Article

Lyapunov Functionals for Delay Differential Equations Model of Viral Infections.

SIAM Journal on Applied Mathematics (Impact Factor: 1.41). 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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    Rotich K Titus, Lagat C Robert
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    ABSTRACT: Dynamics of infectious diseases is one of the most important theoretical methods in studying infectious disease models. The age-structured epidemic model and it’s global properties is an important subject in dynamic system’s research. This thesis mainly utilize the framework of global Lyapunov functionals constructed to obtain stability of infectious diseases models with age-structure. This research is important and practical meaningful to estimate the development trend of infectious diseases. In this paper, two classes of infectious disease models with age-structure will be discussed: one is HIV viral infection model. The global stability of equilibria is determined by a sharp threshold parameter, called basic reproduction number. The other is epidemic model. We focus on how to construct a suitable Lyapunov function to solve the problem of global stability of equilibria. On the other hand, one framework of Lyapunov functional method to solve the problem of the stability of the age-structured model, providing a series of simple and effective determined theorems, can play a important role in preventing and control of disease spread, which may provide decision-making basis and theoretical reference.
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