamical behavior of a higher order stochastically perturbed {SIRI} epidemic model with relapse and media coverage, Chaos, Solitons $\&$ Fractals, 139, 110013.
  52. [52]  Liu, Q., Jiang, D., Hayat, T., and Alsaedi, A. (2019), Dynamics of a stochastic {SIR} epidemic model with distributed delay and degenerate diffusion, Journal of the Franklin Institute, 356, 7347-7370.
  53. [53]  Zhou, Y., Zuo, W., Jiang, D., and Song, M. (2020), Stationary distribution and extinction of a stochastic model of syphilis transmission in an MSM population with telegraph noises, Journal of Applied Mathematics and Computing, 66, 645-672.
  54. [54]  Liu, Q. and Jiang, D. (2020), Dynamical behavior of a higher order stochastically perturbed HIV/AIDS model with differential infectivity and amelioration, Chaos, Solitons $\&$ Fractals, 141, 110333.
  55. [55]  Cheng, Y., Li, M., and Zhang, F. (2019), A dynamics stochastic model with hiv infection of CD4+ t-cells driven by Levy noise, Chaos, Solitons $\&$ Fractals, 129, 62-70.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates