Global stability of an HIV pathogenesis model with cure rate

Liu, Xiangdong; Wang, Hui; Hu, Zhixing; Ma, Wanbiao

doi:10.1016/j.nonrwa.2011.04.016

Cited by 12 publications

(35 citation statements)

References 40 publications

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“…Theorem A If R 0 >1, then there exists a unique infected equilibrium E * =( x * , y * , z * ) of system , which is globally asymptotically stable, if

d \leq r < d + \frac{2 rδ}{K}, that is, d \leq r < \frac{d}{1 - 2 δ / K},

where δ is a positive constant that does not depend on the initial conditions such that

{liminf}_{t \to + \infty} x (t) \geq δ, {liminf}_{t \to + \infty} y (t) \geq δ and {liminf}_{t \to + \infty} z (t) \geq δ .

We note that Liu et al ., show the existence of such δ > 0 on the permanence of system , while they did not refer the explicit expression for δ > 0.…”

Section: Introductionmentioning

confidence: 95%

“…Motivated by the results in , we are interested in the global dynamics of the following virus infection model with cure rate of infected CD4 + T cells in Liu et al ., .

({, \begin{matrix} \frac{dx (t)}{dt} = s - dx (t) - βx (t) z (t) + rx (t) (1 - \frac{x (t) + y (t)}{K}) + ρy (t), \\ \frac{dy (t)}{dt} = βx (t) z (t) + ry (t) (1 - \frac{x (t) + y (t)}{K}) - (a + ρ) y (t), \\ \frac{dz (t)}{dt} = aηy (t) - γz (t), \end{matrix})

where x ( t ), y ( t ), and z ( t ) represent the concentration of susceptible CD4 + T cells, infected CD4 + T cells, and free HIV virus particles in the blood at time t , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, system always has an uninfected equilibrium E 0 =( x 0 ,0,0), where x 0 is a positive root of

\frac{r}{K} {(x^{0})}^{2} - (r - d) x^{0} - s = 0,

that is, by d K ≥ s ,

x^{0} = x^{0} (r) = \frac{K}{2 r} ({}, (r - d) + \sqrt{{(r - d)}^{2} + \frac{4 rs}{K}}) \leq K .

Moreover, from , we have

\frac{r}{K} x^{0} - (r - d) - \frac{s}{x^{0}} = 0,

which is equivalent to

a + ρ - r ((), 1 - \frac{x^{0}}{K}) = \frac{s}{x^{0}} + (a - d) + ρ > 0 .

The basic reproduction number of system can be defined as

R_{0} \equiv \frac{βaη x^{0}}{γ (\frac{s}{x^{0}} + a - d + ρ)} .

By and , we note that R 0 >1 is equivalent to N > N c r i t in Liu et al . , Section 2] and also

ρ < r - a + ((), \frac{βaη}{γ} - \frac{r}{K}) x^{0} = d - a + ((), \frac{βaη}{γ} x^{0} - s)

…”

Section: Introductionmentioning

confidence: 99%

“…For system , by applying the approach devoted in Li and Muldowney , a sufficient condition of the global stability of an infected equilibrium is obtained in Liu et al . , Theorems 2.1 and 5.1] as follows:…”

Section: Introductionmentioning

confidence: 99%

“…We obtain the following practical result on Liu et al . , Theorem 5.1]. Theorem For R 0 >1, assume that

r > a + ρ - ((), \frac{βaη}{γ} - \frac{r}{K}) \frac{s}{\frac{s}{x^{0}} + β \frac{aη x^{0}}{γ}}, if r < \min ((), \frac{βaη}{γ} K, a + ρ) .

Then, system is permanent and

({, \begin{matrix} {liminf}_{t \to + \infty} x (t) \geq δ_{x} : = \frac{s}{\frac{s}{x^{0}} + β \frac{aη x^{0}}{γ}} > 0, \\ {liminf}_{t \to + \infty} y (t) \geq δ_{y} : = \{\begin{matrix} \frac{K}{r} ({}, r - (a + ρ) + ((), \frac{βaη}{γ} - \frac{r}{K}) δ_{x}) > 0, if r < \frac{βaη}{γ} K, \\ \frac{K}{r} ({}, r - (a + ρ) + ((), \frac{βaη}{γ} - \frac{r}{K}) x^{0}) > 0, if r \geq \frac{βaη}{γ} K, \end{matrix} \\ {liminf}_{t \to + \infty} z (t) \geq δ_{z} : = \frac{aη}{γ} δ_{y} > 0, \end{matrix})

and there exists a unique infected equilibrium that is globally asymp...…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On global stability of an HIV pathogenesis model with cure rate

Muroya

Enatsu

2014

Math Methods in App Sciences

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In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Theorem A If R 0 >1, then there exists a unique infected equilibrium E * =( x * , y * , z * ) of system , which is globally asymptotically stable, if

d \leq r < d + \frac{2 rδ}{K}, that is, d \leq r < \frac{d}{1 - 2 δ / K},

where δ is a positive constant that does not depend on the initial conditions such that

{liminf}_{t \to + \infty} x (t) \geq δ, {liminf}_{t \to + \infty} y (t) \geq δ and {liminf}_{t \to + \infty} z (t) \geq δ .

