Motivated by a recent field study [Nat. Commun. 7(2016), 10698] on the impact of fear of large carnivores on the populations in a cascading ecosystem of food chain type with the large carnivores as the top predator, in this paper we propose two model systems in the form of ordinary differential equations to mechanistically explore the cascade of such a fear effect. The models are of the Lotka-Volterra type, one is three imensional and the other four dimensional. The 3-D model only considers the cost of the anti-predation response reflected in the decrease of the production, while the 4-D model considers also the benefit of the response in reducing the predation rate, in addition to the cost by reducing the production. We perform a thorough analysis on the dynamics of the two models. The results reveal that the 3-D model and 4-model demonstrate opposite patterns for trophic cascade in terms of the dependence of population sizes for each species at the co-existence equilibrium on the anti-predation response level parameter, and such a difference is attributed to whether or not there is a benefit for the anti-predation response by the meso-carnivore species.
最近的一项实地研究[Nat. common . 7(2016)， 10698]关于大型食肉动物的恐惧对食物链型级联生态系统中大型食肉动物作为顶级捕食者的种群的影响，在本文中，我们提出了两个常微分方程形式的模型系统，以机械地探索这种恐惧效应的级联。模型是Lotka-Volterra型，一个是三维的，另一个是四维的。三维模型只考虑了反捕食反应的成本，即产量的减少，而4d模型除了考虑减少产量的成本外，还考虑了响应在减少捕食率方面的收益。我们对这两个模型的动力学进行了彻底的分析。结果表明，3-D模型和4- d模型在共存平衡下各物种种群规模对反捕食反应水平参数的依赖性方面表现出相反的营养级联模式，这种差异归因于中食性物种的反捕食反应是否有利。
{"title":"On mechanisms of trophic cascade caused by anti-predation response in food chain systems","authors":"Yang Wang, X. Zou","doi":"10.5206/mase/10739","DOIUrl":"https://doi.org/10.5206/mase/10739","url":null,"abstract":"Motivated by a recent field study [Nat. Commun. 7(2016), 10698] on the impact of fear of large carnivores on the populations in a cascading ecosystem of food chain type with the large carnivores as the top predator, in this paper we propose two model systems in the form of ordinary differential equations to mechanistically explore the cascade of such a fear effect. The models are of the Lotka-Volterra type, one is three imensional and the other four dimensional. The 3-D model only considers the cost of the anti-predation response reflected in the decrease of the production, while the 4-D model considers also the benefit of the response in reducing the predation rate, in addition to the cost by reducing the production. We perform a thorough analysis on the dynamics of the two models. The results reveal that the 3-D model and 4-model demonstrate opposite patterns for trophic cascade in terms of the dependence of population sizes for each species at the co-existence equilibrium on the anti-predation response level parameter, and such a difference is attributed to whether or not there is a benefit for the anti-predation response by the meso-carnivore species. \u0000 \u0000","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45658194","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 discuss the effects of movement and spatial heterogeneity on population dynamics via reaction–diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplanktongrowth, and spatial spread of epidemic disease. Open problems and conjectures are presented.P arts of this survey are adopted from the materials in [89,108,109], and some very recent progress are also included.
{"title":"Selected Topics on Reaction-Diffusion-Advection Models from Spatial Ecology","authors":"King-Yeung Lam, Shuang Liu, Y. Lou","doi":"10.5206/mase/10644","DOIUrl":"https://doi.org/10.5206/mase/10644","url":null,"abstract":"We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction–diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplanktongrowth, and spatial spread of epidemic disease. Open problems and conjectures are presented.P arts of this survey are adopted from the materials in [89,108,109], and some very recent progress are also included.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43441601","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}
Replicating oncolytic viruses provide promising treatment strategies against cancer. However, the success of these viral therapies depends mainly on the complex interactions between the virus particles and the host immune cells. Among these immune cells, macrophages represent one of the first line of defence against viral infections. In this paper, we consider a mathematical model that describes the interactions between a commonly-used oncolytic virus, the Vesicular Stomatitis Virus (VSV), and two extreme types of macrophages: the pro-inflammatory M1 cells (which seem to resist infection with VSV) and the anti-inflammatory M2 cells (which can be infected with VSV). We first show the existence of bounded solutions for this differential equations model. Then we investigate the long-term behaviour of the model by focusing on steady states and limit cycles, and study changes in this long-term dynamics as we vary different model parameters. Moreover, through sensitivity analysis we show that the parameters that have the highest impact on the level of virus particles in the system are the viral burst size (from infected macrophages), the virus infection rate, the M1$to$M2 polarisation rate, and the M1-induced anti-viral immunity.
