{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T01:27:37Z","timestamp":1780363657465,"version":"3.54.1"},"reference-count":74,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T00:00:00Z","timestamp":1699315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100018227","name":"National Research Foundation of Ukraine","doi-asserted-by":"publisher","award":["2021.01\/0311"],"award-info":[{"award-number":["2021.01\/0311"]}],"id":[{"id":"10.13039\/100018227","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The review is devoted to an analysis of mathematical models used for describing epidemic processes. Our main focus is on the models that are based on partial differential equations (PDEs), especially those that were developed and used for the COVID-19 pandemic modeling. Most of our attention is given to the studies in which not only results of numerical simulations are presented but analytical results as well. In particular, traveling fronts (waves), exact solutions, and the estimation of key epidemic parameters of the epidemic models with governing PDEs (typically reaction\u2013diffusion equations) are discussed. The review may serve as a valuable resource for researchers and practitioners in the field of mathematical modeling in epidemiology.<\/jats:p>","DOI":"10.3390\/sym15112025","type":"journal-article","created":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T11:23:39Z","timestamp":1699356219000},"page":"2025","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Reaction\u2013Diffusion Equations in Mathematical Models Arising in Epidemiology"],"prefix":"10.3390","volume":"15","author":[{"given":"Vasyl\u2019","family":"Davydovych","sequence":"first","affiliation":[{"name":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, 01601 Kyiv, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vasyl\u2019","family":"Dutka","sequence":"additional","affiliation":[{"name":"Bakul Institute for Superhard Materials, National Academy of Sciences of Ukraine, 2 Avtozavods\u2019ka Street, 04074 Kyiv, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1733-5240","authenticated-orcid":false,"given":"Roman","family":"Cherniha","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs\u2019ka Street, 01601 Kyiv, Ukraine"},{"name":"School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Brauer, F., and Castillo-Chavez, C. (2012). Mathematical Models in Population Biology and Epidemiology, Springer.","DOI":"10.1007\/978-1-4614-1686-9"},{"key":"ref_2","unstructured":"Diekmann, O., and Heesterbeek, J.A.P. (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley & Sons."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Keeling, M.J., and Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals, Princeton University Press.","DOI":"10.1515\/9781400841035"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (1989). Mathematical Biology, Springer.","DOI":"10.1007\/978-3-662-08539-4"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (2003). Mathematical Biology, II: Spatial Models and Biomedical Applications, Springer.","DOI":"10.1007\/b98869"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hadeler, K.P. (2017). Topics in Mathematical Biology, Berlin.","DOI":"10.1007\/978-3-319-65621-2"},{"key":"ref_7","unstructured":"Bailey, N.T.J. (1975). The Mathematical Theory of Infectious Diseases and Its Applications, Charles Griffin & Company."},{"key":"ref_8","first-page":"700","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc. A"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Dietz, K. (1976). The Incidence of Infectious Diseases Under the Influence of Seasonal Fluctuations, Springer. Lecture Notes in Biomathematics 11.","DOI":"10.1007\/978-3-642-93048-5_1"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1126\/science.7063839","article-title":"Directly transmitted infectious diseases: Control by vaccination","volume":"215","author":"Anderson","year":"1982","journal-title":"Science"},{"key":"ref_11","first-page":"55","article-title":"Contributions to the mathematical theory of epidemics. II\u2014The problem of endemicity","volume":"138","author":"Kermack","year":"1932","journal-title":"Proc. R. Soc. A"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Lin, F., Muthuraman, K., and Lawley, M. (2010). An optimal control theory approach to non-pharmaceutical interventions. BMC Infect. Dis., 10.","DOI":"10.1186\/1471-2334-10-32"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"100501","DOI":"10.1016\/j.epidem.2021.100501","article-title":"Rational evaluation of various epidemic models based on the COVID-19 data of China","volume":"37","author":"Yang","year":"2021","journal-title":"Epidemics"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Cherniha, R., and Davydovych, V. (2020). A mathematical model for the COVID-19 outbreak. arXiv.","DOI":"10.3390\/sym12060990"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Cherniha, R., and Davydovych, V. (2020). A