F. Najm, R. Yafia, M. A. Aziz Alaoui, A. Aghriche, A. Moussaoui
Motivated by some biological and ecological problems given by reaction-diffusion systems with delays and boundary conditions of Neumann type and knowing their associated Lyapunov functions for delay ordinary differential equations, we consider a method for determining their Lyapunov functions to establish the local/global stability. The method is essentially based on adding integral terms to the corresponding Lyapunov function for ordinary differential equations. The new approach is not general but it is applicable in a wide variety of delays reaction-diffusion models with one discrete delay or more, distributed delay, and a combination of both of them. To illustrate our results, we present the method application to a reaction-diffusion epidemiological model with time delay (latency period) and indirect transmission effect.
{"title":"A survey on constructing Lyapunov functions for reaction-diffusion systems with delay and their application in biology","authors":"F. Najm, R. Yafia, M. A. Aziz Alaoui, A. Aghriche, A. Moussaoui","doi":"10.23939/mmc2023.03.965","DOIUrl":"https://doi.org/10.23939/mmc2023.03.965","url":null,"abstract":"Motivated by some biological and ecological problems given by reaction-diffusion systems with delays and boundary conditions of Neumann type and knowing their associated Lyapunov functions for delay ordinary differential equations, we consider a method for determining their Lyapunov functions to establish the local/global stability. The method is essentially based on adding integral terms to the corresponding Lyapunov function for ordinary differential equations. The new approach is not general but it is applicable in a wide variety of delays reaction-diffusion models with one discrete delay or more, distributed delay, and a combination of both of them. To illustrate our results, we present the method application to a reaction-diffusion epidemiological model with time delay (latency period) and indirect transmission effect.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135799238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we are interested in the existence, uniqueness, and numerical simulation of weak periodic solutions for some semilinear elliptic equations with data measures and with arbitrary growth of nonlinearities. Since the data are not very regular and the growths are arbitrary, a new approach is needed to analyze these types of equations. Finally, a suitable numerical discretization scheme is presented. Several numerical examples are given which show the robustness of our algorithm.
{"title":"Semilinear periodic equation with arbitrary nonlinear growth and data measure: mathematical analysis and numerical simulation","authors":"M. El Ghabi, H. Alaa, N. E. Alaa","doi":"10.23939/mmc2023.03.956","DOIUrl":"https://doi.org/10.23939/mmc2023.03.956","url":null,"abstract":"In this work, we are interested in the existence, uniqueness, and numerical simulation of weak periodic solutions for some semilinear elliptic equations with data measures and with arbitrary growth of nonlinearities. Since the data are not very regular and the growths are arbitrary, a new approach is needed to analyze these types of equations. Finally, a suitable numerical discretization scheme is presented. Several numerical examples are given which show the robustness of our algorithm.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135799477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The global analysis of a spatio-temporal fractional order SEIR infection epidemic model is studied and analyzed in this paper. The dynamics of the infection is described by four partial differential equations with a fractional derivative order and with diffusion. The equations of our model describe the evolution of the susceptible, the exposed, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. At first, we will prove the existence and uniqueness of the solution using the results of the fixed point theorem, and the equilibrium points are established and presented according to R0. Next, the bornitude and the positivity of the solutions of the proposed model are established. Using the Lyapunov direct method it has been proved that the global stability of the each equilibrium depends mainly on the basic reproduction number R0. Finally, numerical simulations are performed to validate the theoretical results.
