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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
Recent ecological studies on predator-prey interactions has concentrated on determining the impacts of antipredator behavior due to fear of predators. These studies are mainly confined into one predator-one prey system. But in case of multiple predator attack on single prey species, fear mechanism is still unknown. The combined impact of multiple predator often cannot be anticipated from their independent effects. So coexistence of multiple predators and prey’s fitness becomes an important issue from an ecological point of view. Based on the above observations, we proposed and analyzed a model consisting of two competing predator sharing a common prey where prey’s reproduction rate is affected due to fear generated by the predators. We first study the boundedness, uniform persistence, stability and Hopf bifurcation of the deterministic model. Thereafter, we have investigated the existence and uniqueness of the global positive solution, boundedness, asymptotic stability of the stochastic model. Numerical examples are provided to support our obtained results.
{"title":"Effect of fear on two predator-one prey model in deterministic and fluctuating environment","authors":"D. Mukherjee","doi":"10.5206/MASE/13541","DOIUrl":"https://doi.org/10.5206/MASE/13541","url":null,"abstract":"Recent ecological studies on predator-prey interactions has concentrated on determining the impacts of antipredator behavior due to fear of predators. These studies are mainly confined into one predator-one prey system. But in case of multiple predator attack on single prey species, fear mechanism is still unknown. The combined impact of multiple predator often cannot be anticipated from their independent effects. So coexistence of multiple predators and prey’s fitness becomes an important issue from an ecological point of view. Based on the above observations, we proposed and analyzed a model consisting of two competing predator sharing a common prey where prey’s reproduction rate is affected due to fear generated by the predators. We first study the boundedness, uniform persistence, stability and Hopf bifurcation of the deterministic model. Thereafter, we have investigated the existence and uniqueness of the global positive solution, boundedness, asymptotic stability of the stochastic model. Numerical examples are provided to support our obtained results.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43359311","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 : 2021-02-12DOI: 10.1101/2021.02.10.21251500
Q. Griette, J. Demongeot, P. Magal
We provide a new method to analyze the COVID-19 cumulative reported cases data based on a two-step process: first we regularize the data by using a phenomenological model which takes into account the endemic or epidemic nature of the time period, then we use a mathematical model which reproduces the epidemic exactly. This allows us to derive new information on the epidemic parameters and to compute the effective basic reproductive ratio on a daily basis. Our method has the advantage of identifying robust trends in the number of new infectious cases and produces an extremely smooth reconstruction of the epidemic. The number of parameters required by the method is parsimonious: for the French epidemic between February 2020 and January 2021 we use only 11 parameters in total.
{"title":"A robust phenomenological approach to investigate COVID-19 data for France","authors":"Q. Griette, J. Demongeot, P. Magal","doi":"10.1101/2021.02.10.21251500","DOIUrl":"https://doi.org/10.1101/2021.02.10.21251500","url":null,"abstract":"We provide a new method to analyze the COVID-19 cumulative reported cases data based on a two-step process: first we regularize the data by using a phenomenological model which takes into account the endemic or epidemic nature of the time period, then we use a mathematical model which reproduces the epidemic exactly. This allows us to derive new information on the epidemic parameters and to compute the effective basic reproductive ratio on a daily basis. Our method has the advantage of identifying robust trends in the number of new infectious cases and produces an extremely smooth reconstruction of the epidemic. The number of parameters required by the method is parsimonious: for the French epidemic between February 2020 and January 2021 we use only 11 parameters in total.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44597590","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 traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.
{"title":"Traveling waves in cooperative predation: relaxation of sublinearity","authors":"Srijana Ghimire, Xiang-Sheng Wang","doi":"10.5206/MASE/13393","DOIUrl":"https://doi.org/10.5206/MASE/13393","url":null,"abstract":"In this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42185749","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}
Natural disasters occur across the globe, resulting in billions of dollars of damage each year. Effective preparation before a disaster can help to minimize damages, economic impact, and loss of human life. This paper uses a game theory framework to set up a leader-followers model for resource distribution to several geographic zones before an adverse event. The researchers model population members who may choose to prepare in advance of an event by acquiring supplies, whereas others may wait until the last minute. Failure to prepare in advance could result in a significant loss due to the chance that supplies may no longer be available. Numerical simulations are run to determine how the leader should distribute supplies to maximize the preparedness of the overall population. It was found that population size is a significant factor for supply distribution, but the behaviour of individuals within a zone is also important. Much of the current resource allocation research focuses on the logistics and economics of supply distribution, but this paper demonstrates that social aspects should also be considered.
{"title":"A leader-followers game of emergency preparedness for adverse events","authors":"M. Nahirniak, M. Cojocaru, T. Migot","doi":"10.5206/MASE/11093","DOIUrl":"https://doi.org/10.5206/MASE/11093","url":null,"abstract":"Natural disasters occur across the globe, resulting in billions of dollars of damage each year. Effective preparation before a disaster can help to minimize damages, economic impact, and loss of human life. This paper uses a game theory framework to set up a leader-followers model for resource distribution to several geographic zones before an adverse event. The researchers model population members who may choose to prepare in advance of an event by acquiring supplies, whereas others may wait until the last minute. Failure to prepare in advance could result in a significant loss due to the chance that supplies may no longer be available. Numerical simulations are run to determine how the leader should distribute supplies to maximize the preparedness of the overall population. It was found that population size is a significant factor for supply distribution, but the behaviour of individuals within a zone is also important. Much of the current resource allocation research focuses on the logistics and economics of supply distribution, but this paper demonstrates that social aspects should also be considered.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45815678","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 ecological communities, the behaviour of individuals and the interaction between species may change between seasons, yet this seasonal variation is often not represented explicitly in mathematical models. As global change is predicted to alter season length and other climatic aspects, such seasonal variation needs to be included in models in order to make reasonable predictions for community dynamics. The resulting mathematical descriptions are nonautonomous models with a large number of parameters, and are therefore challenging to analyze. We present a model for two predators and one prey, whereby one predator switches hunting behaviour to seasonally include alternative prey when available. We use a combination of temporal averaging and invasion analysis to derive simplified models and determine the behaviour of the system, in particular to gain insight into conditions under which the two predators can coexist in a changing climate. We compare our results with numerical simulations of the temporally varying model.
{"title":"Seasonal dynamics of a generalist and a specialist predator on a single prey","authors":"N. Bolohan, V. LeBlanc, F. Lutscher","doi":"10.5206/mase/13569","DOIUrl":"https://doi.org/10.5206/mase/13569","url":null,"abstract":"In ecological communities, the behaviour of individuals and the interaction between species may change between seasons, yet this seasonal variation is often not represented explicitly in mathematical models. As global change is predicted to alter season length and other climatic aspects, such seasonal variation needs to be included in models in order to make reasonable predictions for community dynamics. The resulting mathematical descriptions are nonautonomous models with a large number of parameters, and are therefore challenging to analyze. We present a model for two predators and one prey, whereby one predator switches hunting behaviour to seasonally include alternative prey when available. We use a combination of temporal averaging and invasion analysis to derive simplified models and determine the behaviour of the system, in particular to gain insight into conditions under which the two predators can coexist in a changing climate. We compare our results with numerical simulations of the temporally varying model.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70664502","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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