{"title":"A mathematical model for the role of macrophages in the persistence and elimination of oncolytic viruses","authors":"Nada Almuallem, R. Eftimie","doi":"10.5206/mase/8543","DOIUrl":"https://doi.org/10.5206/mase/8543","url":null,"abstract":"Replicating oncolytic viruses provide promising treatment strategies against cancer. However, the success of these viral therapies depends mainly on the complex interactions between the virus particles and the host immune cells. Among these immune cells, macrophages represent one of the first line of defence against viral infections. In this paper, we consider a mathematical model that describes the interactions between a commonly-used oncolytic virus, the Vesicular Stomatitis Virus (VSV), and two extreme types of macrophages: the pro-inflammatory M1 cells (which seem to resist infection with VSV) and the anti-inflammatory M2 cells (which can be infected with VSV). We first show the existence of bounded solutions for this differential equations model. Then we investigate the long-term behaviour of the model by focusing on steady states and limit cycles, and study changes in this long-term dynamics as we vary different model parameters. Moreover, through sensitivity analysis we show that the parameters that have the highest impact on the level of virus particles in the system are the viral burst size (from infected macrophages), the virus infection rate, the M1$to$M2 polarisation rate, and the M1-induced anti-viral immunity. ","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42219865","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}
For a variety of tick species, the resistance, behavioural and immunological response of hosts has been reported in the biological literature but its impact on tick population dynamics has not been mathematically formulated and analyzed using dynamical models reflecting the full biological stages of ticks. Here we develop and simulate a delay differential equation model, with a particular focus on resistance resulting in grooming behaviour. We calculate the basic reproduction number using the spectral analysis of delay differential equations with positive feedback, and establish the existence and uniqueness of a positive equilibrium when the basic reproduction number exceeds unit. We also conduct numerical and sensitivity analysis about the dependence of this positive equilibrium on the the parameter relevant to grooming behaviour. We numerically obtain the relationship between grooming behaviour and equilibrium value at different stages.
{"title":"Modeling the impact of host resistance on structured tick population dynamics","authors":"Mahnaz Alavinejad, J. Sadiku, Jianhong Wu","doi":"10.5206/mase/10508","DOIUrl":"https://doi.org/10.5206/mase/10508","url":null,"abstract":"For a variety of tick species, the resistance, behavioural and immunological response of hosts has been reported in the biological literature but its impact on tick population dynamics has not been mathematically formulated and analyzed using dynamical models reflecting the full biological stages of ticks. Here we develop and simulate a delay differential equation model, with a particular focus on resistance resulting in grooming behaviour. We calculate the basic reproduction number using the spectral analysis of delay differential equations with positive feedback, and establish the existence and uniqueness of a positive equilibrium when the basic reproduction number exceeds unit. We also conduct numerical and sensitivity analysis about the dependence of this positive equilibrium on the the parameter relevant to grooming behaviour. We numerically obtain the relationship between grooming behaviour and equilibrium value at different stages.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47181988","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}
M. Al-arydah, S. Greenhalgh, J. Munganga, Robert J. Smith
The law of mass action is used to govern interactions between susceptible and infected individuals in a variety of infectious disease models. However, the commonly used version is a simplification of the version originally used to describe chemical reactions. We reformulate a general disease model using the chemical-reaction definition of mass action incorporating both an altered transmission term and an altered recovery term in the form of positive exponents. We examine the long-term outcome as these exponents vary. For many scenarios, the reproduction number is either 0 or $infty$, while it obtains finite values only for certain combinations. We found conditions under which endemic equilibria exist and are unique for a variety of possible exponents. We also determined circumstances under which backward bifurcations are possible or do not occur. The simplified form of mass action may be masking generalised behaviour that may result in practice if these exponents ``fluctuate'' around 1. This may lead to a loss of predictability in some models.
