This paper is devoted to a nonlocal dispersal logistic model with seasonal succession in one-dimensional bounded habitat, where the seasonal succession accounts for the effect of two different seasons. Firstly, we provide the persistence-extinction criterion for the species, which is different from that for local diffusion model. Then we show the asymptotic profile of the time-periodic positive solution as the species persists in long run.
{"title":"Long-time behavior of a nonlocal dispersal logistic model with seasonal succession","authors":"Zhenzhen Li, B. Dai","doi":"10.5206/mase/15415","DOIUrl":"https://doi.org/10.5206/mase/15415","url":null,"abstract":"This paper is devoted to a nonlocal dispersal logistic model with seasonal succession in one-dimensional bounded habitat, where the seasonal succession accounts for the effect of two different seasons. Firstly, we provide the persistence-extinction criterion for the species, which is different from that for local diffusion model. Then we show the asymptotic profile of the time-periodic positive solution as the species persists in long run.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42810332","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}
W. Fitzgibbon, J. Morgan, Geoffrey I. Webb, Yixiang Wu
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.
{"title":"A diffusive SEIR model for community transmission of Covid-19 epidemics: application to Brazil","authors":"W. Fitzgibbon, J. Morgan, Geoffrey I. Webb, Yixiang Wu","doi":"10.5206/mase/14150","DOIUrl":"https://doi.org/10.5206/mase/14150","url":null,"abstract":"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.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47530564","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}
We extend the classical reaction-diffusion model for spatial population dynamics of spruce budworm on a finite domain with hostile boundary conditions by including an advection term representing biased unidirectional movement of individuals due to a prevailing wind. We use phase-plane techniques to establish existence of a critical value of advection speed that prevents outbreak solutions on any finite domain while possibly allowing an endemic solution. We obtain lower and upper bounds for this critical advection value in terms of biological parameters involved in the reaction term. We also perform numerical simulations to illustrate the effect of advection on the dependence of the domain size on the maximal population density of a steady state solution and on critical domain sizes for endemic and outbreak solutions. The results are also applicable to other ecological settings (rivers, climate change) where a logistically growing population is subject to predation by a generalist, diffusion and biased movement.
{"title":"Prevailing winds and spruce budworm outbreaks: a reaction-diffusion-advection model","authors":"Abby Anderson, O. Vasilyeva","doi":"10.5206/mase/14112","DOIUrl":"https://doi.org/10.5206/mase/14112","url":null,"abstract":"We extend the classical reaction-diffusion model for spatial population dynamics of spruce budworm on a finite domain with hostile boundary conditions by including an advection term representing biased unidirectional movement of individuals due to a prevailing wind. We use phase-plane techniques to establish existence of a critical value of advection speed that prevents outbreak solutions on any finite domain while possibly allowing an endemic solution. We obtain lower and upper bounds for this critical advection value in terms of biological parameters involved in the reaction term. We also perform numerical simulations to illustrate the effect of advection on the dependence of the domain size on the maximal population density of a steady state solution and on critical domain sizes for endemic and outbreak solutions. The results are also applicable to other ecological settings (rivers, climate change) where a logistically growing population is subject to predation by a generalist, diffusion and biased movement.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43133315","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}
Background: The SARS-CoV-2 pandemic is spreading with a greater intensity across the globe. The synchrony of public health interventions and epidemic waves signify the importance of evaluation of the underline interventions. Method: We developed a mathematical model to present the transmission dynamics of SARS-CoV-2 and to analyze the impact of key nonpharmaceutical interventions such as isolation and screening program on the disease outcomes to the people of New Jersey, USA. We introduced a dynamic isolation of susceptible population with a constant (imposed) and infection oriented interventions. Epidemiological and demographic data are used to estimate the model parameters. The baseline case was explored further to showcase several critical and predictive scenarios. Results and analysis: The model simulations are in good agreement with the infection data for the period of 5 March 2020 to 31 January 2021. Dynamic isolation and screening program are found to be potential measures that can alter the course of epidemic. A 7% increase in isolation rate may result in a 31% reduction of epidemic peak whereas a 3 times increase in screening rate may reduce the epidemic peak by 35%. The model predicts that nearly 9.7% to 12% of the total population of New Jersey may become infected within the middle of July 2021 along with 24.6 to 27.3 thousand cumulative deaths. Within a wide spectrum of probable scenarios, there is a possibility of third wave Conclusion: Our findings could be informative to the public health community to contain the pandemic in the case of economy reopening under a limited or no vaccine coverage. Additional epidemic waves can be avoided by appropriate screening and isolation plans.