mathematical model for the COVID-19 outbreak and its applications. Symmetry, 12.","DOI":"10.3390\/sym12060990"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Nesteruk, I. (2021). COVID19 Pandemic Dynamics, Springer Nature.","DOI":"10.1007\/978-981-33-6416-5"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"109761","DOI":"10.1016\/j.chaos.2020.109761","article-title":"Analysis and forecast of COVID-19 spreading in China, Italy and France","volume":"134","author":"Fanelli","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"104370","DOI":"10.1016\/j.rinp.2021.104370","article-title":"A study on the efficiency of the estimation models of COVID-19","volume":"26","author":"Alenezi","year":"2021","journal-title":"Results Phys."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"El Jai, M., Zhar, M., Ouazar, D., Akhrif, I., and Saidou, N. (2022). Socio-economic analysis of short-term trends of COVID-19: Modelling and data analytics. BMC Public Health, 22.","DOI":"10.1186\/s12889-022-13788-4"},{"key":"ref_20","first-page":"72","article-title":"Revisiting classical SIR modelling in light of the COVID-19 pandemic","volume":"8","author":"Kalachev","year":"2023","journal-title":"Infect. Dis. Model."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"19662","DOI":"10.1038\/s41598-020-76710-1","article-title":"Mathematical modelling of the dynamics and containment of COVID-19 in Ukraine","volume":"10","author":"Kyrychko","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"151","DOI":"10.3961\/jpmph.20.076","article-title":"Estimate of the basic reproduction number for COVID-19: A systematic review and meta-analysis","volume":"53","author":"Alimohamadi","year":"2020","journal-title":"J. Prev. Med. Public. Health"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"617841","DOI":"10.3389\/fevo.2020.617841","article-title":"Effects of demographic and weather parameters on COVID-19 basic reproduction number","volume":"8","author":"Salom","year":"2021","journal-title":"Front. Ecol. Environ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"132540","DOI":"10.1016\/j.physd.2020.132540","article-title":"Accurate closed-form solution of the SIR epidemic model","volume":"408","author":"Barlow","year":"2020","journal-title":"Phys. D"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10700-020-09342-9","article-title":"Numerical solution and parameter estimation for uncertain SIR model with application to COVID-19","volume":"20","author":"Chen","year":"2021","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zhu, X., Gao, B., Zhong, Y., Gu, C., and Choi, K.S. (2021). Extended Kalman filter based on stochastic epidemiological model for COVID-19 modelling. Comput. Biol. Med., 137.","DOI":"10.1016\/j.compbiomed.2021.104810"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1016\/j.jmaa.2003.10.044","article-title":"An integrable SIS model","volume":"290","author":"Nucci","year":"2004","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","first-page":"184","article-title":"Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates","volume":"236","author":"Harko","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2022.1.38","article-title":"Exact solution of the Susceptible-Infectious-Recovered-Deceased (SIRD) epidemic model","volume":"38","author":"Yoshida","year":"2022","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.jde.2023.01.017","article-title":"Existence of exact solution of the Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model","volume":"355","author":"Yoshida","year":"2023","journal-title":"J. Diff. Equ."},{"key":"ref_31","first-page":"653","article-title":"The explicit series solution of SIR and SIS epidemic models","volume":"215","author":"Khan","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"110949","DOI":"10.1016\/j.chaos.2021.110949","article-title":"Dynamics of SEIR epidemic model by optimal auxiliary functions method","volume":"147","author":"Marinca","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_33","unstructured":"Kendall, D.G. (1965). Mathematical models of the spread of infection. Math. Comput. Sci. Biol. Med., 213\u2013225."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"703","DOI":"10.2307\/3212374","article-title":"The initial geographical spread of host-vector and carrier-borne epidemics","volume":"10","author":"Radcliffe","year":"1973","journal-title":"J. Appl. Prob."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1038\/250726a0","article-title":"Geographic and temporal development of plagues","volume":"250","author":"Noble","year":"1974","journal-title":"Nature"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/S0022-5193(85)80276-9","article-title":"A simple model for the spatial spread and control of rabies","volume":"116","author":"Arcuri","year":"1985","journal-title":"J. Theor. Biol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1016\/0362-546X(84)90107-X","article-title":"Thresholds and travelling waves in an epidemic model for rabies","volume":"8","year":"1984","journal-title":"Nonlinear Anal."