{"title":"Applying the chemical-reaction definition of mass action to infectious disease modelling","authors":"M. Al-arydah, S. Greenhalgh, J. Munganga, Robert J. Smith","doi":"10.5206/mase/9372","DOIUrl":"https://doi.org/10.5206/mase/9372","url":null,"abstract":"The law of mass action is used to govern interactions between susceptible and infected individuals in a variety of infectious disease models. However, the commonly used version is a simplification of the version originally used to describe chemical reactions. We reformulate a general disease model using the chemical-reaction definition of mass action incorporating both an altered transmission term and an altered recovery term in the form of positive exponents. We examine the long-term outcome as these exponents vary. For many scenarios, the reproduction number is either 0 or $infty$, while it obtains finite values only for certain combinations. We found conditions under which endemic equilibria exist and are unique for a variety of possible exponents. We also determined circumstances under which backward bifurcations are possible or do not occur. The simplified form of mass action may be masking generalised behaviour that may result in practice if these exponents ``fluctuate'' around 1. This may lead to a loss of predictability in some models.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44296914","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}
Jun Chen, K. Messan, Marisabel Rodriguez Messan, G. DeGrandi-Hoffman, Dingyong Bai, Yun Kang
Western honeybees (Apis Mellifera) serve extremely important roles in our ecosystem and economics as they are responsible for pollinating $ 215 billion dollars annually over the world. Unfortunately, honeybee population and their colonies have been declined dramatically. The purpose of this article is to explore how we should model honeybee population with age structure and validate the model using empirical data so that we can identify different factors that lead to the survival and healthy of the honeybee colony. Our theoretical study combined with simulations and data validation suggests that the proper age structure incorporated in the model and seasonality are important for modeling honeybee population. Specifically, our work implies that the model assuming that (1) the adult bees are survived from the egg population rather than the brood population; and (2) seasonality in the queen egg laying rate, give the better fit than other honeybee models. The related theoretical and numerical analysis of the most fit model indicate that (a) the survival of honeybee colonies requires a large queen egg-laying rate and smaller values of the other life history parameter values in addition to proper initial condition; (b) both brood and adult bee populations are increasing with respect to the increase in the egg-laying rate and the decreasing in other parameter values; and (c) seasonality may promote/suppress the survival of the honeybee colony.
{"title":"How to model honeybee population dynamics: stage structure and seasonality","authors":"Jun Chen, K. Messan, Marisabel Rodriguez Messan, G. DeGrandi-Hoffman, Dingyong Bai, Yun Kang","doi":"10.5206/mase/10559","DOIUrl":"https://doi.org/10.5206/mase/10559","url":null,"abstract":"Western honeybees (Apis Mellifera) serve extremely important roles in our ecosystem and economics as they are responsible for pollinating $ 215 billion dollars annually over the world. Unfortunately, honeybee population and their colonies have been declined dramatically. The purpose of this article is to explore how we should model honeybee population with age structure and validate the model using empirical data so that we can identify different factors that lead to the survival and healthy of the honeybee colony. Our theoretical study combined with simulations and data validation suggests that the proper age structure incorporated in the model and seasonality are important for modeling honeybee population. Specifically, our work implies that the model assuming that (1) the adult bees are survived from the egg population rather than the brood population; and (2) seasonality in the queen egg laying rate, give the better fit than other honeybee models. The related theoretical and numerical analysis of the most fit model indicate that (a) the survival of honeybee colonies requires a large queen egg-laying rate and smaller values of the other life history parameter values in addition to proper initial condition; (b) both brood and adult bee populations are increasing with respect to the increase in the egg-laying rate and the decreasing in other parameter values; and (c) seasonality may promote/suppress the survival of the honeybee colony. ","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47224548","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}
Mycobacterium tuberculosis infection can lead to different disease outcomes, we analyze awith-in host tuberculosis infection model considering interactions between macrophages, T lym-phocytes, and tuberculosis bacteria to understand the dynamics of disease progression. Fourcoexisting equilibria that reflect TB disease dynamics are present: clearance, latency, and pri-mary disease, with low and high pathogen loads. We also derive the conditions for backwardand forward bifurcations and for global stable disease free equilibrium, which affect how thedisease progresses. Numerical bifurcation analysis and simulations elucidate the dynamics offast and slow disease progression.
