Pub Date : 2017-01-01DOI: 10.1080/23737867.2016.1264870
L. Allen, S. Jang, L. Roeger
Models of exponential growth, logistic growth and epidemics are common applications in undergraduate differential equation courses. The corresponding stochastic models are not part of these courses, although when population sizes are small their behaviour is often more realistic and distinctly different from deterministic models. For example, the randomness associated with births and deaths may lead to population extinction even in an exponentially growing population. Some background in continuous-time Markov chains and applications to populations, epidemics and cancer are presented with a goal to introduce this topic into the undergraduate mathematics curriculum that will encourage further investigation into problems on conservation, infectious diseases and cancer therapy. MATLAB programs for graphing sample paths of stochastic models are provided in the Appendix.
{"title":"Predicting population extinction or disease outbreaks with stochastic models","authors":"L. Allen, S. Jang, L. Roeger","doi":"10.1080/23737867.2016.1264870","DOIUrl":"https://doi.org/10.1080/23737867.2016.1264870","url":null,"abstract":"Models of exponential growth, logistic growth and epidemics are common applications in undergraduate differential equation courses. The corresponding stochastic models are not part of these courses, although when population sizes are small their behaviour is often more realistic and distinctly different from deterministic models. For example, the randomness associated with births and deaths may lead to population extinction even in an exponentially growing population. Some background in continuous-time Markov chains and applications to populations, epidemics and cancer are presented with a goal to introduce this topic into the undergraduate mathematics curriculum that will encourage further investigation into problems on conservation, infectious diseases and cancer therapy. MATLAB programs for graphing sample paths of stochastic models are provided in the Appendix.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"1 - 22"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2016.1264870","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47741939","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 : 2017-01-01DOI: 10.1080/23737867.2017.1300075
P. Salomonsky, R. Segal
In this work, we present a mathematical model, which accounts for two fundamental processes involved in the repair of an acute dermal wound. These processes include the inflammatory response and fibroplasia. Our system describes each of these events through the time evolution of four primary species or variables. These include the density of initial damage, inflammatory cells, fibroblasts and deposition of new collagen matrix. Since it is difficult to populate the equations of our model with coefficients that have been empirically derived, we fit these constants by carrying out a large number of simulations until there is reasonable agreement between the time response of the variables of our system and those reported by the literature for normal healing. Once a suitable choice of parameters has been made, we then compare simulation results with data obtained from clinical investigations. While more data is desired, we have a promising first step towards describing the primary events of wound repair within the confines of an implantable system.
{"title":"A mathematical system for human implantable wound model studies","authors":"P. Salomonsky, R. Segal","doi":"10.1080/23737867.2017.1300075","DOIUrl":"https://doi.org/10.1080/23737867.2017.1300075","url":null,"abstract":"In this work, we present a mathematical model, which accounts for two fundamental processes involved in the repair of an acute dermal wound. These processes include the inflammatory response and fibroplasia. Our system describes each of these events through the time evolution of four primary species or variables. These include the density of initial damage, inflammatory cells, fibroblasts and deposition of new collagen matrix. Since it is difficult to populate the equations of our model with coefficients that have been empirically derived, we fit these constants by carrying out a large number of simulations until there is reasonable agreement between the time response of the variables of our system and those reported by the literature for normal healing. Once a suitable choice of parameters has been made, we then compare simulation results with data obtained from clinical investigations. While more data is desired, we have a promising first step towards describing the primary events of wound repair within the confines of an implantable system.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"100 - 77"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1300075","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48486625","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 : 2017-01-01DOI: 10.1080/23737867.2017.1331712
D. M. Chan, C. Kent, D. Johnson
Abstract In this study, we use a discrete, two-patch population model of an Allee species to examine different methods in managing invasions. We first analytically examine the model to show the presence of the strong Allee effect, and then we numerically explore the model to test the effectiveness of different management strategies. As expected invasion is facilitated by lower Allee thresholds, greater carrying capacities and greater proportions of dispersers. These effects are interacting, however, and moderated by population growth rate. Using the gypsy moth as an example species, we demonstrate that the effectiveness of different invasion management strategies is context-dependent, combining complementary methods may be preferable, and the preferred strategy may differ geographically. Specifically, we find methods for restricting movement to be more effective in areas of contiguous habitat and high Allee thresholds, where methods involving mating disruptions and raising Allee thresholds are more effective in areas of high habitat fragmentation.
