Influence of Human Behavior on COVID-19 Dynamics Based on a Reaction-Diffusion Model.

To investigate the influence of human behavior on the spread of COVID-19, we propose a reaction-diffusion model that incorporates contact rate functions related to human behavior. The basic reproduction number R0 is derived and a threshold-type result on its global dynamics in terms of R0 is established. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if R0≤1; while there exists a positive stationary solution and the disease is uniformly persistent if R0>1. By the numerical simulations of the analytic results, we find that human behavior changes may lower infection levels and reduce the number of exposed and infected humans.

On a reaction–diffusion system modelling infectious diseases without lifetime immunity

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

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.

A class of anomalous diffusion epidemic models based on CTRW and distributed delay

In recent years, the epidemic model with anomalous diffusion has gained popularity in the literature. However, when introducing anomalous diffusion into epidemic models, they frequently lack physical explanation, in contrast to the traditional reaction-diffusion epidemic models. The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable. Specifically, based on the continuous-time random walk (CTRW), starting from two stochastic processes of the waiting time and the step length, time-fractional space-fractional diffusion, time-fractional reaction-diffusion and fractional-order diffusion can all be naturally introduced into the SIR (S: susceptible, I: infectious and R: recovered) epidemic models, respectively. The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays. Distributed time delay systems can also be reduced to existing models, such as the standard SIR model, the fractional infectivity model and others, within the proper bounds. Meanwhile, as an application of the above stochastic modeling method, the physical meaning of anomalous diffusion is also considered by taking the SEIR (E: exposed) epidemic model as an example. Similar methods can be used to build other types of epidemic models, including SIVRS (V: vaccine), SIQRS (Q: quarantined) and others. Finally, this paper describes the transmission of infectious disease in space using the real data of COVID-19. © 2023 World Scientific Publishing Company.

Well-posedness for a diffusion–reaction compartmental model simulating the spread of COVID-19

This paper is concerned with the well-posedness of a diffusion–reaction system for a susceptible-exposed-infected-recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio-temporal spread of the COVID-19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions. © 2023 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.

Epidemic modeling with heterogeneity and social diffusion.

We propose and analyze a family of epidemiological models that extend the classic Susceptible-Infectious-Recovered/Removed (SIR)-like framework to account for dynamic heterogeneity in infection risk. The family of models takes the form of a system of reaction-diffusion equations given populations structured by heterogeneous susceptibility to infection. These models describe the evolution of population-level macroscopic quantities S, I, R as in the classical case coupled with a microscopic variable f, giving the distribution of individual behavior in terms of exposure to contagion in the population of susceptibles. The reaction terms represent the impact of sculpting the distribution of susceptibles by the infection process. The diffusion and drift terms that appear in a Fokker-Planck type equation represent the impact of behavior change both during and in the absence of an epidemic. We first study the mathematical foundations of this system of reaction-diffusion equations and prove a number of its properties. In particular, we show that the system will converge back to the unique equilibrium distribution after an epidemic outbreak. We then derive a simpler system by seeking self-similar solutions to the reaction-diffusion equations in the case of Gaussian profiles. Notably, these self-similar solutions lead to a system of ordinary differential equations including classic SIR-like compartments and a new feature: the average risk level in the remaining susceptible population. We show that the simplified system exhibits a rich dynamical structure during epidemics, including plateaus, shoulders, rebounds and oscillations. Finally, we offer perspectives and caveats on ways that this family of models can help interpret the non-canonical dynamics of emerging infectious diseases, including COVID-19.

Pathogen diversity in meta-population networks

The pathogen diversity means that multiple strains coexist, and widely exist in the biology systems. The new mutation of SARS-CoV-2 leading to worldwide pathogen diversity is a typical example. What are the main factors of inducing the pathogen diversity? Previous studies indicated the pathogen mutation is the most important reason for inducing the pathogen diversity. The traffic network and gene network are crucial in shaping the dynamics of pathogen contagion, while their roles for the pathogen diversity still lacking a theoretical study. To this end, we propose a reaction–diffusion process of pathogens with mutations on meta-population networks, which includes population movement and strain mutation. We extend the Microscopic Markov Chain Approach (MMCA) to describe the model. Traffic networks make pathogen diversity more likely to occur in cities with lower infection densities. The likelihood of pathogen diversity is low in cities with short effective distances in the traffic network. Star-type gene network is more likely to lead to pathogen diversity than lattice-type and chain-type gene networks. When pathogen localization is present, infection is localized to strains that are at the endpoints of the gene network. Both the increased probability of movement and mutation promote pathogen diversity. The results also show that the population tends to move to cities with short effective distances, resulting in the infection density is high. © 2022 Elsevier Ltd

Computational Modeling of Reaction-Diffusion COVID-19 Model Having Isolated Compartment

