Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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STABILITY OF A CLASS OF DISCRETE-TIME PATHOGEN INFECTION MODELS WITH LATENTLY INFECTED CELLS

  • ELAIW, A.M. (DEPARTMENT OF MATHEMATICS, KING ABDULAZIZ UNIVERSITY) ;
  • ALSHAIKH, M.A. (DEPARTMENT OF MATHEMATICS, KING ABDULAZIZ UNIVERSITY)
  • Received : 2018.02.23
  • Accepted : 2018.12.11
  • Published : 2018.12.25
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Abstract

This paper studies the global stability of a class of discrete-time pathogen infection models with latently infected cells. The rate of pathogens infect the susceptible cells is taken as bilinear, saturation and general. The continuous-time models are discretized by using nonstandard finite difference scheme. The basic and global properties of the models are established. The global stability analysis of the equilibria is performed using Lyapunov method. The theoretical results are illustrated by numerical simulations.

Keywords

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FIGURE 1. The simulation of susceptible cells of system (5.1)-(5.4) for Case(I).

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FIGURE 2. The simulation of latently infected cells of system (5.1)-(5.4) for Case(I).

E1TAAE_2018_v22n4_253_f0003.png 이미지

FIGURE 3. The simulation of actively infected cells of system (5.1)-(5.4) for Case(I).

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FIGURE 4. The simulation of pathogens of system (5.1)-(5.4) for Case(I).

E1TAAE_2018_v22n4_253_f0005.png 이미지

FIGURE 5. The simulation of susceptible cells of system (5.1)-(5.4) for Case(II).

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FIGURE 6. The simulation of latently infected cells of system (5.1)-(5.4) for Case(II).

E1TAAE_2018_v22n4_253_f0007.png 이미지

FIGURE 7. The simulation of actively infected cells of system (5.1)-(5.4) for Case(II).

E1TAAE_2018_v22n4_253_f0008.png 이미지

FIGURE 8. The simulation of pathogens of system (5.1)-(5.4) for Case(II).

E1TAAE_2018_v22n4_253_f0009.png 이미지

FIGURE 9. The simulation of susceptible cells of system (5.1)-(5.4) for Case(III).

E1TAAE_2018_v22n4_253_f0010.png 이미지

FIGURE 10. The simulation of latently infected cells of system (5.1)-(5.4) for Case(III).

E1TAAE_2018_v22n4_253_f0011.png 이미지

FIGURE 11. The simulation of actively infected cells of system (5.1)-(5.4) for Case(III).

E1TAAE_2018_v22n4_253_f0012.png 이미지

FIGURE 12. The simulation of pathogens of system (5.1)-(5.4) for Case(III).

TABLE 1. The values of R0 for system (5.1)-(5.4) with different values of λ.

