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Biological materials and systems are hierarchically organized.The main motivation for hierarchical biomechanics is that the wide variability of mechanical properties encountered at the macroscopic scale may be traced back to just a few universal. i.e. tissue-invariant, mechanical properties of elementary components at a sufficiently small scale (such as collagen, elastin, and water in case of soft tissues; complemented by hydroxyapatite in case of hard tissues), and to the nano and microstructures which the latter build up. This challenging task requires a physically rigorous and mathematically sound basis, as provided by Finite Element and Fast Fourier Transform methods, as well as by continuum micromechanics resting on (semi-)analytical solutions for Eshelby-type matrix-inclusion problems. Corresponding numerical and analytical mathematical models have undergone diligent experimental validation, by means of data stemming from a variety of biophysical, biochemical, and biomechanical testing methods, such as light and electron microscopy, ultrasonic testing and scanning acoustic microscopy, as well as physico-chemical tests associated with dehydration, demineralization, decollagenization, ashing, and weighing in air and fluid. While elastic scale transition and homogenization methods have attained a high maturity level, the hierarchical nature of dissipative (i.e. viscous or strength) properties is still a vibrant field of research. This applies even more to hierarchical approaches elucidating the interface between biological cells and extracellular matrices, and to the highly undiscovered mechanics unfolding within biological cells.

Countless research contributions reflect two major concepts for modeling the spread of the COVID-19 pandemic: (i) ordinary differential equations for population compartments, such as infected or deceased persons (these approaches often exhibit limited predictive capabilities); and (ii) rules applied to digitally realized agents in the populations (these approaches often lack reliable input data and may become computationally overly expensive). As a remedy, we here introduce and discuss convolutional integrodifferential equations adapted from Boltzmann's hereditary mechanics, so as to predict COVID-19 fatality trends from the evolutions of newly infected persons. Replacing the classical statistical reasoning by deliberations arising from the notion of “virus loads” and the corresponding compliance of the infected population to these loads, model errors with respect to data recorded in 102 countries, territories, or US states can be drastically reduced, namely, up to 98% when compared to the traditional kinetics equation of Kermack and McKendrick. The coefficients of determination between model predictions and recorded data range from 94% to 100%, a precision hitherto unachieved in equation-based epidemic modeling.

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