Cases of COVID-19 and its variant omicron are raised all across the world. The most lethal form and effect of COVID-19 are the omicron version, which has been reported in tens of thousands of cases daily in numerous nations. Following WHO (World health organization) records on 30 December 2021, the cases of COVID-19 were found to be maximum for which boarding individuals were found 1,524,266, active, recovered, and discharge were found to be 82,402 and 34,258,778, respectively. While there were 160,989 active cases, 33,614,434 cured cases, 456,386 total deaths, and 605,885,769 total samples tested. So far, 1,438,322,742 individuals have been vaccinated. The coronavirus or COVID-19 is inciting panic for several reasons. It is a new virus that has affected the whole world. Scientists have introduced certain ways to prevent the virus. One can lower the danger of infection by reducing the contact rate with other persons. Avoiding crowded places and social events with many people reduces the chance of one being exposed to the virus. The deadly COVID-19 spreads speedily. It is thought that the upcoming waves of this pandemic will be even more dreadful. Mathematicians have presented several mathematical models to study the pandemic and predict future dangers. The need of the hour is to restrict the mobility to control the infection from spreading. Moreover, separating affected individuals from healthy people is essential to control the infection. We consider the COVID-19 model in which the population is divided into five compartments. The present model presents the population's diffusion effects on all susceptible, exposed, infected, isolated, and recovered compartments. The reproductive number, which has a key role in the infectious models, is discussed. The equilibrium points and their stability is presented. For numerical simulations, finite difference (FD) schemes like nonstandard finite difference (NSFD), forward in time central in space (FTCS), and Crank Nicolson (CN) schemes are implemented. Some core characteristics of schemes like stability and consistency are calculated. © 2023 Tech Science Press. All rights reserved.

Mathematical modeling of respiratory viral infection and applications to SARS-CoV-2 progression.

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 infection 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 tissue 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.

Modeling and multi-objective optimal control of reaction-diffusion COVID-19 system due to vaccination and patient isolation.

In this paper, a reaction-diffusion COVID-19 model is proposed to explore how vaccination-isolation strategies affect the development of the epidemic. First, the basic dynamical properties of the system are explored. Then, the system's asymptotic distributions of endemic equilibrium under different conditions are studied. Further, the global sensitivity analysis of R 0 is implemented with the aim of determining the sensitivity for these parameters. In addition, the optimal vaccination-isolation strategy based on the optimal path is proposed. Meantime, social cost C ( m , σ ) , social benefit B ( m , σ ) , threshold R 0 ( m , σ ) three objective optimization problem based on vaccination-isolation strategy is explored, and the maximum social cost ( M S C ) and maximum social benefit ( M S B ) are obtained. Finally, the instance prediction of the Lhasa epidemic in China on August 7, 2022, is made by using the piecewise infection rates ß 1 ( t ) , ß 2 ( t ) , and some key indicators are obtained as follows: (1) The basic reproduction numbers of each stage in Lhasa, China are R 0 ( 1 : 8 ) = 0.4678 , R 0 ( 9 : 20 ) = 2.7655 , R 0 ( 21 : 30 ) = 0.3810 and R 0 ( 31 : 100 ) = 0.7819 ; (2) The daily new cases of this epidemic will peak at 43 on the 20th day (August 26, 2022); (3) The cumulative cases in Lhasa, China will reach about 640 and be cleared about the 80th day (October 28, 2022). Our research will contribute to winning the war on epidemic prevention and control.

A class of anomalous diffusion epidemic models based on CTRW and distributed delay

In recent years, the epidemic model with anomalous diffusion has gained popularity in the literature. However, when introducing anomalous diffusion into epidemic models, they frequently lack physical explanation, in contrast to the traditional reaction-diffusion epidemic models. The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable. Specifically, based on the continuous-time random walk (CTRW), starting from two stochastic processes of the waiting time and the step length, time-fractional space-fractional diffusion, time-fractional reaction-diffusion and fractional-order diffusion can all be naturally introduced into the SIR (S: susceptible, I: infectious and R: recovered) epidemic models, respectively. The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays. Distributed time delay systems can also be reduced to existing models, such as the standard SIR model, the fractional infectivity model and others, within the proper bounds. Meanwhile, as an application of the above stochastic modeling method, the physical meaning of anomalous diffusion is also considered by taking the SEIR (E: exposed) epidemic model as an example. Similar methods can be used to build other types of epidemic models, including SIVRS (V: vaccine), SIQRS (Q: quarantined) and others. Finally, this paper describes the transmission of infectious disease in space using the real data of COVID-19.

Mathematical Modelling of the Spatial Distribution of a COVID-19 Outbreak with Vaccination Using Diffusion Equation.

The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction−diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction−diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh−Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate R0<1 and R0>1. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model's effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.