We note that Liu et al ., show the existence of such δ > 0 on the permanence of system , while they did not refer the explicit expression for δ > 0.…”

Section: Introductionmentioning

confidence: 95%

“…Motivated by the results in , we are interested in the global dynamics of the following virus infection model with cure rate of infected CD4 + T cells in Liu et al ., .

({, \begin{matrix} \frac{dx (t)}{dt} = s - dx (t) - βx (t) z (t) + rx (t) (1 - \frac{x (t) + y (t)}{K}) + ρy (t), \\ \frac{dy (t)}{dt} = βx (t) z (t) + ry (t) (1 - \frac{x (t) + y (t)}{K}) - (a + ρ) y (t), \\ \frac{dz (t)}{dt} = aηy (t) - γz (t), \end{matrix})

where x ( t ), y ( t ), and z ( t ) represent the concentration of susceptible CD4 + T cells, infected CD4 + T cells, and free HIV virus particles in the blood at time t , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, system always has an uninfected equilibrium E 0 =( x 0 ,0,0), where x 0 is a positive root of

\frac{r}{K} {(x^{0})}^{2} - (r - d) x^{0} - s = 0,

that is, by d K ≥ s ,

x^{0} = x^{0} (r) = \frac{K}{2 r} ({}, (r - d) + \sqrt{{(r - d)}^{2} + \frac{4 rs}{K}}) \leq K .

Moreover, from , we have

\frac{r}{K} x^{0} - (r - d) - \frac{s}{x^{0}} = 0,

which is equivalent to

a + ρ - r ((), 1 - \frac{x^{0}}{K}) = \frac{s}{x^{0}} + (a - d) + ρ > 0 .

The basic reproduction number of system can be defined as

R_{0} \equiv \frac{βaη x^{0}}{γ (\frac{s}{x^{0}} + a - d + ρ)} .

By and , we note that R 0 >1 is equivalent to N > N c r i t in Liu et al . , Section 2] and also

ρ < r - a + ((), \frac{βaη}{γ} - \frac{r}{K}) x^{0} = d - a + ((), \frac{βaη}{γ} x^{0} - s)

…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We obtain the following practical result on Liu et al . , Theorem 5.1]. Theorem For R 0 >1, assume that

r > a + ρ - ((), \frac{βaη}{γ} - \frac{r}{K}) \frac{s}{\frac{s}{x^{0}} + β \frac{aη x^{0}}{γ}}, if r < \min ((), \frac{βaη}{γ} K, a + ρ) .

Then, system is permanent and

({, \begin{matrix} {liminf}_{t \to + \infty} x (t) \geq δ_{x} : = \frac{s}{\frac{s}{x^{0}} + β \frac{aη x^{0}}{γ}} > 0, \\ {liminf}_{t \to + \infty} y (t) \geq δ_{y} : = \{\begin{matrix} \frac{K}{r} ({}, r - (a + ρ) + ((), \frac{βaη}{γ} - \frac{r}{K}) δ_{x}) > 0, if r < \frac{βaη}{γ} K, \\ \frac{K}{r} ({}, r - (a + ρ) + ((), \frac{βaη}{γ} - \frac{r}{K}) x^{0}) > 0, if r \geq \frac{βaη}{γ} K, \end{matrix} \\ {liminf}_{t \to + \infty} z (t) \geq δ_{z} : = \frac{aη}{γ} δ_{y} > 0, \end{matrix})

and there exists a unique infected equilibrium that is globally asymp...…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On global stability of an HIV pathogenesis model with cure rate

Muroya

Enatsu

2014

Math Methods in App Sciences

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On HIV Model with Adaptive Immune Response, Two Saturated Rates and Therapy

Allali

Tabit

Harroudi

2017

Math. Model. Nat. Phenom.

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The mathematical model of the human immunodeficiency virus (HIV) pathogenesis with the adaptive immune response, two saturated rates and therapy is presented and studied in this work. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies; the two saturated rates describe the viral infection and the CTL proliferation. Two kinds of treatments are incorporated into the model; the objective of the first one is to reduce the number of infected cells, while the aim of the second one is to block the free viruses. The positivity and boundedness of solutions are established. The local stability of the disease free steady state and the infection steady states are studied. Numerical simulations are performed to show the behavior of solutions and the effectiveness of the incorporated therapy in controlling the HIV replication which can improves significantly the patient's life quality.

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