Bulletin of Mathematical Biology最新文献

From Trait-Structured Within-Host Dynamics to SIR Models: A Multiscale Framework With Re-Exposure. 从特征结构的宿主动力学到SIR模型：一个多尺度框架与再暴露。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-05-06 DOI: 10.1007/s11538-026-01647-x

Cyrille Kenne, Pascal Zongo

{"title":"From Trait-Structured Within-Host Dynamics to SIR Models: A Multiscale Framework With Re-Exposure.","authors":"Cyrille Kenne, Pascal Zongo","doi":"10.1007/s11538-026-01647-x","DOIUrl":"10.1007/s11538-026-01647-x","url":null,"abstract":"We present a threshold-based multiscale framework that links mechanistic within-host infection dynamics to a structured, SIR-like population model. Starting from a two-variable system for pathogen load and immune response that includes inoculum (Allee-like) thresholds and nonlinear immune activation, we derive mapping rules that classify continuous trajectories into four states: susceptible (S), infected with low immune protection ( <math><msup><mi>I</mi> <mo>-</mo></msup> </math> ), infected with high immune protection ( <math><msup><mi>I</mi> <mo>+</mo></msup> </math> ), and recovered (R). Here, \"infected\" refers to individuals with a detectable pathogen load. Unlike previous multiscale approaches, our framework integrates both scales into a single system: population compartments emerge by direct projection of within-host trajectories, avoiding ad hoc linking functions. We derive a next-generation operator for trait-structured re-exposure (local vs. global mixing) and an explicit expression for <math><msub><mi>R</mi> <mn>0</mn></msub> </math> under global mixing. Simulations reveal sharp clearance-persistence transitions driven by inoculum size and immune trait, and an emergent <math><mrow><mi>S</mi> <mspace></mspace> <mo>→</mo> <mspace></mspace> <msup><mi>I</mi> <mo>-</mo></msup> <mspace></mspace> <mo>→</mo> <mspace></mspace> <msup><mi>I</mi> <mo>+</mo></msup> <mspace></mspace> <mo>→</mo> <mspace></mspace> <mi>R</mi></mrow> </math> cascade. Under sharp thresholds and activation, chronic within-host equilibria can sustain infection even when <math> <mrow><msub><mi>R</mi> <mn>0</mn></msub> <mo><</mo> <mn>1</mn></mrow> </math> , producing backward-bifurcation-like behavior at the population level. The framework provides a consistent route from immunological heterogeneity to epidemic indicators, with implications for identifying chronic reservoirs, interpreting dose-response data, and estimating control thresholds directly from within-host measurements.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13149680/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147834030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Observer-Based Source Localization in Tree Infection Networks via Laplace Transforms. 基于拉普拉斯变换的树感染网络观测器源定位。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-27 DOI: 10.1007/s11538-026-01640-4

Graham Kesler O'Connor, Julia M Jess, Devlin Costello, Manuel E Lladser

引用次数: 0

Analysis of the Modulation of RAF Signaling by 14-3-3 Proteins. 14-3-3蛋白对RAF信号的调控分析。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-21 DOI: 10.1007/s11538-026-01634-2

Peter Carlip, Edward C Stites

{"title":"Analysis of the Modulation of RAF Signaling by 14-3-3 Proteins.","authors":"Peter Carlip, Edward C Stites","doi":"10.1007/s11538-026-01634-2","DOIUrl":"https://doi.org/10.1007/s11538-026-01634-2","url":null,"abstract":"The regulation of cellular biochemical signaling reactions includes the modulation of protein activity through a variety of processes. For example, signaling by the RAF kinases, which are key transmitters of extracellular growth signals downstream from the RAS GTPases, is modulated by dimerization, protein conformational changes, post-translational modifications, and protein-protein interactions. 14-3-3 proteins are known to play an important role in RAF signal regulation, and have the ability to stabilize both inactive (monomeric) and active (dimeric) states of RAF. It is poorly understood how these antagonistic roles ultimately modulate RAF signaling. To investigate, we develop a mathematical model of RAF activation with both roles of 14-3-3, perform algebraic and numeric analyses, and compare with available experimental data. We derive the conditions necessary to explain experimental observations that 14-3-3 overexpression activates RAF, and we show that even arbitrarily strong binding of 14-3-3 to RAF dimers alone could not necessarily explain this observation. Our integrated analysis also suggests that RAF-14-3-3 binding is relatively weak (significant amounts of RAF would remain unbound if only the first affinity were a factor), and instead that changing avidity more directly controls the bound fraction. Lastly we consider the limit at which RAF-14-3-3 interactions are driven solely by avidity, which allows for significant simplifications to the interaction model. Overall, our work presents a mathematical model that can serve as a foundational piece for future, extended, studies of signaling reactions involving regulated RAF kinase activity.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13099701/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147763331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Statistical and Structural Bias in Birth-Death Models. 出生-死亡模型中的统计和结构偏差。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-17 DOI: 10.1007/s11538-026-01644-0