{"title":"Global dynamic of spatio-temporal fractional order SEIR model","authors":"C. Bounkaicha, K. Allali, Y. Tabit, J. Danane","doi":"10.23939/mmc2023.02.299","DOIUrl":"https://doi.org/10.23939/mmc2023.02.299","url":null,"abstract":"The global analysis of a spatio-temporal fractional order SEIR infection epidemic model is studied and analyzed in this paper. The dynamics of the infection is described by four partial differential equations with a fractional derivative order and with diffusion. The equations of our model describe the evolution of the susceptible, the exposed, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. At first, we will prove the existence and uniqueness of the solution using the results of the fixed point theorem, and the equilibrium points are established and presented according to R0. Next, the bornitude and the positivity of the solutions of the proposed model are established. Using the Lyapunov direct method it has been proved that the global stability of the each equilibrium depends mainly on the basic reproduction number R0. Finally, numerical simulations are performed to validate the theoretical results.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Y. A. Adi, N. Irsalinda, A. Wiraya, S. Sugiyarto, Z. A. Rafsanjani
In this paper, we introduce a two-strain SIR epidemic model with viral mutation and vaccine administration. We discuss and analyze the existence and stability of equilibrium points. This model has three types of equilibrium points, namely disease-free equilibrium, dominance equilibrium point of strain two, and coexistence endemic equilibrium point. The local stability of the dominance equilibrium point of strain two and coexistence endemic equilibrium point are verified by using the Routh--Hurwitz criteria, while for the global stability of the dominance equilibrium point of strain two, we used a suitable Lyapunov function. We also carried out the bifurcation analysis using the application of center manifold theory, and we obtained that the system near the disease-free equilibrium point always has supercritical bifurcation. Finally, the numerical simulations are provided to validate the theoretical results. Continuation of the supercritical bifurcation point results in two Hopf bifurcations indicating a local birth of chaos and quasi-periodicity.
{"title":"An epidemic model with viral mutations and vaccine interventions","authors":"Y. A. Adi, N. Irsalinda, A. Wiraya, S. Sugiyarto, Z. A. Rafsanjani","doi":"10.23939/mmc2023.02.311","DOIUrl":"https://doi.org/10.23939/mmc2023.02.311","url":null,"abstract":"In this paper, we introduce a two-strain SIR epidemic model with viral mutation and vaccine administration. We discuss and analyze the existence and stability of equilibrium points. This model has three types of equilibrium points, namely disease-free equilibrium, dominance equilibrium point of strain two, and coexistence endemic equilibrium point. The local stability of the dominance equilibrium point of strain two and coexistence endemic equilibrium point are verified by using the Routh--Hurwitz criteria, while for the global stability of the dominance equilibrium point of strain two, we used a suitable Lyapunov function. We also carried out the bifurcation analysis using the application of center manifold theory, and we obtained that the system near the disease-free equilibrium point always has supercritical bifurcation. Finally, the numerical simulations are provided to validate the theoretical results. Continuation of the supercritical bifurcation point results in two Hopf bifurcations indicating a local birth of chaos and quasi-periodicity.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135126584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On March 2, 2020, the Moroccan Ministry of Health announced the first case of COVID-19 in the city of Casablanca for a Moroccan tourist who came from Italy. The SARS-COV-2 virus has spread throughout the Kingdom of Morocco. In this paper, we study the spatiotemporal transmission of the COVID-19 virus in the Kingdom of Morocco. By supporting a SIWIHR partial differential equation for the spread of the COVID-19 pandemic in Morocco as a case study. Our main goal is to characterize the optimum order of controlling the spread of the COVID-19 pandemic by adopting a vaccination strategy, the aim of which is to reduce the number of susceptible and infected individuals without vaccination and to maximize the recovered individuals by reducing the cost of vaccination using one of the vaccines approved by the World Health Organization. To do this, we proved the existence of a pair of control. It provides a description of the optimal controls in terms of state and auxiliary functions. Finally, we provided numerical simulations of data related to the transmission of the COVID-19 pandemic. Numerical results are presented to illustrate the effectiveness of the adopted approach.
{"title":"A spatiotemporal spread of COVID-19 pandemic with vaccination optimal control strategy: A case study in Morocco","authors":"A. Kouidere, M. Elhia, O. Balatif","doi":"10.23939/mmc2023.01.171","DOIUrl":"https://doi.org/10.23939/mmc2023.01.171","url":null,"abstract":"On March 2, 2020, the Moroccan Ministry of Health announced the first case of COVID-19 in the city of Casablanca for a Moroccan tourist who came from Italy. The SARS-COV-2 virus has spread throughout the Kingdom of Morocco. In this paper, we study the spatiotemporal transmission of the COVID-19 virus in the Kingdom of Morocco. By supporting a SIWIHR partial differential equation for the spread of the COVID-19 pandemic in Morocco as a case study. Our main goal is to characterize the optimum order of controlling the spread of the COVID-19 pandemic by adopting a vaccination strategy, the aim of which is to reduce the number of susceptible and infected individuals without vaccination and to maximize the recovered individuals by reducing the cost of vaccination using one of the vaccines approved by the World Health Organization. To do this, we proved the existence of a pair of control. It provides a description of the optimal controls in terms of state and auxiliary functions. Finally, we provided numerical simulations of data related to the transmission of the COVID-19 pandemic. Numerical results are presented to illustrate the effectiveness of the adopted approach.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135127084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A kinetic approach based on a modified chain of BBGKI equations for nonequilibrium particle distribution functions was used to describe the ion transfer processes in the ionic solution – porous medium system. A generalized kinetic equation of the revised Enskog–Vlasov–Landau theory for the nonequilibrium ion distribution function in the model of charged solid spheres is obtained, taking into account attractive short-range interactions for the ionic solution – porous medium system.
