Interplay between reaction and diffusion processes in governing the dynamics of virus infections

The impact of immune cell interactions on virus quasi-species formation

The process of viral infection spreading in tissues was influenced by various factors, including virus replication within host cells, transportation, and the immune response. Reaction-diffusion systems provided a suitable framework for examining this process. In this work, we studied a nonlocal reaction-diffusion system of equations that modeled the distribution of viruses based on their genotypes and their interaction with the immune response. It was shown that the infection may persist at a certain level alongside a chronic immune response, exhibiting spatially uniform or oscillatory behavior. Finally, the immune cells may become entirely depleted, leading to a high viral load persisting in the tissue. Numerical simulations were employed to elucidate the nonlinear dynamics and pattern formation inherent in the nonlocal model.

Stability of a diffusive-delayed HCV infection model with general cell-to-cell incidence function incorporating immune response and cell proliferation

In this work, we analyse the dynamics of a five-dimensional hepatitis C virus infection mathematical model including the spatial mobility of hepatitis C virus particles, the transmission of hepatitis C virus infection by mitosis process of infected hepatocytes with logistic growth, time delays, antibody response and cytotoxic T lymphocyte (CTL) immune response with general incidence functions for both modes of infection transmission, namely virus-to-cell as well as cell-to-cell. Firstly, we prove rigorously the existence, the uniqueness, the positivity and the boundedness of the solution of the initial value and boundary problem associated with the new constructed model. Secondly, we found that the basic reproductive number is the sum of the basic reproduction number determined by cell-free virus infection, determined by cell-to-cell infection and determined by proliferation of infected cells. It is proved the existence of five spatially homogeneous equilibria known as infection-free, immune-free, antibody response, CTL response and antibody and CTL responses. By using the linearization methods, the local stability of the latter is established under some rigorous conditions. Finally, we proved the existence of periodic solutions by highlighting the occurrence of a Hopf bifurcation for a certain threshold value of one delay.

The influence of immune cells on the existence of virus quasi-species

This article investigate a nonlocal reaction-diffusion system of equations modeling virus distribution with respect to their genotypes in the interaction with the immune response. This study demonstrates the existence of pulse solutions corresponding to virus quasi-species. The proof is based on the Leray-Schauder method, which relies on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions. Furthermore, linear stability analysis of a spatially homogeneous stationary solution identifies the critical conditions for the emergence of spatial and spatiotemporal structures. Finally, numerical simulations are used to illustrate nonlinear dynamics and pattern formation in the nonlocal model.

Emergence and competition of virus variants in respiratory viral infections

The emergence of new variants of concern (VOCs) of the SARS-CoV-2 infection is one of the main factors of epidemic progression. Their development can be characterized by three critical stages: virus mutation leading to the appearance of new viable variants; the competition of different variants leading to the production of a sufficiently large number of copies; and infection transmission between individuals and its spreading in the population. The first two stages take place at the individual level (infected individual), while the third one takes place at the population level with possible competition between different variants. This work is devoted to the mathematical modeling of the first two stages of this process: the emergence of new variants and their progression in the epithelial tissue with a possible competition between them. The emergence of new virus variants is modeled with non-local reaction–diffusion equations describing virus evolution and immune escape in the space of genotypes. The conditions of the emergence of new virus variants are determined by the mutation rate, the cross-reactivity of the immune response, and the rates of virus replication and death. Once different variants emerge, they spread in the infected tissue with a certain speed and viral load that can be determined through the parameters of the model. The competition of different variants for uninfected cells leads to the emergence of a single dominant variant and the elimination of the others due to competitive exclusion. The dominant variant is the one with the maximal individual spreading speed. Thus, the emergence of new variants at the individual level is determined by the immune escape and by the virus spreading speed in the infected tissue.

Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave

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.

Nonlocal Reaction-Diffusion Equations in Biomedical Applications

Nonlocal reaction-diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction-diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance.

Viral Infection Spreading and Mutation in Cell Culture

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.

Analysis of the initiation of viral infection under flow conditions with applications to transmission in feed

While kinetic models are widely used to describe viral infection at various levels, most of them are focused on temporal aspects and understanding of corresponding spatio-temporal aspects remains limited. In this work, our attention is focused on the initial stage of infection of immobile cells by virus particles ("virions") under flow conditions with diffusion. A practical example of this scenario occurs when humans or animals consume food from virion-containing sources. Mathematically, such situations can be described by using a model constructed in analogy with those employed in chemical engineering for analysis of the function of a plug-flow reactor with dispersion. As in the temporal case, the corresponding spatio-temporal model predicts either the transition to a steady state or exponential growth of the populations of virions and infected cells. The spatial distributions of these species are similar in both of these regimes. In particular, the maximums of the populations are shifted to the upper boundary of the infected region. The results illustrating these conclusions were obtained analytically and by employing numerical calculations without and with the dependence of the kinetic parameters on the coordinate. The model proposed has also been used in order to illustrate the effect of antiviral feed additives on feedborne infection towards curbing disease transmission.