The present study investigates essential steps in build-up of models for description of the spread of infectious diseases. Combining these modules, a SEIRSD model will be developed, which can take into account a possible passive immunisation by vaccination as well as different durations of latent and incubation periods. Besides, infectious persons with and without symptoms can be distinguished. Due to the current world-wide SARS-CoV-2 pandemic (COVID-19 pandemic) models for description of the spread of infectious diseases and their application for forecasts have become into the focus of the scientific community as well as of broad public more than usual. Currently, many papers and studies have appeared and appear dealing with the virus SARS-CoV-2 and the COVID-19 disease caused by it. This occurs under medical, virological, economic, sociological and further aspects as well as under mathematical points of view. Concerning the last-mentioned point, the main focus lies on the application of existing models and their adaptation to data about the course of infection available at the current time. Clearly, the aim is to predict the possible further development, for instance in Germany. It is of particular interest to investigate how will be the influence of political and administrative measures like contact restrictions, closing or rather re-opening of schools, restaurants, hotels etc. on the course of infection. The steps considered here for building up suitable models are well-known for long time. However, understandably they will not be dealt with in an extended way in current application-oriented works. Therefore, it is the aim of this study to present some existing steps of modelling without any pretension of completeness. Thus, on the one hand we give assistance and, on the other hand, we develop a model capable to take already known properties of COVID-19 as well as a later possible passive immunisation by vaccination and a possible loss of immunity of recovered persons into account.

Growth and remodeling of soft tissues is a dynamic process and several theoretical frameworks have been developed to analyze the time-dependent, mechanobiological and/or biomechanical responses of these tissues to changes in external loads. Importantly, general processes can often be conveniently separated into truly non-steady contributions and steady-state ones. Depending on characteristic times over which the external loads are applied, time-dependent models can sometimes be specialized to respective time-independent formulations that simplify the mathematical treatment without compromising the goodness of the particularized solutions. Very few studies have analyzed the long-term, steady-state responses of soft tissue growth and remodeling following a direct approach. Here, we derive a mechanobiologically equilibrated formulation that arises from a general constrained mixture model. We see that integral-type evolution equations that characterize these general models can be written in terms of an equivalent set of time-independent, nonlinear algebraic equations that can be solved efficiently to yield long-term outcomes of growth and remodeling processes in response to sustained external stimuli. We discuss the mathematical conditions, in terms of orders of magnitude, that yield the particularized equations and illustrate results numerically for general arterial mechano-adaptations.

We have developed a micromechanics based model for chemically active saturated fibrous media that incorporates fiber network microstructure, chemical potential driven fluid flow, and micro-poromechanics. The stress-strain relationship of the dry fibrous media is first obtained by considering the fiber behavior. The constitutive relationships applicable to saturated media are then derived in the poromechanics framework using Hill's volume averaging. The advantage of this approach is that the resultant continuum model accounts for the discrete nature of the individual fibers while retaining a form suitable for porous materials. As a result, the model is able to predict the influence of micro-scale phenomena, such as the fiber pre-strain caused by osmotic effects and evolution of fiber network structure with loading, on the overall behavior and in particular, on the poromechanics parameters. Additionally, the model can describe fluid-flow related rate-dependent behavior under confined and unconfined conditions and varying chemical environments. The significance of the approach is demonstrated by simulating unconfined drained monotonic uniaxial compression under different surrounding fluid bath molarity, and fluid-flow related creep and relaxation at different loading-levels and different surrounding fluid bath molarity. The model predictions conform to the experimental observations for saturated soft fibrous materials. The method can potentially be extended to other porous materials such as bone, clays, foams and concrete.