A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data

Pawelek, Kasia A.; Liu, Shengqiang; Pahlevani, Faranak; Liang, Rong

doi:10.1016/j.mbs.2011.11.002

Cited by 147 publications

(104 citation statements)

References 65 publications

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“…Ciupe et al [8] included a time delay representing the time needed to activate the CD8+ T cell response in a model. Pawelek et al [33] incorporated two delays, one the time needed for infected cells to produce virus and the other the time needed for CD8+ T cells to emerge to kill infected cells. The model was fit to the viral load data from patients during primary infection.…”

Section: Introductionmentioning

confidence: 99%

“…The model was fit to the viral load data from patients during primary infection. Some special cases of the model with two delays have been mathematically analysed [33]. However, the stability switching curves for the infected steady state when both delays are positive have not been investigated.…”

Section: Introductionmentioning

confidence: 99%

“…The second model of HIV immune response with two delays is the same as that in ref. [33]. Some special cases of the model have been studied before [33].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of HIV models with two time delays

Alshorman

Wang

Meyer

et al. 2016

Journal of Biological Dynamics

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Time delays can affect the dynamics of HIV infection predicted by mathematical models. In this paper, we studied two mathematical models each with two time delays. In the first model with HIV latency, one delay is the time between viral entry into a cell and the establishment of HIV latency, and the other delay is the time between cell infection and viral production. We defined the basic reproductive number and showed the local and global stability of the steady states. Numerical simulations were performed to evaluate the influence of time delays on the dynamics. In the second model with HIV immune response, one delay is the time between cell infection and viral production, and the other delay is the time needed for the adaptive immune response to emerge to control viral replication. With two positive delays, we obtained the stability crossing curves for the model, which were shown to be a series of open-ended curves. ARTICLE HISTORY

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of HIV models with two time delays

Alshorman

Wang

Meyer

et al. 2016

Journal of Biological Dynamics

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“…Biologically, it has been proved that the prime target of HIV-1 is CD4 + T lymphocytes (large portion of white blood cells in the immune system). Mathematical models in epidemiology and immunology have given considerable contribution to understand the behaviour of disease transmission (Rong et al, 2007;Sedaghat et al, 2007;Wedagedera, 2011;Pawelek et al, 2012). Hence, these results play a noticeable role to improve various kinds of drug therapies.…”

Section: Introductionmentioning

confidence: 99%

Stability properties of a delayed HIV model with nonlinear functional response and absorption effect

Pradeep¹,

Ma²,

Guo³

2015

J. Natn. Sci. Foundation Sri Lanka

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This study investigated the impact of latent and maturation delay on the qualitative behaviour of a human immunodeficiency virus-1 (HIV-1) infection model with nonlinear functional response and absorption effect. Basic reproduction number (R 0 ), which is defined as the average number of infected cells produced by one infected cell after inserting it into a fully susceptible cell population is calculated for the proposed model. As R o (threshold) depends on the negatively exponential function of time delay, these parameters are responsible to predict the future propagation behaviour of the infection. Therefore, for smaller positive values of delay and larger positive values of infection rate, the infection becomes chronic. Besides, infection dies out with larger delays and lower infection rates. To make the model biologically more sensible, we used the functional form of response function that plays an important role rather than the bilinear response function. Existence of equilibria and stability behaviour of the proposed model totally depend on R o . Local stability properties of both infection free and chronic infection equilibria are established by utilising the characteristic equation. As it is crucially important to study the global behaviour at equilibria rather than the local behaviour, we used the method of Liapunov functional. By constructing suitable Liapunov functionals and applying LaSalle's invariance principle for delay differential equations, we established that infection free equilibrium is globally asymptotically stable if R 0 ≤ 1, which biologically means that infection dies out. Moreover, sufficient condition is derived for global stability of chronic infection equilibrium if R 0 > 1 , which biologically means that infection becomes chronic. Numerical simulations are given to illustrate the theoretical results.

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“…In addition, stability analysis may determine the basin of attraction of each equilibrium, that is, the region of coexistence and the region of extinction. Aside from predator-prey systems, some research works on virus infection of CD4 + cells have been performed in [16][17][18][19][20]. In their works, mathematical models describing infection dynamics have infection-free equilibrium and chronic-infection equilibrium.…”

Section: Introductionmentioning

confidence: 99%

On the Stability of Critical Point for Positive Systems and Its Applications to Biological Systems

Lee¹,

Jo²,

Shim³

et al. 2013

Journal of Electrical Engineering and Technology

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-The coexistence and extinction of species are important concepts for biological systems and can be distinguished by an investigation of stability. When determining local stability of nonlinear systems, Lyapunov indirect method based on the Jacobian linearization has been widely employed due to its simplicity. Despite such popularity, it is not applicable to singular systems whose Jacobian has at least one eigenvalue that is equal to zero. In such singular cases, an appropriate Lyapunov function should be sought to determine the stability of systems, which is rather difficult and quite involved. In this paper, we seek for a simple criterion to determine stability of the equilibrium that is located at the boundary of the positive orthant, when one of eigenvalues of the Jacobian is zero. The goal of the paper is to present a generalized condition for the equilibrium to attract all trajectories that starting from initial condition in the positive orthant and near the equilibrium. Unlike the Lyapunov direct method, the proposed method requires just a simple algebraic computation for checking the stability of the critical point. Our approach is applied to various biological systems to show the effectiveness of the proposed method.

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A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data

Cited by 147 publications

References 65 publications

Analysis of HIV models with two time delays

Analysis of HIV models with two time delays

Stability properties of a delayed HIV model with nonlinear functional response and absorption effect

On the Stability of Critical Point for Positive Systems and Its Applications to Biological Systems

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