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.


Available from: Gang Huang, Apr 25, 2014
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    Rotich K Titus, Lagat C Robert
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    ABSTRACT: The current use of Highly Active Anti-Retroviral Therapy (HAART) strategy to control Human Immunodeficiency Virus (HIV) and Acquired Immune Deficiency Syndrome (AIDS) is inefficient in eradicating HIV/AIDS due to inadequate understanding of the dynamics relating to interaction between the immune system components and HIV. As a result, a pool of potential transmitters is continuously created and thus HIV has remained a pandemic. In this paper, we formulate a mathematical model using differential equations to study the effects of time lag due to cellular latency and pharmacological delays and chemotherapy on the control strategy of AIDS epidemic. Equilibrium points of the model are computed and used to determine the reproductive ratio. This important threshold parameter is then used to determine the critical bounds of time lag and therapeutic window that is, the bounds; above Minimum Effect Concentration (MEC) and below Minimum Toxic Concentration (MTC), where drug plasma concentration should lie for effective maintenance of low levels of viral load and reduction of drug toxicity. The mathematical model gives qualitative understanding of HIV prognostic information which is a means of rejuvenating the existing Antiretroviral drugs (ARV's). Numerical simulations show that a stable and persistent endemic equilibrium state of low viral load is achieved when these thresholds and are satisfied. This persistent equilibrium state will lead to eventual eradication of HIV/AIDS. This paper contributes the first logical analysis on the effect of intracellular delay of HIV viral infection on drug efficacy. The study originates new formula for finding the optimal therapeutic window of HAART necessary to control the reproductive ratio of HIV to less than one and evade the risk of drug toxicity.
  • SIAM Journal on Applied Dynamical Systems 01/2015; 14(1):1-24. DOI:10.1137/140971683 · 1.25 Impact Factor
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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.
    03/2015, Degree: Master of Science, Supervisor: Jinliang Wang