Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models proportional to either geo-max-stable distributions (log-logistic and backward log-logistic) or to other max-stable distributions (Fréchet or max-Weibull). We show that the former arise when in the hyper-logistic Blumberg equation, connected to the Beta $(p,q)$ function, we use fractional exponents $p-1=1\mp 1/\alpha$ and $q-1=1\pm 1/\alpha$, and the latter when in the hyper-Gompertz-Turner equation, the exponents of the logarithmic factor are real and eventually fractional. The use of a BetaBoop function establishes interesting connections to Probability Theory, Riemann–Liouville’s fractional integrals, higher-order monotonicity and convexity and generalized unimodality, and the logistic map paradigm inspires the investigation of the dynamics of the hyper-logistic and hyper-Gompertz maps.

Modeling the vulnerabilities lifecycle and exploitation frequency are at the core of security of networks evaluation. Pareto, Weibull, and log-normal models have been widely used to model the exploit and patch availability dates, the time to compromise a system, the time between compromises, and the exploitation volumes. Random samples (systematic and simple random sampling) of the time from publication to update of cybervulnerabilities disclosed in 2021 and in 2022 are analyzed to evaluate the goodness-of-fit of the traditional Pareto and log-normal laws. As censoring and thinning almost surely occur, other heavy-tailed distributions in the domain of attraction of extreme value or geo-extreme value laws are investigated as suitable alternatives. Goodness-of-fit tests, the Akaike information criterion (AIC), and the Vuong test, support the statistical choice of log-logistic, a geo-max stable law in the domain of attraction of the Fréchet model of maxima, with hyperexponential and general extreme value fittings as runners-up. Evidence that the data come from a mixture of differently stretched populations affects vulnerabilities scoring systems, specifically the common vulnerabilities scoring system (CVSS).

Machado-Joseph disease (MJD) is a dominant neurodegenerative disease caused by an expanded CAG repeat in the ATXN3 gene encoding the ataxin-3 protein. Several cellular processes, including transcription and apoptosis, are disrupted in MJD. To gain further insights into the extent of dysregulation of mitochondrial apoptosis in MJD and to evaluate if expression alterations of specific apoptosis genes/proteins can be used as transcriptional biomarkers of disease, the expression levels of BCL2, BAX and TP53 and the BCL2/BAX ratio (an indicator of susceptibility to apoptosis) were assessed in blood and post-mortem brain samples from MJD subjects and MJD transgenic mice and controls. While patients show reduced levels of blood BCL2 transcripts, this measurement displays low accuracy to discriminate patients from matched controls. However, increased levels of blood BAX transcripts and decreased BCL2/BAX ratio are associated with earlier onset of disease, indicating a possible association with MJD pathogenesis. Post-mortem MJD brains show increased BCL2/BAX transcript ratio in the dentate cerebellar nucleus (DCN) and increased BCL2/BAX insoluble protein ratio in the DCN and pons, suggesting that in these regions, severely affected by degeneration in MJD, cells show signs of apoptosis resistance. Interestingly, a follow-up study of 18 patients further shows that blood BCL2 and TP53 transcript levels increase over time in MJD patients. Furthermore, while the similar levels of blood BCL2, BAX, and TP53 transcripts observed in preclinical subjects and controls is mimicked by pre-symptomatic MJD mice, the expression profile of these genes in patient brains is partially replicated by symptomatic MJD mice. Globally, our findings indicate that there is tissue-specific vulnerability to apoptosis in MJD subjects and that this tissue-dependent behavior is partially replicated in a MJD mouse model.