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1007\/s00028-019-00544-2","article-title":"Time periodic traveling wave solutions for a Kermack\u2013McKendrick epidemic model with diffusion and seasonality","volume":"20","author":"Zhang","year":"2020","journal-title":"J. Evol. Equ."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"8334","DOI":"10.3934\/mbe.2022388","article-title":"Modeling epidemic flow with fluid dynamics","volume":"19","author":"Cheng","year":"2022","journal-title":"Math. Biosci. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1007\/s12346-023-00810-2","article-title":"Influence of human behavior on COVID-19 dynamics based on a reaction-diffusion model","volume":"22","author":"Zhi","year":"2023","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"106617","DOI":"10.1016\/j.aml.2020.106617","article-title":"Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion","volume":"111","author":"Viguerie","year":"2021","journal-title":"Appl. Math. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1007\/s00466-020-01888-0","article-title":"Diffusion-reaction models in a continuum mechanics framework with application to COVID-19 modelling","volume":"66","author":"Viguerie","year":"2020","journal-title":"Comput. Mech."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1007\/s00466-021-01986-7","article-title":"Adaptive mesh refinement and coarsening for diffusion-reaction epidemiological models","volume":"67","author":"Grave","year":"2021","journal-title":"Comput. Mech."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"115541","DOI":"10.1016\/j.cma.2022.115541","article-title":"Modeling nonlocal behavior in epidemics via a reaction-diffusion system incorporating population movement along a network","volume":"401","author":"Grave","year":"2022","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"5897","DOI":"10.3934\/mbe.2019295","article-title":"Influence of spatial heterogeneous environment on long-term dynamics of a reaction-diffusion SVIR epidemic model with relapse","volume":"16","author":"Zhu","year":"2019","journal-title":"Math. Biosci. Eng."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.apm.2023.02.002","article-title":"Modeling and multi-objective optimal control of reaction-diffusion COVID-19 system due to vaccination and patient isolation","volume":"118","author":"Tu","year":"2023","journal-title":"Appl. Math. Model."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1515\/cmb-2020-0104","article-title":"A reaction-diffusion system to better comprehend the unlockdown: Application of SEIR-type model with diffusion to the spatial spread of COVID-19 in France","volume":"8","author":"Mammeri","year":"2020","journal-title":"Comput. Math. Biophys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1017\/S0956792521000231","article-title":"On a reaction-diffusion system modelling infectious diseases without lifetime immunity","volume":"33","author":"Yin","year":"2022","journal-title":"Euro. J. Appl. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/BF00160168","article-title":"Asymptotic behaviour of reaction-diffusion systems in population and epidemic models: The role of cross diffusion","volume":"32","author":"Capasso","year":"1994","journal-title":"J. Math. Biol."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1007\/s00028-010-0074-y","article-title":"A reaction-diffusion system with cross-diffusion modelling the spread of an epidemic disease","volume":"10","author":"Bendahmane","year":"2010","journal-title":"J. Evol. Equ."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1017\/S095679252100022X","article-title":"A reaction-diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications in the pandemic modelling","volume":"33","author":"Cherniha","year":"2022","journal-title":"Euro. J. Appl. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/0022-5193(71)90051-8","article-title":"Traveling bands of chemotactic bacteria: A Theoretical Analysis","volume":"30","author":"Keller","year":"1971","journal-title":"J. Theor. Biol."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0022-5193(79)90258-3","article-title":"Spatial segregation of interacting species","volume":"79","author":"Shigesada","year":"1979","journal-title":"J. Theoret. Biol."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"107313","DOI":"10.1016\/j.cnsns.2023.107313","article-title":"The Shigesada\u2013Kawasaki\u2013Teramoto model: Conditional symmetries, exact solutions and their properties","volume":"124","author":"Cherniha","year":"2023","journal-title":"Comm. Nonlinear Sci. Numer. Simulat."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.nonrwa.2014.10.006","article-title":"Existence result for an age-structured SIS epidemic model with spatial diffusion","volume":"23","author":"Kuniya","year":"2015","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s13160-018-0300-5","article-title":"Global behavior of SIS epidemic models with age structure and spatial heterogeneity","volume":"35","author":"Kuniya","year":"2018","journal-title":"Jpn J. Ind. Appl. Math."