{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:51:51Z","timestamp":1760143911934,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,2]],"date-time":"2024-03-02T00:00:00Z","timestamp":1709337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"],"award-info":[{"award-number":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"]}]},{"name":"Science and Technology Project of Education Department of Gansu Province","award":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"],"award-info":[{"award-number":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"]}]},{"name":"Gansu Provincial Science and Technology Plan Project, China","award":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"],"award-info":[{"award-number":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"]}]},{"name":"Special Funds for Guiding Local Scientific and Technological Development by the Central Government","award":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"],"award-info":[{"award-number":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"]}]},{"name":"Natural Science Foundation of Gansu Province, China","award":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"],"award-info":[{"award-number":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"]}]},{"name":"Support Project for Youth Doctor in Colleges and Universities of Gansu Province","award":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"],"award-info":[{"award-number":["62062049","62366028","2023-25","21ZD8RA008","22ZY1QA005","23JRRA1688","2023QB-038"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In order to investigate the impact of two immunization strategies\u2014vaccination targeting susceptible individuals to reduce their infection rate and clinical medical interventions targeting infected individuals to enhance their recovery rate\u2014on the spread of infectious diseases in complex networks, this study proposes a bilinear SIR infectious disease model that considers bidirectional immunization. By analyzing the conditions for the existence of endemic equilibrium points, we derive the basic reproduction numbers and outbreak thresholds for both homogeneous and heterogeneous networks. The epidemic model is then reconstructed and extensively analyzed using continuous-time Markov chain (CTMC) methods. This analysis includes the investigation of transition probabilities, transition rate matrices, steady-state distributions, and the transition probability matrix based on the embedded chain. In numerical simulations, a notable concordance exists between the outcomes of CTMC and mean-field (MF) simulations, thereby substantiating the efficacy of the CTMC model. Moreover, the CTMC-based model adeptly captures the inherent stochastic fluctuation in the disease transmission, which is consistent with the mathematical properties of Markov chains. We further analyze the relationship between the system\u2019s steady-state infection density and the immunization rate through MCS. The results suggest that the infection density decreases with an increase in the immunization rate among susceptible individuals. The current research results will enhance our understanding of infectious disease transmission patterns in real-world scenarios, providing valuable theoretical insights for the development of epidemic prevention and control strategies.<\/jats:p>","DOI":"10.3390\/e26030227","type":"journal-article","created":{"date-parts":[[2024,3,4]],"date-time":"2024-03-04T05:31:12Z","timestamp":1709530272000},"page":"227","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Dynamical Analysis of an Improved Bidirectional Immunization SIR Model in Complex Network"],"prefix":"10.3390","volume":"26","author":[{"given":"Shixiang","family":"Han","sequence":"first","affiliation":[{"name":"School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China"},{"name":"Key Laboratory of Media Convergence Technology and Communication, Lanzhou 730070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanghui","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China"},{"name":"Key Laboratory of Media Convergence Technology and Communication, Lanzhou 730070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huayan","family":"Pei","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China"},{"name":"Key Laboratory of Media Convergence Technology and Communication, Lanzhou 730070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenwen","family":"Chang","sequence":"additional","affiliation":[{"name":"School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China"},{"name":"Key Laboratory of Media Convergence Technology and Communication, Lanzhou 730070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1098\/rspa.1927.0118","article-title":"A contribution to the mathematical theory of epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1098\/rspa.1932.0171","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. Lond. Ser. A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/S0025-5564(01)00049-9","article-title":"Modeling and analysis of a predator\u2013prey model with disease in the prey","volume":"171","author":"Xiao","year":"2001","journal-title":"Math. Biosci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/S0025-5564(01)00057-8","article-title":"Models