Lyapunov Functionals for Delay Differential Equations Model of Viral Infections

Huang, Gang; Takeuchi, Yasuhiro; Ma, Wanbiao

doi:10.1137/090780821

Cited by 213 publications

(136 citation statements)

References 35 publications

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“…The analytical approach is motivated by the construction methods in [9,11,15,24,29]. The hypotheses (A 1 )-(A 4 ), including not only a bilinear incidence rate f(x, v)v = kxv, separable incidence rate f(x, v)v = F(x)G(v) but also (non-separable) standard incidence rate f(x, v)v = xv x+v and Beddington-DeAngelis functional response f(x, v)v = xv 1+α 1 x+α 2 v , play a crucial role in determining the suitable Lyapunov functionals.…”

Section: Discussionmentioning

confidence: 99%

“…On the other hand, in the literature of epidemiology, it has increasingly been asserted that the incidence rate, characterizing the rate of newly infected cells, should be written by not bilinear but general nonlinear functions (see, e.g., [8,9,26]). We point out that since both the model (1.1) does not consider cell-mediated cytotoxic T lymphocytes (CTLs) immune response and model (1.2) does not incorporate non-separable incidence rate, in this paper, we combine these two considerations to formulate our model.…”

Section: Introductionmentioning

confidence: 99%

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Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses

Enatsu¹,

Wang²,

Kuniya³

2017

J. Nonlinear Sci. Appl.

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In this paper, we study the dynamical behavior of a virus model into which cell-mediated and humoral immune responses are incorporated. The global stability of an infection-free equilibrium and four infected equilibria is established via a Lyapunov functional approach. The present construction methods are applicable to a wide range of incidence rates that are monotone increasing with respect to concentration of uninfected cells and concave with respect to the concentration of free virus particles. In addition, when the incidence rate is monotone increasing with respect to concentration of free virus particles, the functional approach plays an important role in determining the global stability of each of the four infected equilibria. This implies that the dynamical behavior of virus prevalence would be determined by basic reproduction numbers when the "saturation effect" for free virus particles appears. We point out that the incidence rate includes not only separable incidence rate but also non-separable incidence rate such as standard incidence and Beddington-DeAngelis functional response.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses

Enatsu¹,

Wang²,

Kuniya³

2017

J. Nonlinear Sci. Appl.

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“…However, such bilinear form is imperfect to depict the dynamical behavior of the viral pathogen infection in detail (see Refs. [40][41][42]. Saturated incidence rate is suitable to model the nonlinear dynamics of the pathogen especially when its concentration is high.…”

Section: Dt = γS(t)p(t) − I(t) − υX(t)i(t)mentioning

confidence: 99%

Stability of CTL immunity pathogen dynamics model with capsids and distributed delay

2017

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In this paper, a pathogen dynamics model with capsids and saturated incidence has been proposed and analyzed. Cytotoxic T Lymphocyte (CTL) immune response and two distributed time delays have been incorporated into the model. The nonnegativity and boundedness of the solutions of the proposed model have been shown. Two threshold parameters which fully determine the existence and stability of the three steady states of the model have been computed. Using the method of Lyapunov function, the global stability of the steady states of the model has been established. The theoretical results have been confirmed by numerical simulations.

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“…The classical HIV infection divided the cells population into following three main compartments depending on cell status, target cells, infected cells producing viruses, matured virus particles, and its concentrations are denoted by x(t), y(t) and v(t), respectively. Dynamics behaviors for the interactions between target cells, infected cells producing viruses, matured viruses and immune cells are investigated by many authors in the literature, [1,8,10,11,12,17,18,19,23,26,28]. These models focused on two dynamics scenarios: (i) whether the viruses are cleared; (ii) whether life long infection-free can be achieved in the host, or sustained oscillatory viral loads phenomenon occurs.…”

Section: Introductionmentioning

confidence: 99%

“…To model the realistic situation, many authors are devoted to nonlinear incidence rate, the rate of contact between viruses and uninfected target cells, The following incidence rate is proposed in the literature, e.g., Holling type II functional response, βxv 1+αx [13], Beddington-DeAngelis functional response, βxv 1+ax+bv [9,16], c(x)f (v) [7], and F (x, v) [10,19].…”

Section: Introductionmentioning

confidence: 99%

Constructing Lyapunov functionals for a delayed viral infection model with multitarget cells, nonlinear incidence rate, state-dependent removal rate

Wang¹,

Lang²,

Li³

2016

J. Nonlinear Sci. Appl.

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For a viral infection model with multitarget cells, nonlinear incidence rate, state-dependent removal rate and distributed delays, we analyze the global asymptotic behavior of its solutions. In this model, the rate of contact between viruses and uninfected target cells and state-dependent removal rate of infected cells depend on general nonlinear functions. The basic reproduction number for the model is discussed. Under certain assumptions, it is shown that if 0 ≤ 1, then the infection-free equilibrium P 0 is globally stable and the viruses are cleared; If 0 > 1, then there is a unique infection equilibrium, which is globally stable implying the infection becomes chronic. The global stability results are achieved by appealing to the direct Lyapunov method.

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Lyapunov Functionals for Delay Differential Equations Model of Viral Infections

Cited by 213 publications

References 35 publications

Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses

Impact of non-separable incidence rates on global dynamics of virus model with cell-mediated, humoral immune responses

Stability of CTL immunity pathogen dynamics model with capsids and distributed delay

Constructing Lyapunov functionals for a delayed viral infection model with multitarget cells, nonlinear incidence rate, state-dependent removal rate

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