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Abstract: The purpose of this note is twofold: First we highlight the importance of an implicit assumption in [Murat Adıvar, Youssef N. Raffoul, Stability and periodicity in dynamic delay equations, Computers and Mathematics with Applications 58 (2009) 264–272]. Second we emphasize one consequence of the bijectivity assumption which enables ruling out the commutativity condition on the delay function. [Copyright &y& Elsevier]

Abstract: A sex-structured mathematical model for heterosexual transmission of HIV/AIDS with explicit incubation period is presented as a system of discrete delay differential equations. The epidemic threshold and equilibria for the model are determined and stabilities are examined. The disease-free equilibrium is shown to be locally and globally stable when the basic reproductive number is less than unity. We use the Lyapunov functional approach to show that the endemic equilibrium is locally asymptotically stable. Further comprehensive qualitative analysis of the model including persistence and permanence are investigated. [Copyright &y& Elsevier]

Abstract: Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function. [Copyright &y& Elsevier]

Abstract: In this work, we investigate the stability of a class of impulsive functional differential equations. Some general stability theorems are obtained. Our results can be applied to finite delay impulsive systems or infinite delay impulsive systems or impulsive systems involving both finite and infinite delays, in a unified way. Examples are also given to illustrate that applying our theorems yields better conclusions than the results in the literature. [Copyright &y& Elsevier]