for transmission of disease with immigration of infectives","volume":"171","author":"Brauer","year":"2001","journal-title":"Math. Biosci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1016\/S1468-1218(03)00019-1","article-title":"Stability analysis for differential infectivity epidemic models","volume":"4","author":"Ma","year":"2003","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1137\/S0036144500371907","article-title":"The mathematics of infectious diseases","volume":"42","author":"Hethcote","year":"2000","journal-title":"SIAM Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0025-5564(00)00025-0","article-title":"An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations","volume":"167","author":"Hyman","year":"2000","journal-title":"Math. Biosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1007\/s002850050194","article-title":"Interaction of maturation delay and nonlinear birth in population and epidemic models","volume":"39","author":"Cooke","year":"1999","journal-title":"J. Math. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3200","DOI":"10.1103\/PhysRevLett.86.3200","article-title":"Epidemic spreading in scale-free networks","volume":"86","author":"Vespignani","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"066117","DOI":"10.1103\/PhysRevE.63.066117","article-title":"Epidemic dynamics and endemic states in complex networks","volume":"63","author":"Vespignani","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"068301","DOI":"10.1103\/PhysRevLett.124.068301","article-title":"Predicting the speed of epidemics spreading in networks","volume":"124","author":"Moore","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1042","DOI":"10.1016\/j.cnsns.2013.08.033","article-title":"Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks","volume":"19","author":"Li","year":"2014","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/j.physa.2018.02.065","article-title":"Global stability of an SIR model with differential infectivity on complex networks","volume":"499","author":"Yuan","year":"2018","journal-title":"Phys. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107206","DOI":"10.1016\/j.cnsns.2023.107206","article-title":"Stability analysis and optimal control in an epidemic model on directed complex networks with nonlinear incidence","volume":"121","author":"Yang","year":"2023","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1140\/epjb\/s10051-023-00513-2","article-title":"Impact of contact rate on epidemic spreading in complex networks","volume":"96","author":"Pei","year":"2023","journal-title":"Eur. Phys. J. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"107116","DOI":"10.1016\/j.cnsns.2023.107116","article-title":"Dynamical analysis of an immumo-epidemiological coupled system on complex networks","volume":"119","author":"Yang","year":"2023","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.physa.2016.10.004","article-title":"Dynamics of epidemic spreading with vaccination: Impact of social pressure and engagement","volume":"467","author":"Pires","year":"2017","journal-title":"Phys. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"050202","DOI":"10.1088\/1674-1056\/ac3d82","article-title":"Correlation and trust mechanism-based rumor propagation model in complex social networks","volume":"31","author":"Sun","year":"2022","journal-title":"Chin. Phys. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"118904","DOI":"10.1088\/1674-1056\/23\/11\/118904","article-title":"Global stability of a susceptible-infected-susceptible epidemic model on networks with individual awareness","volume":"23","author":"Li","year":"2014","journal-title":"Chin. Phys. B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"107183","DOI":"10.1016\/j.cnsns.2023.107183","article-title":"SIR dynamics with infection age in complex heterogeneous networks","volume":"121","author":"Li","year":"2023","journal-title":"Commun. Nonlinear Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"121953","DOI":"10.1016\/j.physa.2019.121953","article-title":"Dynamical analysis of a rumor spreading model with self-discrimination and time delay in complex networks","volume":"533","author":"Zhu","year":"2019","journal-title":"Phys. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.physa.2015.12.078","article-title":"Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks","volume":"448","author":"Huo","year":"2016","journal-title":"Phys. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.physa.2014.05.034","article-title":"Global stability and optimal control of an SIRS epidemic model on heterogeneous networks","volume":"410","author":"Chen","year":"2014","journal-title":"Phys. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"062805","DOI":"10.1103\/PhysRevE.88.062805","article-title":"Behavior of susceptible-vaccinated\u2013infected\u2013recovered epidemics with diversity in the infection rate of individuals","volume":"88","author":"Cai","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.chaos.2014.04.005","article-title":"Effect of vaccination strategies on the dynamic behavior of epidemic spreading and vaccine coverage","volume":"62","author":"Cai","year":"2014","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","first-page":"128","article-title":"A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis","volume":"2","author":"Allen","year":"2017","journal-title":"Infect. Dis. Model."