Jeremy M Beaulieu, Brian C O'Meara

{"title":"Statistical and Structural Bias in Birth-Death Models.","authors":"Jeremy M Beaulieu, Brian C O'Meara","doi":"10.1007/s11538-026-01644-0","DOIUrl":"10.1007/s11538-026-01644-0","url":null,"abstract":"Accurate estimation of speciation ( <math><mi>λ</mi></math> ) and extinction ( <math><mi>μ</mi></math> ) rates from phylogenetic trees is central to studies of diversification, yet it remains unclear whether commonly used estimators are unbiased. Here we examine two sources of error: (1) statistical bias in the estimators themselves, and the (2) structural bias introduced by how small trees are handled in likelihood calculations. For the Yule process, we re-derive the expected bias of the standard estimator, showing that <math><mover><mi>λ</mi> <mo>^</mo></mover> </math> underestimates <math><mi>λ</mi></math> by a factor of <math><mrow><mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> <mo>/</mo> <mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </math> . Extending to the general birth-death model, we use symbolic regression to find functional forms that minimize the bias in both <math><mi>λ</mi></math> and <math><mi>μ</mi></math> . The best-performing correction for <math><mi>λ</mi></math> is identical to the Yule result, while the bias in <math><mi>μ</mi></math> depends on both sample size and the estimated extinction fraction ( <math><mrow><mi>μ</mi> <mo>/</mo> <mi>λ</mi></mrow> </math> ). Applying these corrections substantially improves the fit between the estimated and generating values. When these corrected estimators are used to derive other diversification-related parameters, turnover is nearly unbiased, but net diversification ( <math><mrow><mi>λ</mi> <mo>-</mo> <mi>μ</mi></mrow> </math> ) remains systematically underestimated due to the slight overestimation of <math><mi>μ</mi></math> . On the whole, these results begin to clarify the statistical and structural sources of bias in diversification rate estimation and provide a general framework for improving inference under birth-death models.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13090234/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147715855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Stochastic Analysis of Taxis and Kinesis Properties of Colonial Protozoa. 群体原生动物趋向性和运动特性的随机分析。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-13 DOI: 10.1007/s11538-026-01638-y

Yonatan L Ashenafi, Peter R Kramer

{"title":"Stochastic Analysis of Taxis and Kinesis Properties of Colonial Protozoa.","authors":"Yonatan L Ashenafi, Peter R Kramer","doi":"10.1007/s11538-026-01638-y","DOIUrl":"https://doi.org/10.1007/s11538-026-01638-y","url":null,"abstract":"Protozoan colonies undergo stimulus driven motion for purposes such as and nutrient acquisition. Colonial response to a stimulus is mediated through a mechanical aggregation of the response properties of members of the colony. We develop and apply asymptotic analysis to stochastic models for two separate classes of stimulus driven response of the constituent cells - taxis and kinesis. We investigate in particular the maintenance of effectiveness of taxis and kinesis in the transition from unicellular to multicellular organisms, using experimental observations of chemotaxis and aerotaxis of protozoa as a reference. Our taxis model based on a steering response of individual cells actually leads to a counterproductive drift of the colony down the stimulus gradient, together with a constructive drift up the gradient which is proportional to a measure of asymmetry of the flagellar placement. The strength of taxis drift up the stimulus gradient decreases with colony size while the counterproductive term does not, indicating a failure for colonial taxis based on a steering response of individual cells. Under a kinesis response of the cellular flagellar motion, enhancing the noise as the cell is facing away from the stimulus gradient, the colony does drift up the gradient with a speed independent of colony size, even under a completely symmetric placement of flagella.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147670708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Mathematical Model for Smooth Muscle Cell Phenotype Switching In Atherosclerotic Plaque. 动脉粥样硬化斑块中平滑肌细胞表型转换的数学模型

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-11 DOI: 10.1007/s11538-026-01645-z

