{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:22:39Z","timestamp":1760059359698,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Fund for Scientific Researches of Republic of Bulgaria","award":["KP-06-N82\/4"],"award-info":[{"award-number":["KP-06-N82\/4"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We discuss the spread of epidemics caused by two viruses which cannot infect the same individual at the same time. The mathematical modeling of this epidemic leads to a kind of SIIRR model with two groups of infected individuals and two groups of recovered individuals. An additional assumption is that after recovering from one of the viruses, the individual cannot be infected by the other virus. The mathematical model consists of five equations which can be reduced to a system of three differential equations for the susceptible and for the recovered individuals. This system has analytical solutions for the case when one of the viruses infects many more individuals than the other virus. Cases which are more complicated than this one can be studied numerically. The theory is applied to the study of waves of popularity of an individual\/groups of individuals or of an idea\/group of ideas in the case of the presence of two opposite opinions about the popularity of the corresponding individual\/group of individuals or idea\/group of ideas. We consider two cases for the initial values of the infected individuals: (a) the initial value of the individuals infected with one of the viruses is much larger than the initial values of the individuals infected by the second virus, and (b) the two initial values of the infected individuals are the same. The following effects connected to the evolution of the numbers of infected individuals are observed: 1. arising of bell-shaped profiles of the numbers of infected individuals; 2. suppression of popularity; 3. faster increase\u2013faster decrease effect for the peaks of the bell-shaped profiles; 4. peak shift in the time; 5. effect of forgetting; 6. window of dominance; 7. short-term win\u2013long-term loss effect; 8. effect of the single peak. The proposed SIIRR model is used to build a mathematical theory of popularity waves where a person or idea can have positive and negative popularity at the same time and these popularities evolve with time.<\/jats:p>","DOI":"10.3390\/e27060611","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T08:22:34Z","timestamp":1749457354000},"page":"611","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Sic Transit Gloria Mundi: A Mathematical Theory of Popularity Waves Based on a SIIRR Model of Epidemic Spread"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6209-547X","authenticated-orcid":false,"given":"Nikolay K.","family":"Vitanov","sequence":"first","affiliation":[{"name":"Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria"}]},{"given":"Zlatinka I.","family":"Dimitrova","sequence":"additional","affiliation":[{"name":"Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,9]]},"reference":[{"key":"ref_1","unstructured":"Miller, J.H., and Page, S.E. (2007). Complex Adaptive Systems: An Introduction to Computational Models of Social Life, Princeton University Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Latora, V., Nicosia, V., and Russo, G. (2017). Complex Networks. Principles, Methods, and Applications, Cambridge University Press.","DOI":"10.1017\/9781316216002"},{"key":"ref_3","unstructured":"Chian, A.C.-L. (2007). Complex Systems Approach to Economic Dynamics, Springer."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Vitanov, N.K. (2016). Science Dynamics and Research Production. Indicators, Indexes, Statistical Laws and Mathematical Models, Springer.","DOI":"10.1007\/978-3-319-41631-1"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Treiber, M., and Kesting, A. (2013). Traffic Flow Dynamics: Data, Models, and Simulation, Springer.","DOI":"10.1007\/978-3-642-32460-4"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Dimitrova, Z.I. (2022). Flows of Substances in Networks and Network Channels: Selected Results and Applications. Entropy, 24.","DOI":"10.3390\/e24101485"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Drazin, P.G. (1992). Nonlinear Systems, Cambridge University Press.","DOI":"10.1017\/CBO9781139172455"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ganji, D.D., Sabzehmeidani, Y., and Sedighiamiri, A. (2018). Nonlinear Systems in Heat Transfer, Elsevier.","DOI":"10.1016\/B978-0-12-812024-8.00003-5"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Yoshida, Z. (2010). Nonlinear Science. The Challenge of Complex Systems, Springer.","DOI":"10.1007\/978-3-642-03406-0"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Diekmann, O., Heesterbeek, H., and Britton, T. (2012). Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press.","DOI":"10.23943\/princeton\/9780691155395.001.0001"},{"key":"ref_11","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_12","doi-asserted-by":"crossref","unstructured":"Brauer, F., Castillo-Chavez, C., and Feng, Z. (2019). Mathematcal Models in Epidemiology, Springer.","DOI":"10.1007\/978-1-4939-9828-9"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Li, M.I. (2018). An Introduction to Mathematical Modeling of Infectious Diseases, Springer.","DOI":"10.1007\/978-3-319-72122-4"},{"key":"ref_14","first-page":"113","article-title":"Mathematical Epidemiology: Past, Present and Future","volume":"2","author":"Brauer","year":"2017","journal-title":"Infect. Dis. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1146\/annurev.py.23.090185.002031","article-title":"A Comparison of Simulation Approaches to Epidemic Modeling","volume":"23","author":"Teng","year":"1985","journal-title":"Annu. Rev. Phytopathol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.mbs.2010.01.006","article-title":"Stochastic Epidemic Models: A Survey","volume":"225","author":"Britton","year":"2010","journal-title":"Math. Biosci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Levin, S.A. (1994). A Thousand and One Epidemic Models. Frontiers in Mathematical Biology, Springer.","DOI":"10.1007\/978-3-642-50124-1"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1098\/rsif.2005.0051","article-title":"Networks and Epidemic Models","volume":"2","author":"Keeling","year":"2005","journal-title":"J. R. Soc. Interface"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/0025-5564(78)90006-8","article-title":"A Generalization of the Kermack- McKendrick Deterministic Epidemic Model","volume":"42","author":"Capasso","year":"1978","journal-title":"Math. Biosci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"036603","DOI":"10.1088\/1361-6633\/aa5398","article-title":"Unification of Theoretical Approaches for Epidemic Spreading on Complex Networks","volume":"80","author":"Wang","year":"2017","journal-title":"Rep. Prog. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Cifuentes-Faura, J., Faura-Mart\u00ednez, U., and Lafuente-Lechuga, M. (2022). Mathematical Modeling and the Use of Network Models as Epidemiological Tools. Mathematics, 10.","DOI":"10.3390\/math10183347"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Cui, Q., Qiu, Z., Liu, W., and Hu, Z. (2017). Complex Dynamics of an SIR Epidemic Model with Nonlinear Saturate Incidence and Recovery Rate. Entropy, 19.","DOI":"10.3390\/e19070305"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rahimi, I., Gandomi, A.H., Asteris, P.G., and Chen, F. (2021). Analysis and Prediction of COVID-19 Using SIR, SEIQR, and Machine Learning Models: Australia, Italy, and UK Cases. Information, 12.","DOI":"10.3390\/info12030109"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Trawicki, M.B. (2017). Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity. Mathematics, 5.","DOI":"10.3390\/math5010007"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Frank, T.D. (2022). COVID-19 Epidemiology and Virus Dynamics, Springer.","DOI":"10.1007\/978-3-030-97178-6"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Godio, A., Pace, F., and Vergnano, A. (2020). SEIR Modeling of the Italian Epidemic of SARS-CoV-2 Using Computational Swarm Intelligence. Int. J. Environ. Res. Public Health, 17.","DOI":"10.20944\/preprints202004.0073.v2"},{"key":"ref_27","unstructured":"Scharnhorst, A., Boerner, K., and Besselaar, P. (2010). Knowledge Epidemics and Population Dynamics Models for Describing Idea Diffusion. Models of Science Dynamics, Springer."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Etxeberria-Etxaniz, M., Alonso-Quesada, S., and De la Sen, M. (2020). On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation. Appl. Sci., 10.","DOI":"10.3390\/app10228296"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Al-Shbeil, I., Djenina, N., Jaradat, A., Al-Husban, A., Ouannas, A., and Grassi, G. (2023). A New COVID-19 Pandemic Model including the Compartment of Vaccinated Individuals: Global Stability of the Disease-Free Fixed Point. Mathematics, 11.","DOI":"10.3390\/math11030576"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Lee, S.J., Lee, S.E., Kim, J.