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This paper presents current knowledge about the structure and function of the lymphatic system. Mathematical models of lymph flow in the single lymphangion, the series of lymphangions, the lymph nodes, and the whole lymphatic system are considered. The main results and further perspectives are discussed.

A model of lymph flow in the human lymphatic system in the quasi-one-dimensional approach has been created and investigated under different conditions. The model includes an implementation of contractions and valve influence on lymph flow. We consider contractions of lymphatic vessels and their influence on resulting flow in the whole network of lymphatic vessels and lymph nodes. We have investigated flow with zero pressure gradient and have found parameters, which influence the efficiency of contractions most significantly.

The model of lymph flow in the human lymphatic system in the quasi-one-dimensional approach is considered in the paper. Calculations with natural gravity force influence are presented. Calculations show that in some sets of parameters of lymphatic vessels the physiological flow can be reached in the model with zero global pressure gradient influence only by the “muscle” pump.

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Severe acute respiratory syndrome of coronavirus 2 (SARS-CoV-2) is a respiratory virus that disrupts the functioning of several organ systems. The cardiovascular system represents one of the systems targeted by the novel coronavirus disease (COVID-19). Indeed, a hypercoagulable state was observed in some critically ill COVID-19 patients. The timely prediction of thrombosis risk in COVID-19 patients would help prevent the incidence of thromboembolic events and reduce the disease burden. This work proposes a methodology that identifies COVID-19 patients with a high thromboembolism risk using computational modelling and machine learning. We begin by studying the dynamics of thrombus formation in COVID-19 patients by using a mathematical model fitted to the experimental findings of in vivo clot growth. We use numerical simulations to quantify the upregulation in the size of the formed thrombi in COVID-19 patients. Next, we show that COVID-19 upregulates the peak concentration of thrombin generation (TG) and its endogenous thrombin potential. Finally, we use a simplified 1D version of the clot growth model to generate a dataset containing the hemostatic responses of virtual COVID-19 patients and healthy subjects. We use this dataset to train machine learning algorithms that can be readily deployed to predict the risk of thrombosis in COVID-19 patients.

Replication of viruses in living tissues and cell cultures is a “number game” involving complex biological processes (cell infection, virus replication inside infected cell, cell death, viral degradation) as well as transport processes limiting virus spatial propagation. In epithelial tissues and immovable cell cultures, viral particles are basically transported via Brownian diffusion. Highly non-linear kinetics of viral replication combined with diffusion limitation lead to spatial propagation of infection as a moving front switching from zero to high local viral concentration, the behavior typical of spatially distributed excitable media. We propose a mathematical model of viral infection propagation in cell cultures and tissues under the diffusion limitation. The model is based on the reaction–diffusion equations describing the concentration of uninfected cells, exposed cells (infected but still not shedding the virus), virus-shedding cells, and free virus. We obtain the expressions for the viral replication number, which determines the condition for spatial infection progression, and for the final concentration of uninfected cells. We determine analytically the speed of spatial infection propagation and validate it numerically. We calibrate the model to recent experimental data on SARS-CoV-2 Delta and Omicron variant replication in human nasal epithelial cells. In the case of competition of two virus variants in the same cell culture, the variant with larger individual spreading speed wins the competition and eliminates another one. These results give new insights concerning the emergence of new variants and their spread in the population.