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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    • "Remark 7.1 Recently, this kind of Lyapunov function has been extensively used for the analysis of differential equations with discrete or distributed delay models (see, e.g., Korobeinikov, 2004; McCluskey, 2009, 2010a,b,c; Wang et al., 2011), virus infection models (see, e.g., Huang et al., 2010; Li & Shu, 2010; Liu & Wang, 2010) and human infection models (see, e.g. Huang et al., 2010; Magal et al., 2010; McCluskey, 2012). "
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    ABSTRACT: In this paper, we study the global dynamics of an susceptible, vaccinated, infectious and recovered epidemiological model with infection-age structure. Biologically, we assume that effective contacts between vaccinated individuals and infectious individuals are less than that between susceptible individuals and infectious individuals. Using Lyapunov functions, we show that the global stability of each equilibrium is completely determined by the basic reproduction number R_0: if R_0≤1, then the diseasefree equilibrium is globally asymptotically stable; while if R_0>1, then there exists a unique endemic equilibrium which is globally asymptotically stable.
    Full-text · Article · Jan 2016 · IMA Journal of Applied Mathematics
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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]. "
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    ABSTRACT: In this paper, a class of delay differential equations model of HIV infection dynamics with nonlinear transmissions and apoptosis induced by infected cells is proposed, and then the global properties of the model are considered. It shows that the infection-free equilibrium of the model is globally asymptotically stable if the basic reproduction number R-0 < 1, and globally attractive if R-0 = 1. The positive equilibrium of the model is locally asymptotically stable if R-0 > 1. Furthermore, it also shows that the model is permanent, and some explicit expressions for the eventual lower bounds of positive solutions of the model are given.
    Full-text · Article · Nov 2015 · Discrete and Continuous Dynamical Systems - Series B
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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.
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