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Streszczenie:
Non-uniform distributions of various biological factors can be essential for tissue growth control, morphogenesis or tumor growth. The first model describing the emergence of such distributions was suggested by A. Turing for the explanation of cell differentiation in a growing embryo. In this model, diffusion-driven instability of the homogeneous in space solution appears due to the interaction of two or more morphogens described by a reaction–diffusion system of equations. In this work we suggest another mechanism of the emergence of spatial distributions in biological tissues based on local cell communication and global inhibition, and described by a nonlocal reaction–diffusion equation. Instability of the homogeneous in space solution leads to the emergence of stationary pulses and not of periodic solutions as in the case of Turing instability.

Słowa kluczowe:
Nonlocal reaction–diffusion equation, Instability, Stationary pulses, Biological models

Streszczenie:
We consider the problem of existence and properties of pulse solutions to a system of equations modeling fast calcium waves in long cells. These waves have the speed up to 1000 μm/s. They propagate via the inflow of calcium ions from the extracellular space through the mechanically stimulated membrane channels. The channels open due to mechano-chemical interaction, in which stretching of the cell's membrane at a point opens the calcium channels at neighboring points due to the forces exerted by the actomyosin network. The existence of homoclinic solutions is based on the celebrated exchange lemma, which cannot be applied straightforwardly due to some specificities of the model equations.

Słowa kluczowe:
calcium signaling, exchange lemma, homoclinic traveling waves

Streszczenie:
Viral infection in cell culture and tissue is modeled with delay reaction-diffusion
equations. It is shown that progression of viral infection can be characterized by
the viral replication number, time-dependent viral load, and the speed of infec-
tion spreading. These three characteristics are determined through the original
model parameters including the rates of cell infection and of virus production in
the infected cells. The clinical manifestations of viral infection, depending on tis-
sue damage, correlate with the speed of infection spreading, while the infectivity
of a respiratory infection depends on the viral load in the upper respiratory tract.
Parameter determination from the experiments on Delta and Omicron variants
allows the estimation of the infection spreading speed and viral load. Different
variants of the SARS-CoV-2 infection are compared confirming that Omicron
is more infectious and has less severe symptoms than Delta variant. Within the
same variant, spreading speed (symptoms) correlates with viral load allowing
prognosis of disease progression.

Słowa kluczowe:
reaction-diffusion equations,spreading speed,SARS-CoV-2 variants,viral infection,viral load

Streszczenie:
Persistent epidemic can lead to the emergence of new virus strains due to virus mutations. This work is devoted to the SIR model with strain-dependence of infected individuals due to virus mutations and a continuous strain variable. Characterization of epidemic progression is obtained for a strain-dependent infectivity function in numerical simulations and with some analytical estimates. Taking into account limited infection-induced immunity, transition from recovered to susceptible compartment is considered. Different scenarios of epidemic progression are identified. In the case of a monotonically growing transmission rate as a function of strain, only one epidemic outbreak is observed. However, if the transmission rate is a non-monotonic function, then multiple outbreaks can occur.

Słowa kluczowe:
epidemic model, strain-dependent transmission rate, reinfection

Streszczenie:
A new model of viral infection spreading in cell cultures is proposed taking into account virus mutation. This model represents a reaction-diffusion system of equations with time delay for the concentrations of uninfected cells, infected cells and viral load. Infection progression is characterized by the virus replication number Rv, which determines the total viral load. Analytical formulas for the speed of propagation and for the viral load are obtained and confirmed by numerical simulations. It is shown that virus mutation leads to the emergence of a new virus variant. Conditions of the coexistence of the two variants or competitive exclusion of one of them are found, and different stages of infection progression are identified.