{"title":"Modeling SARS-CoV-2 spread with dynamic isolation","authors":"Md. Azmir Ibne Islam, Sharmin Sultana Shanta, Ashrafur Rahman","doi":"10.5206/mase/13886","DOIUrl":"https://doi.org/10.5206/mase/13886","url":null,"abstract":"Background: The SARS-CoV-2 pandemic is spreading with a greater intensity across the globe. The synchrony of public health interventions and epidemic waves signify the importance of evaluation of the underline interventions. \u0000Method: We developed a mathematical model to present the transmission dynamics of SARS-CoV-2 and to analyze the impact of key nonpharmaceutical interventions such as isolation and screening program on the disease outcomes to the people of New Jersey, USA. We introduced a dynamic isolation of susceptible population with a constant (imposed) and infection oriented interventions. Epidemiological and demographic data are used to estimate the model parameters. The baseline case was explored further to showcase several critical and predictive scenarios.\u0000Results and analysis: The model simulations are in good agreement with the infection data for the period of 5 March 2020 to 31 January 2021. Dynamic isolation and screening program are found to be potential measures that can alter the course of epidemic. A 7% increase in isolation rate may result in a 31% reduction of epidemic peak whereas a 3 times increase in screening rate may reduce the epidemic peak by 35%. The model predicts that nearly 9.7% to 12% of the total population of New Jersey may become infected within the middle of July 2021 along with 24.6 to 27.3 thousand cumulative deaths. Within a wide spectrum of probable scenarios, there is a possibility of third wave \u0000Conclusion: Our findings could be informative to the public health community to contain the pandemic in the case of economy reopening under a limited or no vaccine coverage. Additional epidemic waves can be avoided by appropriate screening and isolation plans. ","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43907866","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}
We consider a model for an N × N lattice network of weakly coupled neural oscilla- tors with periodic boundary conditions (2D square torus), where the coupling between neurons is assumed to be within a von Neumann neighborhood of size r, denoted as von Neumann r-neighborhood. Using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (Ψh, Ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where Ψh and Ψv are not necessarily to be identical. Applying the Kronecker production representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., Ψh,Ψv are equal). We begin our analysis by deriving the precise conditions for stability of such cluster solutions with von Neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von Neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. This developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. Finally, numerical simulations are used to validate the above analytical results for various values of N and r by considering an inhibitory network of Morris-Lecar neurons.
{"title":"Cluster solutions in networks of weakly coupled oscillators on a 2D square torus","authors":"J. Culp","doi":"10.5206/mase/14147","DOIUrl":"https://doi.org/10.5206/mase/14147","url":null,"abstract":"\u0000 \u0000 \u0000We consider a model for an N × N lattice network of weakly coupled neural oscilla- tors with periodic boundary conditions (2D square torus), where the coupling between neurons is assumed to be within a von Neumann neighborhood of size r, denoted as von Neumann r-neighborhood. Using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (Ψh, Ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where Ψh and Ψv are not necessarily to be identical. Applying the Kronecker production representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., Ψh,Ψv are equal). We begin our analysis by deriving the precise conditions for stability of such cluster solutions with von Neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von Neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. This developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. Finally, numerical simulations are used to validate the above analytical results for various values of N and r by considering an inhibitory network of Morris-Lecar neurons. \u0000 \u0000 \u0000","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42125971","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 process of local cancer cell invasion of the surrounding tissue is key for the overall tumour growth and spread within the human body, the past 3 decades witnessing intense mathematical modelling efforts in these regards. However, for a deep understanding of the cancer invasion process these modelling studies require robust data assimilation approaches. While being of crucial importance in assimilating potential clinical data, the inverse problems approaches in cancer modelling are still in their early stages, with questions regarding the retrieval of the characteristics of tumour cells motility, cells mutations, and cells population proliferation, remaining widely open. This study deals with the identification and reconstruction of the usually unknown cancer cell proliferation law in cancer modelling from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. Considering two basic tumour configurations, associated with the case of one cancer cells population and two cancer cells subpopulations that exercise their dynamics within the extracellular matrix, we combine Tikhonov regularisation and gaussian mollification approaches with finite element and finite differences approximations to reconstruct the proliferation laws for each of these sub-populations from both exact and noisy measurements. Our inverse problem formulation is accompanied by numerical examples for the reconstruction of several proliferation laws used in cancer growth modelling.
{"title":"Inverse Reconstruction of Cell Proliferation Laws in Cancer Invasion Modelling","authors":"D. Trucu, Maher Alwuthaynani","doi":"10.5206/mase/13865","DOIUrl":"https://doi.org/10.5206/mase/13865","url":null,"abstract":"The process of local cancer cell invasion of the surrounding tissue is key for the overall tumour growth and spread within the human body, the past 3 decades witnessing intense mathematical modelling efforts in these regards. However, for a deep understanding of the cancer invasion process these modelling studies require robust data assimilation approaches. While being of crucial importance in assimilating potential clinical data, the inverse problems approaches in cancer modelling are still in their early stages, with questions regarding the retrieval of the characteristics of tumour cells motility, cells mutations, and cells population proliferation, remaining widely open. This study deals with the identification and reconstruction of the usually unknown cancer cell proliferation law in cancer modelling from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. Considering two basic tumour configurations, associated with the case of one cancer cells population and two cancer cells subpopulations that exercise their dynamics within the extracellular matrix, we combine Tikhonov regularisation and gaussian mollification approaches with finite element and finite differences approximations to reconstruct the proliferation laws for each of these sub-populations from both exact and noisy measurements. Our inverse problem formulation is accompanied by numerical examples for the reconstruction of several proliferation laws used in cancer growth modelling.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44706962","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 paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.