-O., and Kim, G.B. (2021). Two-Way Contact Network Modeling for Identifying the Route of COVID-19 Community Transmission. Informatics, 8.","DOI":"10.3390\/informatics8020022"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Harjule, P., Poonia, R.C., Agrawal, B., Saudagar, A.K.J., Altameem, A., Alkhathami, M., Khan, M.B., Hasanat, M.H.A., and Malik, K.M. (2022). An Effective Strategy and Mathematical Model to Predict the Sustainable Evolution of the Impact of the Pandemic Lockdown. Healthcare, 10.","DOI":"10.3390\/healthcare10050759"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Chen, J., and Yin, T. (2023). Transmission Mechanism of Post-COVID-19 Emergency Supply Chain Based on Complex Network: An Improved SIR Model. Sustainability, 15.","DOI":"10.3390\/su15043059"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Almeshal, A.M., Almazrouee, A.I., Alenizi, M.R., and Alhajeri, S.N. (2020). Forecasting the Spread of COVID-19 in Kuwait Using Compartmental and Logistic Regression Models. Appl. Sci., 10.","DOI":"10.3390\/app10103402"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Batool, H., Li, W., and Sun, Z. (2023). Extinction and Ergodic Stationary Distribution of COVID-19 Epidemic Model with Vaccination Effects. Symmetry, 15.","DOI":"10.3390\/sym15020285"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Jitsinchayakul, S., Humphries, U.W., and Khan, A. (2023). The SQEIRP Mathematical Model for the COVID-19 Epidemic in Thailand. Axioms, 12.","DOI":"10.3390\/axioms12010075"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Khorev, V., Kazantsev, V., and Hramov, A. (2023). Effect of Infection Hubs in District-Based Network Epidemic Spread Model. Appl. Sci., 13.","DOI":"10.3390\/app13021194"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.3390\/covid2120129","article-title":"Parameters Sensitivity Analysis of COVID-19 Based on the SCEIR Prediction Model","volume":"2","author":"Ni","year":"2022","journal-title":"COVID"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1016\/j.apm.2020.08.057","article-title":"Analytical Features of the SIR Model and their Applications to COVID-19","volume":"90","author":"Kudryashov","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_39","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_40","doi-asserted-by":"crossref","unstructured":"Leonov, A., Nagornov, O., and Tyuflin, S. (2023). Modeling of Mechanisms of Wave Formation for COVID-19 Epidemic. Mathematics, 11.","DOI":"10.3390\/math11010167"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Wang, W., and Xia, Z. (2023). Study of COVID-19 Epidemic Control Capability and Emergency Management Strategy Based on Optimized SEIR Model. Mathematics, 11.","DOI":"10.3390\/math11020323"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Chang, Y.-C., and Liu, C.-T. (2022). A Stochastic Multi-Strain SIR Model with Two-Dose Vaccination Rate. Mathematics, 10.","DOI":"10.3390\/math10111804"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Noeiaghdam, S., and Micula, S. (2021). Dynamical Strategy to Control the Accuracy of the Nonlinear Bio-Mathematical Model of Malaria Infection. Mathematics, 9.","DOI":"10.3390\/math9091031"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Noeiaghdam, S., Micula, S., and Nieto, J.J. (2021). A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library. Mathematics, 9.","DOI":"10.3390\/math9121321"},{"key":"ref_45","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_46","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/S0167-2789(02)00389-5","article-title":"The Effect of Cross-immunity and Seasonal Forcing in a Multi-strain Epidemic Model","volume":"165","author":"Kamo","year":"2002","journal-title":"Phys. D"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Massard, M., Eftimie, R., Perasso, A., and Saussereau, B. (2022). A Multi-strain Epidemic Model for COVID-19 with Infected and Asymptomatic Cases: Application to French Data. J. Theor. Biol., 545.","DOI":"10.1016\/j.jtbi.2022.111117"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Lazebnik, T., and Bunimovich-Mendrazitsky, S. (2022). Generic Approach for Mathematical Model of Multi-strain Pandemics. PLoS ONE, 17.","DOI":"10.1371\/journal.pone.0260683"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/s11071-020-05929-4","article-title":"Global Dynamics of a Multi-strain SEIR Epidemic Model with General Incidence Rates: Application to COVID-19 Pandemic","volume":"102","author":"Khyar","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Arruda, E.F., Das, S.S., Dias, C.M., and Pastore, D.H. (2021). Modelling and Optimal Control of Multi Strain Epidemics, with Application to COVID-19. PLoS ONE, 16.","DOI":"10.1371\/journal.pone.0257512"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Fudolig, M., and Howard, R. (2020). The Local Stability of a Modified Multi-strain SIR Model for Emerging Viral Strains. PLoS ONE, 15.","DOI":"10.1101\/2020.03.19.20039198"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1080\/17513750802638712","article-title":"A Non-autonomous Multi-strain SIS Epidemic Model","volume":"3","author":"Martcheva","year":"2009","journal-title":"J. Biol. Dyn."