Lyapunov Functionals for Delay Differential Equations Model of Viral Infections

SIAM Journal on Applied Mathematics (Impact Factor: 1.43). 01/2010; 70(7):2693-2708. DOI: 10.1137/090780821
Source: DBLP


We study global properties of a class of delay differential equations model for virus infections with nonlinear transmissions. Compared with the typical virus infection dynamical model, this model has two important and novel features. To give a more complex and general infection process, a general nonlinear contact rate between target cells and viruses and the removal rate of infected cells are considered, and two constant delays are incorporated into the model, which describe (i) the time needed for a newly infected cell to start producing viruses and (ii) the time needed for a newly produced virus to become infectious (mature), respectively. By the Lyapunov direct method and using the technology of constructing Lyapunov functionals, we establish global asymptotic stability of the infection-free equilibrium and the infected equilibrium. We also discuss the effects of two delays on global dynamical properties by comparing the results with the stability conditions for the model without delays. Further, we generalize this type of Lyapunov functional to the model described by n-dimensional delay differential equations.

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Available from: Gang Huang, Apr 25, 2014
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    • "bilinear infection rate is replaced with the saturation infection rate [53], Holling type II functional response [25], Beddington-DeAngelis functional response [23], Crowley-Martin functional response [34], or more general nonlinear infection rate [24] [31] [37] [46]. "

    Discrete and Continuous Dynamical Systems - Series B 11/2015; 21(1):103-119. DOI:10.3934/dcdsb.2016.21.103 · 0.77 Impact Factor
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    • "Recently, a great deal of attention has been paid by many researchers on the Beddington-DeAngelis type response function, which has been used in the virus dynamics model (Wang et al., 2010; Huang et al., 2011; Elaiw et al., 2012), and in ecological model (Hsu et al., 2013) and their stability behaviours have been studied. Moreover, a general form of incidence rate function f (x, v) has been considered (Huang et al., 2010). "
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    ABSTRACT: 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 (R0), 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 Ro (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 Ro. 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 R0 ≤ 1, which biologically means that infection dies out. Moreover, sufficient condition is derived for global stability of chronic infection equilibrium if R0 > 1 , which biologically means that infection becomes chronic. Numerical simulations are given to illustrate the theoretical results.
    Journal of the National Science Foundation of Sri Lanka 10/2015; 43(3):235. DOI:10.4038/jnsfsr.v43i3.7953 · 0.20 Impact Factor
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    • "(t) = ln n n * + n * n − 1, and the Volterra–type Lyapunov functional [7] [8] [12] [16] [17] [27] [28] W + (t) = τ 0 n(t − ω) n * − 1 − ln n(t − ω) n * dω. "
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    ABSTRACT: This paper addresses the global stability of a two-species Lotka-Volterra cooperative system with four discrete delays via the method of Lyapunov functionals. We apply our Lyapunov functional techniques to prove stability of Lotka-Volterra models of mutualism with two delays or without delays. We note that, if all the delayed feedbacks are positive then the stability properties are independent of the values of the delays. We compare our results with known results. Keywords: Lotka-Volterra cooperative system, discrete delays, delayed positive feedback, Lyapunov functional, global stability.
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