{"title":"Management of invasive Allee species","authors":"D. M. Chan, C. Kent, D. Johnson","doi":"10.1080/23737867.2017.1331712","DOIUrl":"https://doi.org/10.1080/23737867.2017.1331712","url":null,"abstract":"Abstract In this study, we use a discrete, two-patch population model of an Allee species to examine different methods in managing invasions. We first analytically examine the model to show the presence of the strong Allee effect, and then we numerically explore the model to test the effectiveness of different management strategies. As expected invasion is facilitated by lower Allee thresholds, greater carrying capacities and greater proportions of dispersers. These effects are interacting, however, and moderated by population growth rate. Using the gypsy moth as an example species, we demonstrate that the effectiveness of different invasion management strategies is context-dependent, combining complementary methods may be preferable, and the preferred strategy may differ geographically. Specifically, we find methods for restricting movement to be more effective in areas of contiguous habitat and high Allee thresholds, where methods involving mating disruptions and raising Allee thresholds are more effective in areas of high habitat fragmentation.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"167 - 186"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1331712","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42671092","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 : 2017-01-01DOI: 10.1080/23737867.2017.1302828
T. Beebe, S. Robertson
We develop a host–vector model of West Nile virus (WNV) transmission that incorporates multiple avian host species as well as host stage-structure (juvenile and adult stages), allowing for both species-specific and stage-specific biting rates of vectors on hosts. We use this ordinary differential equation model to explore WNV transmission dynamics that occur between vectors and multiple structured host populations as a result of heterogeneous biting rates on species and/or life stages. Our analysis shows that increased exposure of juvenile hosts generally results in larger outbreaks of WNV infectious vectors when compared to differential host species exposure. We also find that increased juvenile exposure is an important mechanism for determining the effect of species diversity on the disease risk of a community.
{"title":"A two-species stage-structured model for West Nile virus transmission","authors":"T. Beebe, S. Robertson","doi":"10.1080/23737867.2017.1302828","DOIUrl":"https://doi.org/10.1080/23737867.2017.1302828","url":null,"abstract":"We develop a host–vector model of West Nile virus (WNV) transmission that incorporates multiple avian host species as well as host stage-structure (juvenile and adult stages), allowing for both species-specific and stage-specific biting rates of vectors on hosts. We use this ordinary differential equation model to explore WNV transmission dynamics that occur between vectors and multiple structured host populations as a result of heterogeneous biting rates on species and/or life stages. Our analysis shows that increased exposure of juvenile hosts generally results in larger outbreaks of WNV infectious vectors when compared to differential host species exposure. We also find that increased juvenile exposure is an important mechanism for determining the effect of species diversity on the disease risk of a community.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"112 - 132"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1302828","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43367421","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 : 2017-01-01DOI: 10.1080/23737867.2017.1296383
E. Agyingi, T. Wiandt, M. Ngwa
We present a mathematical model of the transmission dynamics of two species of malaria with time lags. The model is equally applicable to two strains of a malaria species. The reproduction numbers of the two species are obtained and used as threshold parameters to study the stability and bifurcations of the equilibria of the model. We find that the model has a disease free equilibrium, which is a global attractor when the reproduction number of each species is less than one. Further, we observe that the non-disease free equilibrium of the model contains stability switches and Hopf bifurcations take place when the delays exceed the critical values.
{"title":"Stability and Hopf bifurcation of a two species malaria model with time delays","authors":"E. Agyingi, T. Wiandt, M. Ngwa","doi":"10.1080/23737867.2017.1296383","DOIUrl":"https://doi.org/10.1080/23737867.2017.1296383","url":null,"abstract":"We present a mathematical model of the transmission dynamics of two species of malaria with time lags. The model is equally applicable to two strains of a malaria species. The reproduction numbers of the two species are obtained and used as threshold parameters to study the stability and bifurcations of the equilibria of the model. We find that the model has a disease free equilibrium, which is a global attractor when the reproduction number of each species is less than one. Further, we observe that the non-disease free equilibrium of the model contains stability switches and Hopf bifurcations take place when the delays exceed the critical values.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"59 - 76"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1296383","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44707709","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 : 2017-01-01DOI: 10.1080/23737867.2017.1411843
H. Highlander
Abstract Independent of institution size and faculty research expectations, a growing number of colleges and universities encourage their undergraduates to engage in some form of research experience. To meet the demand of students seeking such experiences and to ensure these experiences are of high quality, it is imperative to have qualified mentors. While senior faculty rely on years of experience in mentoring research projects, professors stepping into these undergraduate mentoring roles at the graduate student or junior faculty level may not be as equipped to handle the potential hurdles unique to working with teams of undergraduates. This article is aimed at such an audience. Although much of the article is relevant to mentoring projects in any area of mathematics, some comments and suggestions are directed more to working with students in applied mathematics. This article includes advice gleaned from the National Science Foundation-sponsored Center for Undergraduate Research in Mathematics (CURM) faculty workshop in conjunction with personal experiences from the author, a CURM mini-grant recipient. The primary goals of the paper are to answer questions one might have when starting a project with undergraduates and to provide the reader with concrete steps to follow in planning and successfully completing such a project.
