Abstract
In this paper, we formulate an age-space-structured HIV infection model that incorporating infection age, multiple target cells, and nonlocal dispersal. Applying the characteristic line method, we reduce the infection-age model to a delayed integro-differential system. The global well-posedness and boundedness of the semiflow for the system are established. The principal eigenvalue of the nonlocal dispersal problem is formulated, and it plays the same role as the basic reproduction number \(R_0\) (the spectral radius of the next generation operator), which determines the global behavior of the steady states of the system. More precisely, the infection-free steady state is globally asymptotically stable (g.a.s) when \(R_0<1\), the virus is always present and the infected steady state is g.a.s when \(R_0>1\). Numerical simulations are carried out reinforcing these analytical results. In particular, three different kernel functions are given out to study the impact of dispersal form on the HIV infection within the host. Finally, our simulation works show that (i) increasing the dispersal rate and decreasing the intracellular delay will be increasing the final viral loads; (ii) the dispersal kernel function affects the value of \(R_0\) and the final viral loads, and it is revealed that the dispersal form plays a crucial role in the process of HIV infection within the host.
Similar content being viewed by others
References
A.T. Haase, Targeting early infection to prevent HIV-1 mucosal transmission. Nature 464, 217–223 (2010)
J.A. Levy, HIV and the Pathogenesis of AIDS, 3rd edn. (ASM Press, Washington, DC, 2007)
X. Wang, X. Song, S. Tang, L. Rong, Analysis of HIV models with multiple target cell populations and general nonlinear rates of viral infection and cell death. Math. Comput. Simul. 124, 87–103 (2016)
E.C. Manda, F. Chirove, Modelling coupled within host and population dynamics of R5 and X4 HIV infection. J. Math. Biol. 76, 1123–1158 (2018)
X. Wang, Y. Lou, X. Song, Age-structured within-host HIV dynamics with multiple target cells. Studies in Appl. Math. 138, 43–76 (2016)
C. Angel, A. Eric, Global properties of an age-structured virus model with saturated antibody immune response, multi-target cells and general incidence rate. arXiv preprint arXiv:1712.05064 (2017)
C. Cheng, Y. Dong, Y. Takeuchi, An age-structured virus model with two routes of infection in heterogeneous environments. Nonlinear Anal. RWA 39, 464–491 (2018)
X. Ren, Y. Tian, L. Liu, X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission. J. Math. Biol. 76, 1831–1872 (2018)
A.D. Agha, A.M. Elaiw, Stability of a general reaction-diffusion HIV-1 dynamics model with humoral immunity. Eur. Phys. J. Plus 134, 390–408 (2019)
Y. Gao, J. Wang, Threshold dynamics of a delayed nonlocal reaction-diffusion HIV infection model with both cell-free and cell-to-cell transmissions. J. Math. Anal. Appl. 488, 124047 (2020)
W. Wang, X. Wang, Z. Feng, Time periodic reaction-diffusion equations for modeling 2-LTR dynamics in HIV-infected patients. Nonlinear Anal. RWA 57, 103184 (2021)
H. Sun, J. Wang, Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay. Comput. Math. Appl. 77, 284–301 (2019)
W. Wang, W. Ma, Z. Feng, Complex dynamics of a time periodic nonlocal and time-delayed model of reaction-diffusion equations for modeling CD4\(^+\) T cells decline. J. Comput. Appl. Math. 367, 112430 (2020)
G. Zhang, W. Li, Y. Sun, Asymptotic behavior for nonlocal dispersal equations. Nonlinear Anal. 72, 4466–4474 (2010)
L. Liu, P. Weng, A nonlocal diffusion model of a single species with age structure. J. Math. Anal. Appl. 432, 38–52 (2015)
P. Weng, L. Liu, Globally asymptotic stability of a delayed integro-differential equation with nonlocal diffusion. Can. Math. Bull. 60, 4436–448 (2017)
P. Magal, X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37(1), 251–275 (2005)
F. Yang, W. Li, Dynamics of a nonlocal dispersal SIS epidemic model, J. Dyn. Differ. Equ., Revised
T. Kuniya, J. Wang, Global dynamics of an SIR epidemic model with nonlocal diffusion. Nonlinear Anal. RWA 43, 262–282 (2018)
F. Yang, W. Li, S. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions. J. Differ. Equ. 267, 2011–2051 (2019)
G. Zhao, S. Ruan, Spatial and temporal dynamics of a nonlocal viral infection model. SIAM J. Appl. Math. 78(4), 1954–1980 (2018)
X. Wang, Y. Chen, J. Yang, Spatial and temporal dynamics of a viral infection model with two nonlocal effects. Complexity (2019). https://doi.org/10.1155/2019/5842942
L. Liu, R. Xu, Z. Jin, Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion. Appl. Math. Model. 82(5), 150–167 (2020)
X. Lai, X. Zou, Dynamics of evolutionary competition between budding and lytic viral releases strategies. Math. Biol. Eng. 11(5), 1091–1113 (2014)
P. Wu, H. Zhao, Dynamics of an HIV infection model with two infection routes and evolutionary competition between two viral strains. Appl. Math. Model. 84, 240–264 (2020)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, NY, USA, 1983)
G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (CRC Press, Boca Raton, 1985)
X.-Q. Zhao, Dynamical Systems in Population Biology (Springer-Verlag, New York, 2017)
G.M. Jorge, D.R. Julio, On the principle eigenvalue of some nonlocal diffusion problems. J. Differ. Equ. 246(5), 21–38 (2009)
W. Wang, X.-Q. Zhao, Basic reproduction number for reaction-diffusion epidemic models. SIAM J. Appl. Dyn. Syt. 11, 1652–1673 (2012)
P. Bates, G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal. J. Math. Aanl. Appl. 332(1), 428–440 (2007)
H.L. Smith, X.-Q. Zhao, Robust persistence for semidynamical systems. Nonlinear Anal. Theory Methods Appl. 47(9), 6169–6179 (2001)
D.E. Kirschner, G.F. Webb, A model for treatment strategy in the chemotherapy of AIDS. Bull. Math. Biol. 58, 367–390 (1996)
M. Markowitz, M. Louie, A. Hurley et al., A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and T cell decay in vivo. J. Virol. 77(2–3), 5037–5038 (2003)
C.Y. Kao, Y. Lou, W. Shen, Random dispersal vs non-local dispersal. Discrete Contin. Dyn. Syst. 26, 551–596 (2010)
E.C. Manda, F. Chirove, Modelling coupled within host and population dynamics of \(R_5\) and \(X_4\) HIV infection. J. Math. Biol. 76, 1123–1158 (2018)
Acknowledgements
The author is very grateful to Dr. Aaron Sun for his linguistic assistance. Also, the author would like to thank Editor and the anonymous referees for their helpful comments and suggestions which led to an improvement of the original manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, P. Dynamics of a delayed integro-differential HIV infection model with multiple target cells and nonlocal dispersal. Eur. Phys. J. Plus 136, 117 (2021). https://doi.org/10.1140/epjp/s13360-020-01049-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-020-01049-5