Lyapunov Functionals for Delay Differential Equations Model of Viral Infections.

SIAM Journal on Applied Mathematics (Impact Factor: 1.58). 01/2010; 70:2693-2708. DOI: 10.1137/090780821
Source: DBLP
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    ABSTRACT: The use of combination antiretroviral therapy has proven remarkably effective in controlling HIV disease progression and prolonging survival. However, the emergence of drug resistance can occur. It is necessary that we gain a greater understanding of the evolution of drug resistance. Here, we consider an HIV viral dynamical model with general form of target cell density, drug resistance and intracellular delay incorporating antiretroviral therapy. The model includes two strains: wild-type and drug-resistant. The basic reproductive ratio for each strain is obtained for the existence of steady states. Qualitative analysis of the model such as the well-posedness of the solutions and the equilibrium stability is provided. Global asymptotic stability of the disease-free and drug-resistant steady states is shown by constructing Lyapunov functions. Furthermore, sufficient conditions related to the properties of the target cell density are obtained for the local asymptotic stability of the positive steady state. Numerical simulations are conducted to study the impact of target cell density and intracellular delay focusing on the stability of the positive steady state. The occurrence of Hopf bifurcation of periodic solutions is shown to depend on the target cell density.
    Journal of Mathematical Analysis and Applications. 01/2014; 414(2):514–531.
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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. · 2.77 Impact Factor
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    ABSTRACT: In this paper, a class of three delayed viral dynamics models with immune response and saturation infection rate are proposed and studied. By constructing suitable Lyapunov functionals, we derive the basic reproduction number R_0 and the corresponding immune response reproduction numbers for the viral infection models, and establish that the global dynamics are completely determined by the values of the related basic reproduction number and immune response reproduction numbers.


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Jun 3, 2014