Joseph P Ndenda, Michael G Watson, Ashish Misra, Mary R Myerscough

{"title":"A Mathematical Model for Smooth Muscle Cell Phenotype Switching In Atherosclerotic Plaque.","authors":"Joseph P Ndenda, Michael G Watson, Ashish Misra, Mary R Myerscough","doi":"10.1007/s11538-026-01645-z","DOIUrl":"10.1007/s11538-026-01645-z","url":null,"abstract":"Smooth muscle cells (SMCs) play a fundamental role in the development of atherosclerotic plaques. They ingest lipids in a similar way to monocyte-derived macrophages (MDMs) in the plaque. This can stimulate SMCs to undergo a phenotypic switch to a macrophage-like phenotype. We formulate an ordinary differential equation (ODE) model for the populations of SMCs, MDMs and smooth muscle cell-derived macrophages (SDMs) and the internalised lipid load in each population. We use this model to explore the effect on plaque fate of SMC phenotype switching. We find that when SMCs switch to a macrophage-like phenotype, there is an increase in the lipid quantity in the model plaque that is internalised inside cells. Additionally, removal of SMCs from the model plaque via phenotype switching reduces the number of SMCs in the fibrous cap, increases the lipid in the necrotic core, and increases plaque inflammation. These features are hallmarks of vulnerable plaques, whose rupture can cause heart attacks or strokes. When SDMs are highly proliferative or resistant to cell death, the model plaque becomes increasingly pathological. The model suggests that the switch of SMCs to a macrophage-like phenotype may drive the development of unstable and pathological plaques.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13068761/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147653713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Macrophage Polarization Determines Inflammation Amplification and Resolution. 巨噬细胞极化决定炎症放大和消退。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-10 DOI: 10.1007/s11538-026-01641-3

W El Hajj, N El Khatib, V Volpert

{"title":"Macrophage Polarization Determines Inflammation Amplification and Resolution.","authors":"W El Hajj, N El Khatib, V Volpert","doi":"10.1007/s11538-026-01641-3","DOIUrl":"https://doi.org/10.1007/s11538-026-01641-3","url":null,"abstract":"Sterile inflammation is a form of pathogen-free inflammation caused by non-infectious stimuli, such as injury, toxins, or cell stress. It serves to protect tissues, promote repair, and restore homeostasis. Like other forms of inflammation, sterile inflammation has three distinct stages: initiation, amplification, and resolution. The transition between the last two phases is particularly important for restoring tissue balance. If this process is disrupted, it can lead to the development of chronic inflammation and various diseases. Although the exact mechanisms that control this transition in the process of sterile inflammation are not fully understood, they remain a key area of research. In this work, we develop a mathematical model to explore the interplay between pro- and anti-inflammatory mediators in the process of sterile inflammation. The model consists of an integro-differential reaction-diffusion system that captures the spatial and temporal evolution of inflammation within tissue and blood. Specifically, the model tracks the concentrations of seventeen key-players involved in the inflammatory response, including circulating macrophages, DAMPs, RAMPs, inflammatory cytokines, and tissue-resident macrophages (TRMs). We focus on the role of macrophage polarization in determining the outcome of sterile inflammation, whether it leads to resolution or chronicity. According to this model, the anti-inflammatory processes, namely the inhibition in polarization of M1-type macrophages, can lead to the resolution of inflammation.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147644190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Vascular-Coupled Modeling of Treatment Resistance in Tyrosine Kinase Inhibitor Therapy: Parameter Estimation and Phase-Dependent Sensitivity Analysis. 酪氨酸激酶抑制剂治疗耐药的血管耦合建模：参数估计和相依赖敏感性分析。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-09 DOI: 10.1007/s11538-026-01643-1