{"title":"Kinetic description of ion transport in the system \"ionic solution – porous environment\"","authors":"M. Tokarchuk","doi":"10.23939/mmc2022.03.719","DOIUrl":"https://doi.org/10.23939/mmc2022.03.719","url":null,"abstract":"A kinetic approach based on a modified chain of BBGKI equations for nonequilibrium particle distribution functions was used to describe the ion transfer processes in the ionic solution – porous medium system. A generalized kinetic equation of the revised Enskog–Vlasov–Landau theory for the nonequilibrium ion distribution function in the model of charged solid spheres is obtained, taking into account attractive short-range interactions for the ionic solution – porous medium system.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45384960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-12DOI: 10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.2
M. Tokarchuk
Based on a chain of BBGKI equations with a modified boundary condition that takes into account multiparticle correlations, kinetic equations in the approximate "pairs" collisions and in the polarization approximation, taking into account the interaction through the third particle, obtained. The specifics of the model representation of the pair potential of particle interaction through short-range and long-range parts were taken into account. In the case of the short-range potential in the form of the potential of solid spheres, the contribution of Enskog's revised theory to the complete integration of the collision of the kinetic equation is obtained. The collision integrals include paired quasi-equilibrium distribution functions that depend on the nonequilibrium mean values of the particle number density and the inverse temperature. The method of collective variables Yukhnovskii is applied for the calculation of pair quasi-equilibrium distribution function with an allocation of short-range and long-range parts in the potential of the interaction of particles. In this case, the system with short-range interaction is considered as a frame of reference.
{"title":"To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables","authors":"M. Tokarchuk","doi":"10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.2","DOIUrl":"https://doi.org/10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.23939/mmc2022.02.440 10.2","url":null,"abstract":"Based on a chain of BBGKI equations with a modified boundary condition that takes into account multiparticle correlations, kinetic equations in the approximate \"pairs\" collisions and in the polarization approximation, taking into account the interaction through the third particle, obtained. The specifics of the model representation of the pair potential of particle interaction through short-range and long-range parts were taken into account. In the case of the short-range potential in the form of the potential of solid spheres, the contribution of Enskog's revised theory to the complete integration of the collision of the kinetic equation is obtained. The collision integrals include paired quasi-equilibrium distribution functions that depend on the nonequilibrium mean values of the particle number density and the inverse temperature. The method of collective variables Yukhnovskii is applied for the calculation of pair quasi-equilibrium distribution function with an allocation of short-range and long-range parts in the potential of the interaction of particles. In this case, the system with short-range interaction is considered as a frame of reference.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46707205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Ghazdali, M. Hakim, A. Laghrib, N. Mamouni, A. Metrane, A. Ourdou
In this paper, a new Blind Source Separation (BSS) method that handles mixtures of noisy independent/dependent sources is introduced. We achieve that by minimizing a criterion that fuses a separating part, based on Kullback–Leibler divergence for either dependent or independent sources, with a regularization part that employs the bilateral total variation (BTV) for the purpose of denoising the observations. The proposed algorithm utilizes a primal-dual algorithm to remove the noise, while a gradient descent method is implemented to retrieve the signal sources. Our algorithm has shown its effectiveness and efficiency and also surpassed the standard existing BSS algorithms.