Słowa kluczowe:
viral infection, mutation, cell culture, reaction-diffusion equations, time delay

Streszczenie:
In December 2019, the first case of infection with a new virus COVID-19 (SARS-CoV-2), named coronavirus, was reported in the city of Wuhan, China. At that time, almost nobody paid any attention to it. The new pathogen, however, fast proved to be extremely infectious and dangerous, resulting in about 3–5% mortality. Over the few months that followed, coronavirus has spread over entire world. At the end of March, the total number of infections is fast approaching the psychological threshold of one million, resulting so far in tens of thousands of deaths. Due to the high number of lives already lost and the virus high potential for further spread, and due to its huge overall impact on the economies and societies, it is widely admitted that coronavirus poses the biggest challenge to the humanity after the second World war. The COVID-19 epidemic is provoking numerous questions at all levels. It also shows that modern society is extremely vulnerable and unprepared to such events. A wide scientific and public discussion becomes urgent. Some possible directions of this discussion are suggested in this article.

Słowa kluczowe:
COVID-19, epidemic progression, mathematical models, crisis management, open questions

Streszczenie:
Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction–diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction–diffusion wave. The theoretical results are confirmed by the results of numerical simulations.

Słowa kluczowe:
Atherosclerosis, Reaction–diffusion equations, Nonlinear boundary conditions, Existence of travellingwaves, Numerical simulations

Streszczenie:
Atherosclerosis is an inflammatory disease. The atherosclerosis process starts when low-density lipoproteins (LDLs) enter the intima of the blood vessel, where they are oxidized (ox-LDLs). The anti-inflammatory response triggers the recruitment of monocytes. Once in the intima, the monocytes are transformed into macrophages and foam cells, leading to the production of inflammatory cytokines and further recruitment of monocytes. This auto-amplified process leads to the formation of an atherosclerotic plaque and, possibly, to its rupture. In this paper we develop two mathematical models based on reaction–diffusion equations in order to explain the inflammatory process. The first model is one-dimensional: it does not consider the intima’s thickness and shows that low ox-LDL concentrations in the intima do not lead to a chronic inflammatory reaction. Intermediate ox-LDL concentrations correspond to a bistable system, which can lead to a travelling wave that can be initiated by certain conditions, such as infection or injury. High ox-LDL concentrations correspond to a monostable system, and even a small perturbation of the non-inflammatory case leads to travelling-wave propagation, which corresponds to a chronic inflammatory response. The second model we suggest is two-dimensional: it represents a reaction–diffusion system in a strip with nonlinear boundary conditions to describe the recruitment of monocytes as a function of the cytokines’ concentration. We prove the existence of travelling waves and confirm our previous results, which show that atherosclerosis develops as a reaction–diffusion wave. The results of the two models are confirmed by numerical simulations. The latter show that the two-dimensional model converges to the one-dimensional one if the thickness of the intima tends to zero.

Słowa kluczowe:
mathematical modelling, biomathematics, partial differential equations, travelling waves, reaction–diffusion equations, atherosclerosis

Streszczenie:
We study the existence and properties of calcium waves in the presence of buffers. The model represents a reaction–diffusion system of equations with some diffusion coefficients equal to zero. They correspond to immobile buffers. The proof of the existence of travelling waves is carried out by passing to zero in the diffusion coefficients.

Streszczenie:
The existence and structural stability of travelling waves of systems of the free cytosolic calcium concentration in the presence of immobile buffers are studied. The proof is carried out by passing to zero with the diffusion coefficients of buffers. Thus, its method is different from Ref. 13 where the existence is proved straightforwardly.

Słowa kluczowe:
Calcium dynamics, travelling waves, implicit function theorem

Streszczenie:
We study the existence of travelling waves of the free cytosolic calcium concentration in the presence of immobile buers. We also take into account the mechanochemical interaction. In the proof we use the a priori estimates for solutions of systems with diusing buers and pass to zero with their diusion coefficients.

Streszczenie:
Existence of reaction-diffusion-convection waves in unbounded strips is proved in the case of small Rayleigh numbers. In the bistable case the wave is unique, in the monostable case they exist for all speeds greater than the minimal one. The proof uses the implicit function theorem. Its application is based on the Fredholm property, index, and solvability conditions for elliptic problems in unbounded domains.