Global dynamics for an SVEIR epidemic model with diffusion and nonlinear incidence rate

In this paper, we investigate an SVEIR epidemic model with reaction-diffusion and nonlinear incidence. We establish the well-posedness of the solutions and the basic reproduction number R-0. Moreover, we show that the disease-free steady state is globally asymptotically stable when R-0 < 1, whereas the disease will be persistent when R-0 > 1. Furthermore, using the method of Lyapunov functional, we prove the global stability of the positive steady state for the spatial homogeneous model.

Differentiation of COVID-19 Conditions using Mediastinum Shape in Chest X-ray Images

In this work, an attempt has been made to analyze the shape variations in mediastinum for differentiation of Coronavirus Disease-2019 (COVID-19) and normal conditions in chest X-ray images. For this, the images are obtained from a publicly available dataset. Segmentation of mediastinum from the raw images is performed using Reaction Diffusion Level Set (RDLS) method. Shape-based features are extracted from the delineated mediastinum masks and are statistically analyzed. Further, the features are fed to two classifiers, namely, multi-layer perceptron and support vector machine for differentiation of normal and COVID-19 images. From the results, it is observed that the employed RDLS method is able to delineate mediastinum from the raw chest X-ray images. Eight shape features are observed to be statistically significant. The mean values of these features are found to be distinctly higher for COVID-19 images as compared to normal images. Area under the curve of greater than 76.9% is achieved for both the classifiers. It appears that mediastinum could be used as a region of interest for computerized detection and mass screening of the disease. © 2022 The Author(s), published by De Gruyter.

Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion.

In view of the facts in the infection and propagation of COVID-19, a stochastic reaction-diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.

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.

Infection spreading in cell culture as a reaction-diffusion wave: RAIRO: Numerical Analysis

Infection spreading in cell culture occurs due to virus replication in infected cells and its random motion in the extracellular space. Multiplicity of infection experiments in cell cultures are conventionally used for the characterization of viral infection by the number of viral plaques and the rate of their growth. We describe this process with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells, virus, and interferon. Time delay corresponds to the duration of viral replication inside infected cells. We show that infection propagates in cell culture as a reaction-diffusion wave, we determine the wave speed and prove its existence. Next, we carry out numerical simulations and identify three stages of infection progression: infection decay during time delay due to virus replication, explosive growth of viral load when infected cells begin to reproduce it, and finally, wave-like infection progression in cell culture characterized by a constant or slowly growing total viral load. The modelling results are in agreement with the experimental data for the coronavirus infection in a culture of epithelial cells and for some other experiments. The presence of interferon produced by infected cells decreases the viral load but does not change the speed of infection progression in cell culture. In the 2D modelling, the total viral load grows faster than in the 1D case due to the increase of plaque perimeter.

Virus replication and competition in a cell culture: Application to the SARS-CoV-2 variants.

Viral replication in a cell culture is described by a delay reaction-diffusion system. It is shown that infection spreads in cell culture as a reaction-diffusion wave, for which the speed of propagation and viral load can be determined both analytically and numerically. Competition of two virus variants in the same cell culture is studied, and it is shown that the variant with larger individual wave speed out-competes another one, and eliminates it. This approach is applied to the Delta and Omicron variants of the SARS-CoV-2 infection in the cultures of human epithelial and lung cells, allowing characterization of infectivity and virulence of each variant, and their comparison.

Spatial dynamics for an SIRE epidemic model with diffusion and prevention in contaminated environments

In this paper, a reaction-diffusion SIRE epidemic model in contaminated environments is proposed, in which the effect of protection for susceptible individuals is included by the nonlinear incidence functions b(S)E$b(S)E$ and g(S)I$g(S)I$. When the space is heterogeneous, the basic reproduction number R0$\mathcal {R}_{0}$ is derived, by which we find that if R0 <= 1$\mathcal {R}_{0}\le 1$, the disease-free steady state is globally asymptotically stable, while R0>1$\mathcal {R}_{0}>1$, the disease is uniform persistent. Furthermore, when R0>1$\mathcal {R}_{0}>1$ and additional conditions hold, the global asymptotic stability of special endemic steady state is obtained in homogeneous space. Finally, the theoretical results are validated by numerical simulations, some open questions are illustrated.

A DIFFUSIVE SEIR MODEL FOR COMMUNITY TRANSMISSION OF COVID-19 EPIDEMICS: APPLICATION TO BRAZIL

A mathematical model incorporating diffusion is developed to describe the spatial spread of COVID-19 epidemics in geographical regions. The dynamics of the spatial spread are based on community transmission of the virus. The model is applied to the outbreak of the COVID-19 epidemic in Brazil. © 2022 Mathematics in Applied Sciences and Engineering. All rights reserved.