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References

  1. M. A. Nowak and C.R.M. Bangham. Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. https://doi.org/10.1126/science.272.5258.74
  2. D.S. Callaway, and A.S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 29-64. https://doi.org/10.1006/bulm.2001.0266
  3. A. M. Elaiw and N. H. AlShamrani, Global analysis for a delay-distributed viral infection model with antibodies and general nonlinear incidence rate, J. Korean Soc. Ind. Appl. Math., 18(4) (2014), 317-335. https://doi.org/10.12941/jksiam.2014.18.317
  4. A. M. Elaiw, Global threshold dynamics in humoral immunity viral infection models including an eclipse stage of infected cells, J. Korean Soc. Ind. Appl. Math., 19:2 (2015), 137-170. https://doi.org/10.12941/jksiam.2015.19.137
  5. H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses, SIAM Journal of Applied Mathematics, 73(3) (2013), 1280-1302. https://doi.org/10.1137/120896463
  6. P. Georgescu and Y.H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM Journal of Applied Mathematics, 67 (2006), 337-353.
  7. A. M. Elaiw, E. Kh. Elnahary and A. A. Raezah, Effect of cellular reservoirs and delays on the global dynamics of HIV, Advances in Difference Equations, (2018) 2018:85. https://doi.org/10.1186/s13662-018-1523-0
  8. L. Gibelli, A. Elaiw, M.A. Alghamdi and A.M. Althiabi, Heterogeneous population dynamics of active particles: Progression, mutations, and selection dynamics, Mathematical Models and Methods in Applied Sciences, 27, (2017), 617-640. https://doi.org/10.1142/S0218202517500117
  9. A. M. Elaiw, A. A. Raezah and B. S. Alofi, Dynamics of delayed pathogen infection models with pathogenic and cellular infections and immune impairment, AIP Advances, 8 (2018) Article ID 025323.
  10. A. M. Elaiw, A. A. Raezah and S. A. Azoz, Stability of delayed HIV dynamics models with two latent reservoirs and immune impairment, Advances in Difference Equations, (2018) 2018:414. https://doi.org/10.1186/s13662-018-1869-3
  11. K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Analysis: Real World Applications, 13(4) (2012), 1866-1872. https://doi.org/10.1016/j.nonrwa.2011.12.015
  12. M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with antiretroviral treatment, Nonlinear Analysis: Real World Applications, 17 (2014), 147-160. https://doi.org/10.1016/j.nonrwa.2013.11.002
  13. K.Wang, A. Fan, and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Analysis: Real World Applications, 11 (2010), 3131-3138. https://doi.org/10.1016/j.nonrwa.2009.11.008
  14. F. Zhang, J. Li, C. Zheng, L. Wang, Dynamics of an HBV/HCV infection model with intracellular delay and cell proliferation , Communications in Nonlinear Science and Numerical Simulation, 42 (2017), 464-476. https://doi.org/10.1016/j.cnsns.2016.06.009
  15. X. Song and A. Neumann, Global stability and periodic solution of the viral dynamics, Journal of Mathematical Analysis and Applications, 329 (2007), 281-297. https://doi.org/10.1016/j.jmaa.2006.06.064
  16. X. Shi, X. Zhou and X. Song, Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Analysis: Real World Applications, 11 (2010), 1795-1809. https://doi.org/10.1016/j.nonrwa.2009.04.005
  17. A. M. Elaiw and N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Analysis: Real World Applications, 26, (2015), 161-190. https://doi.org/10.1016/j.nonrwa.2015.05.007
  18. A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications, 11 (2010), 2253-2263. https://doi.org/10.1016/j.nonrwa.2009.07.001
  19. A.M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynamics, 69(1-2) (2012), 423-435 https://doi.org/10.1007/s11071-011-0275-0
  20. A. M. Elaiw and N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Mathematical Methods in the Applied Sciences, 39 (2016), 4-31. https://doi.org/10.1002/mma.3453
  21. A. M. Elaiw and N. H. AlShamrani, Stability of a general delay-distributed virus dynamics model with multistaged infected progression and immune response, Mathematical Methods in the Applied Sciences, 40(3) (2017), 699-719. https://doi.org/10.1002/mma.4002
  22. A. M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dynamics in Nature and Society, 2012 (2012) Article ID 253703.
  23. A. M. Elaiw, I. A. Hassanien, S. A. Azoz, Global stability of HIV infection models with intracellular delays, Journal of the Korean Mathematical Society, 49(4) (2012), 779-794. https://doi.org/10.4134/JKMS.2012.49.4.779
  24. A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences, 36 (2013), 383-394. https://doi.org/10.1002/mma.2596
  25. G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM Journal of Applied Mathematics, 70(7) (2010), 2693-2708. https://doi.org/10.1137/090780821
  26. A.S. Perelson, D.E. Kirschner and R.D. Boer, Dynamics of HIV infection of CD4+ T cells, Mathematical Biosciences, 114(1). (1993), 81-125. https://doi.org/10.1016/0025-5564(93)90043-A
  27. B. Buonomo and C. Vargas-De-Leon, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, Journal of Mathematical Analysis and Applications, 385 (2012), 709-720. https://doi.org/10.1016/j.jmaa.2011.07.006
  28. A. M. Elaiw, A. A. Almatrafi, and A. D. Hobiny, Effect of antibodies on pathogen dynamics with delays and two routes of infection, AIP Advances 8 (2018), Article ID 065104.
  29. A. M. Elaiw and N. H. AlShamrani, Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays, Mathematical Methods in the Applied Science, 36 (2018), 125-142.
  30. A. M. Elaiw, T. O. Alade, S. M. Alsulami, Analysis of latent CHIKV dynamics models with general incidence rate and time delays. Journal of Biological Dynamics, 12(1) (2018), 700-730. https://doi.org/10.1080/17513758.2018.1503349
