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

ABSTRACT 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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    • "(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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    • "Such condition is satisfied by several well known infection rates. The DDE model given in [10] without delay is studied by Korobeinikov in [11]. Motivated by the above comments, in the present paper, we proposed a model more realistic by using an infection rate that saturates and a discrete intracellular delay. "
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    Applied Mathematics and Computation 05/2015; 259:293–312. DOI:10.1016/j.amc.2015.02.053 · 1.55 Impact Factor
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    • "Actually, the nonlinear incidence rate Fðx; vÞ and the virus producing rate G i ðy i Þ do not destabilize two equilibria. Comparing the model with discrete delays studied in [11], our results show that distributed delay in initial infection and the distributed delays in infected host have no impact for the global stability of equilibria. By using the method of Lyapunov functionals, LaSalle's invariance principle, a general incidence rate and a general removal rate, we can resolve global dynamics for a class of in host viral infection model. "
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    ABSTRACT: We investigate an in-host model with general incidence and removal rate, as well as distributed delays in virus infections and in productions. By employing Lyapunov functionals and LaSalle’s invariance principle, we define and prove the basic reproductive number R_0 as a threshold quantity for stability of equilibria. It is shown that if R_0>1, then the infected equilibrium is globally asymptotically stable, while if R_0<=1, then the infection free equilibrium is globally asymptotically stable under some reasonable assumptions. Moreover, n+1 distributed delays describe (i) the time between viral entry and the transcription of viral RNA, (ii) the n-1-stage time needed for activated infected cells between viral RNA transcription and viral release, and (iii) the time necessary for the newly produced viruses to be infectious (maturation), respectively. The model can describe the viral infection dynamics of many viruses such as HIV-1, HCV and HBV.
    Communications in Nonlinear Science and Numerical Simulation 01/2015; 20(1):263–272. DOI:10.1016/j.cnsns.2014.04.027 · 2.87 Impact Factor
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