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Kang, H., and Ruan, S. (2021). Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion. J. Math. Biol., 83.","DOI":"10.1007\/s00285-021-01634-x"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s00033-022-01804-x","article-title":"Traveling waves of an epidemic model with general nonlinear incidence rate and infection-age structure","volume":"73","author":"Tian","year":"2022","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Loli Piccolomini, E., and Zama, F. (2020). Monitoring Italian COVID-19 spread by a forced SEIRD model. PLoS ONE, 15.","DOI":"10.1101\/2020.04.03.20049734"},{"key":"ref_60","first-page":"113","article-title":"Notice sur la loi que la population suit dans son accroissement","volume":"10","author":"Verhulst","year":"1838","journal-title":"Corr. Math. Phys."},{"key":"ref_61","unstructured":"(2020, May 01). Available online: https:\/\/www.worldometers.info\/coronavirus."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Cherniha, R., and Davydovych, V. (2017). Nonlinear Reaction-Diffusion Systems\u2014Conditional Symmetry, exact Solutions and Their Applications in Biology, Springer. Lecture Notes in Mathematics 2196.","DOI":"10.1007\/978-3-319-65467-6"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"106579","DOI":"10.1016\/j.cnsns.2022.106579","article-title":"Construction and application of exact solutions of the diffusive Lotka\u2013Volterra system: A review and new results","volume":"113","author":"Cherniha","year":"2022","journal-title":"Comm. Nonlinear Sci. Numer. Simulat."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1007\/s11071-021-06623-9","article-title":"Numerical simulation and stability analysis of a novel reaction-diffusion COVID-19 model","volume":"106","author":"Ahmed","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1080\/17513758.2012.742578","article-title":"Numerical simulation of a susceptible-exposed-infectious space-continuous model for the spread of rabies in raccoons across a realistic landscape","volume":"7","author":"Keller","year":"2013","journal-title":"J. Biol. Dyn."},{"key":"ref_66","first-page":"1046","article-title":"A spatial epidemic model with a moving boundary","volume":"6","author":"Zhuang","year":"2021","journal-title":"Infect. Dis. Model."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1093\/imammb\/4.2.109","article-title":"Analytical results on the stability of age-structured recurrent epidemic models","volume":"4","author":"Greenhalgh","year":"1987","journal-title":"IMA J. Math. Appl. Med. Biol."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1137\/0148085","article-title":"Endemic thresholds and stability in a class of age-structured epidemics","volume":"48","author":"Busenberg","year":"1988","journal-title":"SIAM J. Appl. Math."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/BF00178326","article-title":"Threshold and stability results for an age-structured epidemic model","volume":"28","author":"Inaba","year":"1990","journal-title":"J. Math. Biol."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"109971","DOI":"10.1016\/j.chaos.2020.109971","article-title":"Coronavirus pandemic: A predictive analysis of the peak outbreak epidemic in South Africa, Turkey, and Brazil","volume":"138","author":"Djilali","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.aej.2020.08.053","article-title":"Age-structured modelling of COVID-19 epidemic in the USA, UAE and Algeria","volume":"60","author":"Bentout","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1080\/17513758.2021.2020916","article-title":"Using an age-structured COVID-19 epidemic model and data to model virulence evolution in Wuhan, China","volume":"16","author":"Duan","year":"2022","journal-title":"J. Biol. Dyn."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"5110","DOI":"10.1002\/mma.7096","article-title":"Design and analysis of a discrete method for a time-delayed reaction\u2013diffusion epidemic model","volume":"44","author":"Ahmed","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1007\/s10473-021-0421-9","article-title":"A diffusive SVEIR epidemic model with time delay and general incidence","volume":"41","author":"Zhou","year":"2021","journal-title":"Acta. Math. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/2025\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:18:43Z","timestamp":1760131123000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/2025"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,7]]},"references-count":74,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["sym15112025"],"URL":"https:\/\/doi.org\/10.3390\/sym15112025","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,7]]}}}