{"title":"Keys to successful mentoring of undergraduate research teams with an emphasis in applied mathematics research","authors":"H. Highlander","doi":"10.1080/23737867.2017.1411843","DOIUrl":"https://doi.org/10.1080/23737867.2017.1411843","url":null,"abstract":"Abstract Independent of institution size and faculty research expectations, a growing number of colleges and universities encourage their undergraduates to engage in some form of research experience. To meet the demand of students seeking such experiences and to ensure these experiences are of high quality, it is imperative to have qualified mentors. While senior faculty rely on years of experience in mentoring research projects, professors stepping into these undergraduate mentoring roles at the graduate student or junior faculty level may not be as equipped to handle the potential hurdles unique to working with teams of undergraduates. This article is aimed at such an audience. Although much of the article is relevant to mentoring projects in any area of mathematics, some comments and suggestions are directed more to working with students in applied mathematics. This article includes advice gleaned from the National Science Foundation-sponsored Center for Undergraduate Research in Mathematics (CURM) faculty workshop in conjunction with personal experiences from the author, a CURM mini-grant recipient. The primary goals of the paper are to answer questions one might have when starting a project with undergraduates and to provide the reader with concrete steps to follow in planning and successfully completing such a project.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"244 - 255"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1411843","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49592477","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 : 2017-01-01DOI: 10.1080/23737867.2017.1379890
Olivia Brozek, M. Glomski
Abstract Ebola virus disease (EVD) struck West Africa in 2013–2016 in an epidemic of unprecedented scope, with over 28000 cases and 11000 fatalities in the affected region. The protracted duration of the outbreak – more than two-and-one-half years of active transmission – raises questions about the persistence of EVD. In this brief paper, we qualitatively examine conditions supporting long-running EVD epidemics via a susceptible – exposed – infectious – recovered – deceased-infectious differential equations model that incorporates births and non disease-related deaths. We define an ‘effective epidemiological population’ to include contagious individuals recently deceased from the disease. Under a constant effective epidemiological population condition, we consider the basic reproductive number and use Lyapunov function arguments to establish conditions in the parameter space supporting an exchange of stability from the disease-free equilibrium to an endemic equilibrium.
{"title":"Conditions for endemicity: qualitative population dynamics in a long-running outbreak of Ebola virus disease","authors":"Olivia Brozek, M. Glomski","doi":"10.1080/23737867.2017.1379890","DOIUrl":"https://doi.org/10.1080/23737867.2017.1379890","url":null,"abstract":"Abstract Ebola virus disease (EVD) struck West Africa in 2013–2016 in an epidemic of unprecedented scope, with over 28000 cases and 11000 fatalities in the affected region. The protracted duration of the outbreak – more than two-and-one-half years of active transmission – raises questions about the persistence of EVD. In this brief paper, we qualitatively examine conditions supporting long-running EVD epidemics via a susceptible – exposed – infectious – recovered – deceased-infectious differential equations model that incorporates births and non disease-related deaths. We define an ‘effective epidemiological population’ to include contagious individuals recently deceased from the disease. Under a constant effective epidemiological population condition, we consider the basic reproductive number and use Lyapunov function arguments to establish conditions in the parameter space supporting an exchange of stability from the disease-free equilibrium to an endemic equilibrium.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"207 - 218"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1379890","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46526314","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 : 2017-01-01DOI: 10.1080/23737867.2017.1282843
Erin N. Bodine, A. Yust
The study of the Allee effect on the stability of equilibria of predator-prey systems is of recent interest to mathematicians, ecologists, and conservationists. Many theoretical models that include the Allee effect result in an unstable coexistence equilibrium. However, empirical evidence suggests that predator–prey systems exhibiting density-dependent growth at small population densities still can achieve coexistence in the long term. We review an often cited model that incorporates an Allee effect in the predator population resulting in an unstable coexistence equilibrium, and then present a novel extension to this model which includes a term modeling intraspecific competition within the predator population. The additional term penalizes predator population growth for large predator to prey density ratios. We use equilibrium analysis to define the regions in the parameter space where the coexistence equilibrium is stable, and show that there exist biologically reasonable parameter sets which produce a stable coexistence equilibrium for our model.
