Heumann-Hötzel model for aging revisited

Brain aging and garbage cleaning : Modelling the role of sleep, glymphatic system, and microglia senescence in the propagation of inflammaging

Brain aging is a complex process involving many functions of our body and described by the interplay of a sleep pattern and changes in the metabolic waste concentration regulated by the microglial function and the glymphatic system. We review the existing modelling approaches to this topic and derive a novel mathematical model to describe the crosstalk between these components within the conceptual framework of inflammaging. Analysis of the model gives insight into the dynamics of garbage concentration and linked microglial senescence process resulting from a normal or disrupted sleep pattern, hence, explaining an underlying mechanism behind healthy or unhealthy brain aging. The model incorporates accumulation and elimination of garbage, induction of glial activation by garbage, and glial senescence by over-activation, as well as the production of pro-inflammatory molecules by their senescence-associated secretory phenotype (SASP). Assuming that insufficient sleep leads to the increase of garbage concentration and promotes senescence, the model predicts that if the accumulation of senescent glia overcomes an inflammaging threshold, further progression of senescence becomes unstoppable even if a normal sleep pattern is restored. Inverting this process by "rejuvenating the brain" is only possible via a reset of concentration of senescent glia below this threshold. Our model approach enables analysis of space-time dynamics of senescence, and in this way, we show that heterogeneous patterns of inflammation will accelerate the propagation of senescence profile through a network, confirming a negative effect of heterogeneity.

Analytical properties of a three-compartmental dynamical demographic model

The three-compartmental demographic model by Korotaeyv-Malkov-Khaltourina, connecting population size, economic surplus, and education level, is considered from the point of view of dynamical systems theory. It is shown that there exist two integrals of motion, which enables the system to be reduced to one nonlinear ordinary differential equation. The study of its structure provides analytical criteria for the dominance ranges of the dynamics of Malthus and Kremer. Additionally, the particular ranges of parameters enable the derived general ordinary differential equations to be reduced to the models of Gompertz and Thoularis-Wallace.