Global properties of basic virus dynamics models

Korobeinikov, Andrei

doi:10.1016/j.bulm.2004.02.001

Cited by 410 publications

(269 citation statements)

References 6 publications

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“…Let x 0 , y 0 , z 0 , w 0 > 0 and c = max{Φ(x 0 , y 0 , z 0 , w 0 ), Φ(p)}. As above we see that N = {(x, y, z, w) ∈ (0, ∞) 4 : Φ(x, y, z, w) ≤ c} is closed, invariant, and contains exactly the equilibriump. By La Salle's principle, the solution converges to p * as t → ∞.…”

Section: Proofs Of Theorem 21 and 22mentioning

confidence: 70%

“…As general references for the theory of such problems we recommend the books [3], [5], [7]. The phase space is taken as X = C(G) 4 and the dependent variable is…”

Section: Models With Diffusionmentioning

confidence: 99%

“…In particular, the Lyapunov functions from [9] are available also for the study of the asymptotic behaviour of the basic model of virus dynamics, see also [2], [4]. Thus there arises the natural question whether it is possible to generalize this approach to the case of models taking into account immune response, cf.…”

Section: Introductionmentioning

confidence: 99%

“…So Φ * is a Lyapunov function for (4..2) on (0, ∞) 4 . IfΦ * (x, y, z, w) = 0, then y = y * and w = 0.…”

Section: Proofs Of Theorem 21 and 22mentioning

confidence: 99%

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Global Asymptotic Stability of Equilibria in Models for Virus Dynamics

Prüß

Zacher

Schnaubelt

2008

Math. Model. Nat. Phenom.

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Abstract. In this paper several models in virus dynamics with and without immune response are discussed concerning asymptotic behaviour. The case of immobile cells but diffusing viruses and T-cells is included. It is shown that, depending on the value of the basic reproductive number R 0 of the virus, the corresponding equilibrium is globally asymptotically stable. If R 0 < 1 then the virus-free equilibrium has this property, and in case R 0 > 1 there is a unique disease equilibrium which takes over this property..

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Section: Proofs Of Theorem 21 and 22mentioning

confidence: 70%

“…As general references for the theory of such problems we recommend the books [3], [5], [7]. The phase space is taken as X = C(G) 4 and the dependent variable is…”

Section: Models With Diffusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…So Φ * is a Lyapunov function for (4..2) on (0, ∞) 4 . IfΦ * (x, y, z, w) = 0, then y = y * and w = 0.…”

Section: Proofs Of Theorem 21 and 22mentioning

confidence: 99%

See 2 more Smart Citations

Global Asymptotic Stability of Equilibria in Models for Virus Dynamics

Prüß

Zacher

Schnaubelt

2008

Math. Model. Nat. Phenom.

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“…The pioneering work is done by Nowak and Bangham [16], in which the relation between antiviral immune responses, virus load, and virus diversity are theoretically understood by formulating systems with ordinary differential equations describing the population dynamics of immune responses to virus load. Motivated by their work, many studies have subsequently focused on the global stability of equilibria in models for viral dynamics which play a crucial role in clarifying and evaluating treatment strategies for infections and establishing thresholds for treatment rates [11,12,14,15,20,24,29]. In this paper, we focus our attention on the global stability of steady states for these models, because this should enhance our understanding of virus dynamics, which gives us a detailed information and various insights on whether a disease will die out or not, and the mechanisms of specific immune responses.…”

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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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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The artificial neural network approach for the transmission of malicious codes in wireless sensor networks with Caputo derivative

Khan,

ur Rahman,

Arfan

et al. 2024

Int J Numerical Modelling

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The current manuscript investigates a six compartmental mathematical model for malicious Codes in Wireless Sensor Network is consider for investigation under the fractional operator of Caputo along with their numerical scheme. The six agent nodes of the network sensors are transferable like in infection with in their community of different nodes. With the help of fixed point theory the presentation of existence and uniqueness of solution of the said model are also given. The scheme of numerical solution under fractional format is developed with the choice of fractional orders which increasing the degree of freedom for such type of network analysis. The numerical simulation of all the six agents are given on different fractional orders along with sensitivity of the fractional orders and some used parameters. The new analysis artificial neural network (ANN) method has been utilized for the considered model and compared with Adams–Bashforth (AB) method. We divided the data set into three categories training, testing and validation with ANN method and the analysis is presented in this work.

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Global properties of basic virus dynamics models

Abstract: Abstract. Lyapunov functions for basic virus dynamics models are introduced, and global stability of the models are thereby established.

Cited by 410 publications

References 6 publications

Global Asymptotic Stability of Equilibria in Models for Virus Dynamics

Global Asymptotic Stability of Equilibria in Models for Virus Dynamics

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

The artificial neural network approach for the transmission of malicious codes in wireless sensor networks with Caputo derivative

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