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.mbs.2013.02.006","article-title":"Relations between deterministic and stochastic thresholds for disease extinction in continuous-and discrete-time infectious disease models","volume":"243","author":"Allen","year":"2013","journal-title":"Math. Biosci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"036105","DOI":"10.1103\/PhysRevE.84.036105","article-title":"Nonperturbative heterogeneous mean-field approach to epidemic spreading in complex networks","volume":"84","author":"Moreno","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"052803","DOI":"10.1103\/PhysRevE.90.052803","article-title":"Effective degree Markov-chain approach for discrete-time epidemic processes on uncorrelated networks","volume":"90","author":"Cai","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Watmough","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1021\/j100540a008","article-title":"Exact stochastic simulation of coupled chemical reactions","volume":"81","author":"Gillespie","year":"1977","journal-title":"J. Phys. Chem."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"027103","DOI":"10.1103\/PhysRevE.71.027103","article-title":"Generation of uncorrelated random scale-free networks","volume":"71","author":"Catanzaro","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mislove, A., Viswanath, B., Gummadi, K.P., and Druschel, P. (2010, January 4\u20136). You are who you know: Inferring user profiles in online social networks. Proceedings of the third ACM International Conference on Web Search and Data Mining, New York, NY, USA.","DOI":"10.1145\/1718487.1718519"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/15427951.2009.10129177","article-title":"Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters","volume":"6","author":"Leskovec","year":"2009","journal-title":"Internet Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"8577","DOI":"10.1002\/mma.7285","article-title":"Global stability analysis of a fractional SVEIR epidemic model","volume":"44","author":"Nabti","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1186\/s13662-019-2447-z","article-title":"Global stability analysis for a generalized delayed SIR model with vaccination and treatment","volume":"2019","author":"Elazzouzi","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"122236","DOI":"10.1016\/j.physa.2019.122236","article-title":"Stability analysis of an SDILR model based on rumor recurrence on social media","volume":"535","author":"Yao","year":"2019","journal-title":"Phys. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"125381","DOI":"10.1016\/j.amc.2020.125381","article-title":"Stability analysis and control strategies for a new SIS epidemic model in heterogeneous networks","volume":"383","author":"Xie","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.amc.2017.01.020","article-title":"Knowledge transmission model with consideration of self-learning mechanism in complex networks","volume":"304","author":"Wang","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Huang, C., Cao, J., Wen, F., and Yang, X. (2016). Stability analysis of SIR model with distributed delay on complex networks. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0158813"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"5729","DOI":"10.3934\/mbe.2019286","article-title":"Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks","volume":"16","author":"Hu","year":"2019","journal-title":"Math. Biosci. Eng."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TNSE.2018.2860988","article-title":"Spreading dynamics of an SEIR model with delay on scale-free networks","volume":"7","author":"Kang","year":"2018","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"108455","DOI":"10.1016\/j.aml.2022.108455","article-title":"Stability analysis of a model of epidemic dynamics with nonlinear feedback producing recurrent infection waves","volume":"136","author":"Bulai","year":"2023","journal-title":"Appl. Math. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"095256","DOI":"10.1088\/1402-4896\/acf16f","article-title":"Stability analysis and optimal control strategies of a fractional-order monkeypox virus infection model","volume":"98","author":"Adel","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Yaagoub, Z., and Allali, K. (2023). Global Stability of Multi-Strain SEIR Epidemic Model with Vaccination Strategy. Math. Comput. Appl., 28.","DOI":"10.3390\/mca28010009"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/3\/227\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:08:11Z","timestamp":1760105291000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/3\/227"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,2]]},"references-count":46,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["e26030227"],"URL":"https:\/\/doi.org\/10.3390\/e26030227","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,3,2]]}}}