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Existing HIV infection dynamics models are based on the classical target cell‐infected cell‐free virus compartment model. In fact, in order to facilitate modeling and dynamical analysis, these models assume that the probability of infected cells staying in different infection stages obeys an exponential distribution. However, this does not accurately describe the process of HIV infection from target cells to budding of virus, that is, fusion‐reverse transcription‐reproductivity formation‐budding process. Therefore, to expand the idea of HIV modeling, in this paper, we first establish the HIV integral‐differential equation model that obeys the general distribution, and then degenerate the model into a subclass ODEs model under the assumption of gamma distribution. Finally, an equivalent age structure model is given, and numerical simulations are conducted to describe the integral‐differential equations model. It is worth mentioning that ODEs models with subclasses can provide good numerical implementations for age structure as well as other PDEs HIV models. This is another important purpose of our work. [ABSTRACT FROM AUTHOR]

In recent years various probability models have been proposed for describing lifetime data. Increasing model flexibility is often sought as a means to better describe asymmetric and heavy tail distributions. Such extensions were pioneered by the beta‐G family. However, efficient (GoF) measures for the beta‐G distributions are sought. In this paper, we combine probability weighted moments (PWMs) and the Mellin transform (MT) in order to furnish new qualitative and quantitative GoF tools for model selection within the beta‐G class. We derive PWMs for the Fréchet and Kumaraswamy distributions, and we provide expressions for the MT, and for the log‐cumulants (LC) of the beta‐Weibull, beta‐Fréchet, beta‐Kumaraswamy, and beta‐log‐logistic distributions. Subsequently, we construct LC diagrams and, based on the Hotelling's T2 statistic, we derive confidence ellipses for the LCs. Finally, the proposed GoF measures are applied on five real data sets in order to demonstrate their applicability. [ABSTRACT FROM AUTHOR]