{"title":"Predator–prey dynamics with intraspecific competition and an Allee effect in the predator population","authors":"Erin N. Bodine, A. Yust","doi":"10.1080/23737867.2017.1282843","DOIUrl":"https://doi.org/10.1080/23737867.2017.1282843","url":null,"abstract":"The study of the Allee effect on the stability of equilibria of predator-prey systems is of recent interest to mathematicians, ecologists, and conservationists. Many theoretical models that include the Allee effect result in an unstable coexistence equilibrium. However, empirical evidence suggests that predator–prey systems exhibiting density-dependent growth at small population densities still can achieve coexistence in the long term. We review an often cited model that incorporates an Allee effect in the predator population resulting in an unstable coexistence equilibrium, and then present a novel extension to this model which includes a term modeling intraspecific competition within the predator population. The additional term penalizes predator population growth for large predator to prey density ratios. We use equilibrium analysis to define the regions in the parameter space where the coexistence equilibrium is stable, and show that there exist biologically reasonable parameter sets which produce a stable coexistence equilibrium for our model.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"23 - 38"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1282843","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42015969","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 : 2017-01-01DOI: 10.1080/23737867.2017.1289129
Emek Köse, S. Moore, Chinenye Ofodile, A. Radunskaya, Ellen R. Swanson, Elizabeth Zollinger
Therapeutic vaccines play a large role in the cast of immunotherapies that are now an essential component in most cancer treatment regimes. The complexity of the immune response and the ability of the tumour to mount a counter-offensive to this response have made it difficult to predict who will respond to what treatments, and for clinicians to optimise treatment strategies for individual patients. In this paper, we present a mathematical model that captures the dynamics of the adaptive response to an autologous whole-cell cancer vaccine, without some of the complexities of previous models that incorporate delays. Model simulations are compared to published experimental and clinical data, and used to discuss possible improvements to vaccine design.
{"title":"Immuno-kinetics of immunotherapy: dosing with DCs","authors":"Emek Köse, S. Moore, Chinenye Ofodile, A. Radunskaya, Ellen R. Swanson, Elizabeth Zollinger","doi":"10.1080/23737867.2017.1289129","DOIUrl":"https://doi.org/10.1080/23737867.2017.1289129","url":null,"abstract":"Therapeutic vaccines play a large role in the cast of immunotherapies that are now an essential component in most cancer treatment regimes. The complexity of the immune response and the ability of the tumour to mount a counter-offensive to this response have made it difficult to predict who will respond to what treatments, and for clinicians to optimise treatment strategies for individual patients. In this paper, we present a mathematical model that captures the dynamics of the adaptive response to an autologous whole-cell cancer vaccine, without some of the complexities of previous models that incorporate delays. Model simulations are compared to published experimental and clinical data, and used to discuss possible improvements to vaccine design.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"39 - 58"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1289129","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46696100","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 : 2017-01-01DOI: 10.1080/23737867.2017.1316528
Liang Sun, Junkoo Park, A. Barrera
Abstract Oxygen diffusion for time-dependent diffusion and consumption can be measured for small tissue regions containing a single capillary. An all or none model is reflected by myocardial infarction where necrotic regions are clearly demarcated. However if there is more than one capillary, the problem becomes very difficult; since the boundary of the ischemic area is no longer circular and is not known a priori. A geometric compartmental model using the Fick’s method will be presented for multi-capillary supply. Our method is to approach the steady state by a transient process, which paradoxically may be more efficient than the steady-state problem.
{"title":"A compartmental model for capillary supply","authors":"Liang Sun, Junkoo Park, A. Barrera","doi":"10.1080/23737867.2017.1316528","DOIUrl":"https://doi.org/10.1080/23737867.2017.1316528","url":null,"abstract":"Abstract Oxygen diffusion for time-dependent diffusion and consumption can be measured for small tissue regions containing a single capillary. An all or none model is reflected by myocardial infarction where necrotic regions are clearly demarcated. However if there is more than one capillary, the problem becomes very difficult; since the boundary of the ischemic area is no longer circular and is not known a priori. A geometric compartmental model using the Fick’s method will be presented for multi-capillary supply. Our method is to approach the steady state by a transient process, which paradoxically may be more efficient than the steady-state problem.","PeriodicalId":37222,"journal":{"name":"Letters in Biomathematics","volume":"4 1","pages":"133 - 147"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/23737867.2017.1316528","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60101957","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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