{"title":"Robust approach for blind separation of noisy mixtures of independent and dependent sources","authors":"A. Ghazdali, M. Hakim, A. Laghrib, N. Mamouni, A. Metrane, A. Ourdou","doi":"10.23939/mmc2021.04.761","DOIUrl":"https://doi.org/10.23939/mmc2021.04.761","url":null,"abstract":"In this paper, a new Blind Source Separation (BSS) method that handles mixtures of noisy independent/dependent sources is introduced. We achieve that by minimizing a criterion that fuses a separating part, based on Kullback–Leibler divergence for either dependent or independent sources, with a regularization part that employs the bilateral total variation (BTV) for the purpose of denoising the observations. The proposed algorithm utilizes a primal-dual algorithm to remove the noise, while a gradient descent method is implemented to retrieve the signal sources. Our algorithm has shown its effectiveness and efficiency and also surpassed the standard existing BSS algorithms.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44479514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the presented study, the mathematical model for drying the porous timber beam of a circular cross-section under the action of a convective-heat nonstationary flow of the drying agent is constructed. When solving the problem, a capillary-porous structure of the beam is described in terms of a quasi-homogeneous medium with effective coefficients, which are chosen so that the solution in a homogeneous medium coincides with the solution in the porous medium. The influence of the porous structure is taken into account by introducing into the Stefan–Maxwell equation the effective binary interaction coefficients. The problem of mutual phase distribution is solved using the principle of local phase equilibrium. The given properties of the material (heat capacity, density, thermal conductivity) are considered to be functions of the porosity of the material as well as densities and heat capacities of body components. The solution is obtained for determining the temperature in the beam at an arbitrary time of drying at any coordinate point of the radius, thermomechanical characteristics of the material, and the parameters of the drying agent.
{"title":"Investigation of drying the porous wood of a cylindrical shape","authors":"B. Gayvas, V. Dmytruk","doi":"10.23939/mmc2022.02.399","DOIUrl":"https://doi.org/10.23939/mmc2022.02.399","url":null,"abstract":"In the presented study, the mathematical model for drying the porous timber beam of a circular cross-section under the action of a convective-heat nonstationary flow of the drying agent is constructed. When solving the problem, a capillary-porous structure of the beam is described in terms of a quasi-homogeneous medium with effective coefficients, which are chosen so that the solution in a homogeneous medium coincides with the solution in the porous medium. The influence of the porous structure is taken into account by introducing into the Stefan–Maxwell equation the effective binary interaction coefficients. The problem of mutual phase distribution is solved using the principle of local phase equilibrium. The given properties of the material (heat capacity, density, thermal conductivity) are considered to be functions of the porosity of the material as well as densities and heat capacities of body components. The solution is obtained for determining the temperature in the beam at an arbitrary time of drying at any coordinate point of the radius, thermomechanical characteristics of the material, and the parameters of the drying agent.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68766609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Article demonstrates the applicability of modeling non-stationary non-isothermal gas flow along a linear section of a gas transmission system by means of using various numerically simulated models and sophisticated numerical techniques. There are described several models of non-stationary non-isothermal regimes of gas flow along the pipeline section. They are included in the considered general model and their comparative analysis is carried out by the virtue of numerical simulation. The finite difference algorithm is used to solve the simultaneous equations of the numerically simulated model for the pipeline section. The results of calculating the gas flow parameters using various models are presented: both with and without taking into account kinetic energy, as well as both with and without taking into account the Joule–Thompson effect. The matter of choosing the appropriate model is discussed. The obtained results can be used at the stage of transfer pipeline system operation in order to develop scientifically well-founded recommendations for improving the safety and efficiency of the pipeline transportation system.
{"title":"Mathematical modeling of non-stationary gas flow modes along a linear section of a gas transmission system","authors":"I. Husarova, A. Tevyashev, O. A. Tevyasheva","doi":"10.23939/mmc2022.02.416","DOIUrl":"https://doi.org/10.23939/mmc2022.02.416","url":null,"abstract":"Article demonstrates the applicability of modeling non-stationary non-isothermal gas flow along a linear section of a gas transmission system by means of using various numerically simulated models and sophisticated numerical techniques. There are described several models of non-stationary non-isothermal regimes of gas flow along the pipeline section. They are included in the considered general model and their comparative analysis is carried out by the virtue of numerical simulation. The finite difference algorithm is used to solve the simultaneous equations of the numerically simulated model for the pipeline section. The results of calculating the gas flow parameters using various models are presented: both with and without taking into account kinetic energy, as well as both with and without taking into account the Joule–Thompson effect. The matter of choosing the appropriate model is discussed. The obtained results can be used at the stage of transfer pipeline system operation in order to develop scientifically well-founded recommendations for improving the safety and efficiency of the pipeline transportation system.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68766705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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