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.cam.2012.08.008","article-title":"Bifurcation Analysis of a Family of Multi-strain Epidemiology Models","volume":"252","author":"Kooi","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.jtbi.2010.03.005","article-title":"A General Multi-strain Model with Environmental Transmission: Invasion Conditions for the Disease-free and Endemic States","volume":"264","author":"Breban","year":"2010","journal-title":"J. Theor. Biol."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1098\/rsif.2008.0333","article-title":"Improving the Realism of Deterministic Multi-strain Models: Implications for Modelling Influenza A","volume":"6","author":"Minayev","year":"2009","journal-title":"J. R. Soc. Interface"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Pateras, J., Vaidya, A., and Ghosh, P. (2022). Network Thermodynamics-Based Scalable Compartmental Model for Multi-Strain Epidemics. Mathematics, 10.","DOI":"10.3390\/math10193513"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s12190-012-0580-x","article-title":"A Two-Strain Epidemic Model with Mutant Strain and Vaccination","volume":"40","author":"Cai","year":"2021","journal-title":"J. Appl. Math. Comput."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1016\/j.chaos.2017.11.035","article-title":"Two-Strain Epidemic Model with Two Vaccinations","volume":"106","author":"Baba","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Otunuga, O.M. (2022). Analysis of Multi-Strain Infection of Vaccinated and Recovered Population through Epidemic Model: Application to COVID-19. PLoS ONE, 17.","DOI":"10.1371\/journal.pone.0271446"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1080\/17513758.2010.510213","article-title":"Flu Epidemics: A Two-Strain Fu Model with a Single Vaccination","volume":"5","author":"Zou","year":"2011","journal-title":"J. Biol. Dyn."},{"key":"ref_61","first-page":"7541861","article-title":"Competitive Coexistence in a Two-Strain Epidemic Model with a Periodic Infection Rate","volume":"2020","author":"Li","year":"2020","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Vitanov, N.K., Dimitrova, Z.I., and Vitanov, K.N. (2021). Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods. Entropy, 23.","DOI":"10.1063\/5.0040409"},{"key":"ref_63","first-page":"020003","article-title":"Simple Equations Method (SEsM): Review and New Results","volume":"2459","author":"Vitanov","year":"2022","journal-title":"AIP Conf. Ser."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Vitanov, N.K., and Dimitrova, Z.I. (2021). Simple Equations Method and Non-linear Differential Equations with Non-polynomial Non-linearity. Entropy, 23.","DOI":"10.3390\/e23121624"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Vitanov, N.K., Dimitrova, Z.I., and Vitanov, K.N. (2021). On the Use of Composite Functions in the Simple Equations Method to Obtain Exact Solutions of Nonlinear Differential Equations. Computation, 9.","DOI":"10.3390\/computation9100104"},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Vitanov, N.K. (2022). Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations. Entropy, 24.","DOI":"10.3390\/e24111653"},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Vitanov, N.K., and Vitanov, K.N. (2023). Epidemic Waves and Exact solutions of a Sequence of Nonlinear Differential Equations Connected to the SIR model of Epidemics. Entropy, 25.","DOI":"10.3390\/e25030438"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/611\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:48:44Z","timestamp":1760032124000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/6\/611"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,9]]},"references-count":67,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["e27060611"],"URL":"https:\/\/doi.org\/10.3390\/e27060611","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,6,9]]}}}