  31. A. M. Elaiw, T. O. Alade, S. M. Alsulami, Analysis of within-host CHIKV dynamics models with general incidence rate. International Journal of Biomathematics 11(5) (2018) Article Number: 1850062.
  32. A. Korobeinikov, Global properties of basic virus dynamics models, Bulletin of Mathematical Biology, 66(4). (2004), 879-883. https://doi.org/10.1016/j.bulm.2004.02.001
  33. R. E. Mickens, Nonstandard Finite Difference Models of Differential equations, World Scientific, Singapore, 1994.
  34. R. E. Mickens, Dynamics consistency: a fundamental principle for constructing nonstandard finite difference scheme for differential equation. Journal of Difference Equations and Applications. 9 (2003), 1037-1051. https://doi.org/10.1080/1023619031000146913
  35. D. Ding and X. Ding, Dynamic consistent non-standard numerical scheme for a dengue disease transmission model, Journal of Difference Equations and Applications, 20 (2014), 492-505. https://doi.org/10.1080/10236198.2013.858715
  36. D. Ding, W. Qin, and X. Ding, Lyapunov functions and global stability for a discretized multi-group SIR epidemic model, Discrete and Continuous Dynamical Systems - Series B, 20 (2015), 1971-1981. https://doi.org/10.3934/dcdsb.2015.20.1971
  37. Y. Enatsu, Y. Nakata, Y. Muroya, G. Izzo, and A. Vecchio, Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates, Journal of Difference Equations and Applications, 18 (2012), 1163-1181. https://doi.org/10.1080/10236198.2011.555405
  38. J. Liu, B. Peng, and T. Zhang, Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence, Applied Mathematics Letters, 39 (2015), 60-66. https://doi.org/10.1016/j.aml.2014.08.012
  39. Z. Teng, L. Wang, and L. Nie, Global attractivity for a class of delayed discrete SIRS epidemic models with general nonlinear incidence, Mathematical Methods in the Applied Sciences, 38 (2015), 4741-4759. https://doi.org/10.1002/mma.3389
  40. K. Hattaf and N. Yousfi, Global properties of a discrete viral infection model with general incidence rate. Mathematical Methods in the Applied Sciences, 39 (2016), 998-1004. https://doi.org/10.1002/mma.3536
  41. P. Shi and L. Dong, Dynamical behaviors of a discrete HIV-1 virus model with bilinear infective rate, Mathematical Methods in the Applied Sciences, 37 (2014), 2271-2280. https://doi.org/10.1002/mma.2974
  42. Y. Yang, J. Zhou, X. Ma and T. Zhang, Nonstandard finite difference scheme for a diffusive within-host virus dynamics model both virus-to-cell and cell-to-cell transmissions, Computers and Mathematical with Applications, 72 (2016), 1013-1020. https://doi.org/10.1016/j.camwa.2016.06.015
  43. J. Xu, Y. Geng and J. Hou, A nonstandard finite difference scheme for a delayed and diffusive viral infection model with general nonlinear incidence rate, Computers and Mathematical with Applications, 74 (2017), 1782-1798. https://doi.org/10.1016/j.camwa.2017.06.041
  44. A. Korpusik, A nonstandard finite difference scheme for a basic model of cellular immune response to viral infection, Common Nonlinear Sci Numer Simulat, 43 (2017), 369-384. https://doi.org/10.1016/j.cnsns.2016.07.017
  45. Yu. Yang, Ma. Xinsheng and Li. Yahui, Global stability of a discrete virus dynamics model with Holling type-II infection function, Mathematical Methods in the Applied Sciences, 39 (2016), 2078-2082. https://doi.org/10.1002/mma.3624
  46. J. Wang, Z. Teng and H. Miao, Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response, Advanced in Difference Equations, 143 (2016), DOI 10.1186/s13662-016-0862-y.
  47. Y. Geng, J. Xu, J. Hou, Discretization and dynamic consistency of a delayed and diffusive viral infection model, Applied Mathematics and Computation 316 (2018) 282-295. https://doi.org/10.1016/j.amc.2017.08.041
  48. W. Qin , L. Wang , X. Ding , A non-standard finite difference method for a hepatitis b virus infection model with spatial diffusion, Journal of Difference Equations and Applications, 20 (2014) 1641-1651. https://doi.org/10.1080/10236198.2014.968565
  49. K. Manna and S.P. Chakrabarty, Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids, Journal of Difference Equations and Applications, 21(10) (2015), 918-933. https://doi.org/10.1080/10236198.2015.1056524
  50. Y. Yang and J. Zhou, Global stability of a discrete virus dynamics model with diffusion and general infection function, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2018.1527028.
  51. J. Zhou and Y. Yang, Global dynamics of a discrete viral infection model with time delay, virus-to-cell and cell-to-cell transmissions, Journal of Difference Equations and Applications, 23(11) (2017).
  52. J. Xu, J. Hou, Y. Geng and S. Zhang, Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence, Advances in Difference Equations (2018) 2018:108. https://doi.org/10.1186/s13662-018-1560-8
  53. S. Elaydi, An introduction to Difference Equations 3rd ed, Springer, New York, (2005).
  54. J. N., Blankson, D. Persaud, and R. F. Siliciano, The challenge of viral reservoirs in HIV-1 infection. Annual Review of Medicine, 53 (2002), 557-593. https://doi.org/10.1146/annurev.med.53.082901.104024
  55. J. N., Blankson, D. Persaud, and R. F. Siliciano, The challenge of viral reservoirs in HIV-1 infection. Annual Review of Medicine, 53 (2002), 557-593. https://doi.org/10.1146/annurev.med.53.082901.104024
  56. A. R.M. Carvalho and C. M. A. Pinto, Contributions of the latent reservoir and of the pool of long -lived chronically infected $CD4^+$ T cells in HIV dynamics: a fractional approach, Proceedings of the ENOC2017, June