{"title":"Analysis of solutions and disease progressions for a within-host Tuberculosis model","authors":"Wenjing Zhang, Federico Frascoli, J. Heffernan","doi":"10.5206/mase/10221","DOIUrl":"https://doi.org/10.5206/mase/10221","url":null,"abstract":"Mycobacterium tuberculosis infection can lead to different disease outcomes, we analyze awith-in host tuberculosis infection model considering interactions between macrophages, T lym-phocytes, and tuberculosis bacteria to understand the dynamics of disease progression. Fourcoexisting equilibria that reflect TB disease dynamics are present: clearance, latency, and pri-mary disease, with low and high pathogen loads. We also derive the conditions for backwardand forward bifurcations and for global stable disease free equilibrium, which affect how thedisease progresses. Numerical bifurcation analysis and simulations elucidate the dynamics offast and slow disease progression.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49397674","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}
Mathematical modelling has been proven to be useful in understanding some problems from biological science, provided that it is used properly. However, it has also attracted some criticisms as partially presented in a recent opinion article cite{Gilbert2018} from biological community. This note intends to clarify some confusion and misunderstanding in regard to mathematically modelling by commenting on those critiques raised in cite{Gilbert2018}, with a hope of initiating some further discussion so that both applied mathematicians and biologist can better use mathematical modelling and better understand the results from modelling.
{"title":"Viewpoints on modelling: Comments on \"Achilles and the tortoise: Some caveats to mathematical modelling in biology\"","authors":"J. Lei","doi":"10.5206/mase/10267","DOIUrl":"https://doi.org/10.5206/mase/10267","url":null,"abstract":"Mathematical modelling has been proven to be useful in understanding some problems from biological science, provided that it is used properly. However, it has also attracted some criticisms as partially presented in a recent opinion article cite{Gilbert2018} from biological community. This note intends to clarify some confusion and misunderstanding in regard to mathematically modelling by commenting on those critiques raised in cite{Gilbert2018}, with a hope of initiating some further discussion so that both applied mathematicians and biologist can better use mathematical modelling and better understand the results from modelling.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45875970","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 consider the following system of delayed differentialequations,[left{begin {array}{rcl}frac {dS(t)}{dt} & = & sigma phi-beta S(t)I(t-tau)-eta S(t),frac {dI(t)}{dt} & = & sigma(1-phi)+betaS(t)I(t-tau)-(eta+omega)I(t),end {array}right.]which can be used to model plant diseases. Here $phiin (0,1]$,$tauge 0$, and all other parameters are positive. The case where $phi=1$ is well studiedand there is a threshold dynamics. Thesystem always has a disease-free equilibrium, which is globallyasymptotically stable if the basic reproduction number $R_0triangleqfrac{betasigma}{eta(eta+omega)}le 1$ and is unstable if$R_0>1$; when $R_0>1$, the system also has a unique endemic equilibrium,which is globally asymptotically stable. In this paper, we study thecase where $phiin (0,1)$. It turns out that the system only has anendemic equilibrium, which is globally asymptotically stable. Thelocal stability is established by the linearizationmethod while the global attractivity is obtained by the Lyapunovfunctional approach. The theoretical results are illustrated withnumerical simulations.
{"title":"Global asymptotic stability of a delayed plant disease model","authors":"Yuming Chen, Chongwu Zheng","doi":"10.5206/mase/9451","DOIUrl":"https://doi.org/10.5206/mase/9451","url":null,"abstract":"In this paper, we consider the following system of delayed differentialequations,[left{begin {array}{rcl}frac {dS(t)}{dt} & = & sigma phi-beta S(t)I(t-tau)-eta S(t),frac {dI(t)}{dt} & = & sigma(1-phi)+betaS(t)I(t-tau)-(eta+omega)I(t),end {array}right.]which can be used to model plant diseases. Here $phiin (0,1]$,$tauge 0$, and all other parameters are positive. The case where $phi=1$ is well studiedand there is a threshold dynamics. Thesystem always has a disease-free equilibrium, which is globallyasymptotically stable if the basic reproduction number $R_0triangleqfrac{betasigma}{eta(eta+omega)}le 1$ and is unstable if$R_0>1$; when $R_0>1$, the system also has a unique endemic equilibrium,which is globally asymptotically stable. In this paper, we study thecase where $phiin (0,1)$. It turns out that the system only has anendemic equilibrium, which is globally asymptotically stable. Thelocal stability is established by the linearizationmethod while the global attractivity is obtained by the Lyapunovfunctional approach. The theoretical results are illustrated withnumerical simulations.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49640073","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 statistical theory of non-equilibrium fluctuation in damped Volterra-Lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment has been presented. The method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. The characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response.
{"title":"A prey-predator system with herd behaviour of prey in a rapidly fluctuating environment","authors":"G. Samanta, A. Mondal, D. Sahoo, P. Dolai","doi":"10.5206/mase/8196","DOIUrl":"https://doi.org/10.5206/mase/8196","url":null,"abstract":"A statistical theory of non-equilibrium fluctuation in damped Volterra-Lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment has been presented. The method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. The characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47893584","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：