Mohammed El Hammani, Sidi Mohamed Douiri, Imad El Harraki, Hamza Aguedjig

{"title":"Vascular-Coupled Modeling of Treatment Resistance in Tyrosine Kinase Inhibitor Therapy: Parameter Estimation and Phase-Dependent Sensitivity Analysis.","authors":"Mohammed El Hammani, Sidi Mohamed Douiri, Imad El Harraki, Hamza Aguedjig","doi":"10.1007/s11538-026-01643-1","DOIUrl":"https://doi.org/10.1007/s11538-026-01643-1","url":null,"abstract":"Acquired resistance to tyrosine kinase inhibitors (TKIs) remains the primary obstacle to long-term disease control in targeted cancer therapy, yet whether resistance emerges gradually through clonal selection or abruptly via mutation acquisition remains unclear. We develop a four-dimensional ordinary differential equation model coupling drug-sensitive and drug-resistant tumor populations with dynamic vascular support and explicit TKI pharmacokinetics. Mathematical analysis establishes solution positivity and uniform boundedness, characterizes all equilibrium states, and determines local stability conditions via Jacobian eigenvalue analysis, revealing threshold relationships between drug efficacy and evolutionary outcomes. We perform systematic parameter estimation using differential evolution on longitudinal tumor mass data from a gastrointestinal stromal tumor patient treated with imatinib. Models assuming continuous effective drug pressure fail systematically, with best fit achieving only coefficient of determination R-squared equals 0.721, unable to reproduce the observed 24-fold tumor mass increase during relapse. In striking contrast, incorporating a sigmoid resistance modulation function-where cytotoxicity progressively vanishes due to mutant clonal expansion near day 683-yields near-perfect agreement with R-squared equals 0.999, accurately capturing all three clinical phases. The estimated transition rate implies a rapid 10â€\"90 percent clonal takeover within approximately 2.5 days, providing quantitative evidence that explosive relapse reflects abrupt mutation acquisition rather than gradual selection.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147638045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Revisiting Turing's Chemical Basis of Morphogenesis. 回顾图灵的形态发生的化学基础。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-07 DOI: 10.1007/s11538-026-01629-z

John J Tyson

{"title":"Revisiting Turing's Chemical Basis of Morphogenesis.","authors":"John J Tyson","doi":"10.1007/s11538-026-01629-z","DOIUrl":"10.1007/s11538-026-01629-z","url":null,"abstract":"In a 1952 paper, Alan Turing showed that spatially distributed chemical reactions evolving in time by local kinetic rate laws and in space by unbiased molecular diffusion can develop a stable, time-independent, spatial pattern from a spatially homogeneous, steady state solution subjected to small, spatially periodic perturbations over a critical range of wavelengths. He proposed this mechanism as a potential chemical basis for biological morphogenesis. Although his proposal was initially ignored and remains controversial, Turing's idea still plays a major role in any discussion of spontaneous pattern formation in biological and chemical systems. Nevertheless, it is safe to say that few people have carefully studied his 1952 paper, which is notoriously difficult to read. For this reason, I am 'revisiting' Turing's paper to help new investigators to understand and appreciate his remarkable contribution to mathematical biology. Along the way, we shall resolve several 'peculiarities' of Turing's reaction mechanisms and numerical simulations, and place his work in context with later textbook examples. The relation of stationary Turing patterns to time-dependent traveling waves of chemical activity is also described.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13056747/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147627440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Clonal Diversity at Early Cancer Recurrence. 肿瘤早期复发的克隆多样性。

IF 2.2 4区数学

Bulletin of Mathematical Biology Pub Date : 2026-04-06 DOI: 10.1007/s11538-026-01617-3

Kevin Leder, Zicheng Wang

{"title":"Clonal Diversity at Early Cancer Recurrence.","authors":"Kevin Leder, Zicheng Wang","doi":"10.1007/s11538-026-01617-3","DOIUrl":"https://doi.org/10.1007/s11538-026-01617-3","url":null,"abstract":"Despite initial success, cancer therapies often fail due to the emergence of drug-resistant cells. In this study, we use a mathematical model to investigate how cancer evolves over time, specifically focusing on the state of the tumor when it recurs after treatment. We use a two-type birth-death process to capture the dynamics of both drug-sensitive and drug-resistant cells. Assuming resistant cells have equal fitness, we analyze the clonal diversity of drug-resistant cells at the time of cancer recurrence, which is defined as the first time the population size of drug-resistant cells exceeds a specified proportion of the initial population size of drug-sensitive cells. We examine two clonal diversity indices: the number of clones and the Simpson's Index. We calculate the expected values of these indices at the time of cancer recurrence. Additionally, we examine these two indices conditioned on early recurrence in the special case of a deterministically decaying sensitive population, with the aim of addressing the question of whether early recurrence is driven by a single mutation that generates an unusually large family of drug-resistant cells (corresponding to a low clonal diversity), or if it is due to the presence of an unusually large number of mutations causing drug resistance (corresponding to a high clonal diversity). Our findings, based on both indices, support the latter possibility. Furthermore, we demonstrate that the time of cancer recurrence can serve as a valuable indicator of clonal diversity, providing new insights into the evolutionary dynamics of recurrent cancers.","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 5","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147621994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0