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**Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay.**
*(English)*
Zbl 1203.92051

Summary: We have considered a nonautonomous stage-structured HIV/AIDS epidemic model having two stages of the period of infection according to the developing progress of infection before AIDS defined in, with varying total population size and distributed time delay to become infectious. The infected persons in the different stages have different abilities of transmitting the disease. By all kinds of treatment methods, some people with the symptomatic stages can be transformed into asymptomatic stages. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using the inequality analytical technique. We have obtained an explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values \(R_{0}\) and \(R^{*}\) and further obtained that the disease will be permanent when \(R_{0}>1\) and the disease will be going extinct when \(R^{*}<1\). By the Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Also, we have observed that the time delay decreases the lower bounds of the infective and full-blown AIDS group. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness.

### MSC:

92C60 | Medical epidemiology |

34K20 | Stability theory of functional-differential equations |

34K25 | Asymptotic theory of functional-differential equations |

92D30 | Epidemiology |

65C20 | Probabilistic models, generic numerical methods in probability and statistics |

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\textit{G. P. Samanta}, Nonlinear Anal., Real World Appl. 12, No. 2, 1163--1177 (2011; Zbl 1203.92051)

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