{"title":"$m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection","authors":"Changqing Ji, D. Zhu, Jingli Ren","doi":"10.5206/mase/13996","DOIUrl":"https://doi.org/10.5206/mase/13996","url":null,"abstract":"In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45423603","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 viral infection model with self-proliferation of cytotoxic T lymphocytes (CTLs) is proposed and its global dynamics is obtained. When the per capita self-proliferation rate of CTLs is sufficient large, an infection-free but immunity-activated equilibrium always exists and is globally asymptotically stable if the basic reproduction number of virus is less than a threshold value, which means that the immune effect still exists though virus be eliminated. Qualitative numerical simulations further indicate that the increase of per capita self-proliferation rate may lead to more severe infection outcome, which may provide insight into the failure of immune therapy.
{"title":"Global properties of a virus dynamics model with self-proliferation of CTLs","authors":"Cuicui Jiang, H. Kong, Guohong Zhang, Kaifa Wang","doi":"10.5206/MASE/13822","DOIUrl":"https://doi.org/10.5206/MASE/13822","url":null,"abstract":"A viral infection model with self-proliferation of cytotoxic T lymphocytes (CTLs) is proposed and its global dynamics is obtained. When the per capita self-proliferation rate of CTLs is sufficient large, an infection-free but immunity-activated equilibrium always exists and is globally asymptotically stable if the basic reproduction number of virus is less than a threshold value, which means that the immune effect still exists though virus be eliminated. Qualitative numerical simulations further indicate that the increase of per capita self-proliferation rate may lead to more severe infection outcome, which may provide insight into the failure of immune therapy.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42729522","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}
Generalized Nash Games are a powerful modelling tool, first introduced in the 1950's. They have seen some important developments in the past two decades. Separately, Evolutionary Games were introduced in the 1960's and seek to describe how natural selection can drive phenotypic changes in interacting populations. In this paper, we show how the dynamics of these two independently formulated models can be linked under a common framework and how this framework can be used to expand Evolutionary Games. At the center of this unified model is the Replicator Equation and the relationship we establish between it and the lesser known Projected Dynamical System.
{"title":"The replicator dynamics of generalized Nash games","authors":"Jason Lequyer","doi":"10.5206/MASE/11137","DOIUrl":"https://doi.org/10.5206/MASE/11137","url":null,"abstract":"Generalized Nash Games are a powerful modelling tool, first introduced in the 1950's. They have seen some important developments in the past two decades. Separately, Evolutionary Games were introduced in the 1960's and seek to describe how natural selection can drive phenotypic changes in interacting populations. In this paper, we show how the dynamics of these two independently formulated models can be linked under a common framework and how this framework can be used to expand Evolutionary Games. At the center of this unified model is the Replicator Equation and the relationship we establish between it and the lesser known Projected Dynamical System.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48460533","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 : 2021-04-09DOI: 10.1101/2021.04.06.21255039
A. M. G. Perez, D. A. Oluyori
In this study, we propose and analyze an extended SEIARD model with vaccination. We compute the control reproduction number Rc of our model and study the stability of equilibria. We show that the set of disease-free equilibria is locally asymptotically stable when Rc<1 and unstable when Rc>1, and we provide a sufficient condition for its global stability. Furthermore, we perform numerical simulations using the reported data of COVID-19 infections and vaccination in Mexico to study the impact of different vaccination, transmission and efficacy rates on the dynamics of the disease.
{"title":"An extended SEIARD model for COVID-19 vaccination in Mexico: analysis and forecast","authors":"A. M. G. Perez, D. A. Oluyori","doi":"10.1101/2021.04.06.21255039","DOIUrl":"https://doi.org/10.1101/2021.04.06.21255039","url":null,"abstract":"In this study, we propose and analyze an extended SEIARD model with vaccination. We compute the control reproduction number Rc of our model and study the stability of equilibria. We show that the set of disease-free equilibria is locally asymptotically stable when Rc<1 and unstable when Rc>1, and we provide a sufficient condition for its global stability. Furthermore, we perform numerical simulations using the reported data of COVID-19 infections and vaccination in Mexico to study the impact of different vaccination, transmission and efficacy rates on the dynamics of the disease.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43175073","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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