Modeling of the COVID-19 pandemic in the limit of no acquired immunity

We propose the SEIRS compartmental epidemiology model aimed at modeling the COVID-19 pandemy dynamics.  The limit case of no acquired immunity (neither natural nor via vaccination) is considered mimicking the situation (i) when no effective vaccine being developed or available yet, and (ii) the virus strongly mutates causing massive reinfections.  Therefore, the only means of suppressing the virus spread are via quarantine measures and effective identification and isolation of infected individuals.  We found both the disease-free and the endemic fixed points and examined their stability.  The basic reproduction ratio is obtained and its dependence on the parameters of the model is discussed.  We found the presence of the contact rate threshold beyond which the disease-free fixed point cannot be reached.  Using the numeric solution, the approximate analytic solution of the model, characterized by rescaled contact rate, is obtained.  Several possible "quarantine on"/"quarantine off" scenarios are considered and the one combined with flexible adjustment of the identification and isolation rates is found to be the most effective in bringing the second and consequent waves down.  The study can be interpreted as a reference point for the case when the natural or acquired immunity, as well as vaccination, are taken into account.  It will be a topic of a separate study.

  1. Organization W. H. Coronavirus disease (COVID-19) pandemic 2020 (accessed 14 September, 2020).
  2. Rojas-Vallejos J.  Strengths and limitations of mathematical models in pandemics – the case of COVID-19 in Chile.  Medwave. 20 (3), e7874–e7874 (2020).
  3. Adam D.  Special report: The simulations driving the world's response to COVID-19.  Nature. 580 (7803), 316–318 (2020).
  4. Eubank S., Eckstrand I., Lewis B., Venkatramanan S., Marathe M., Barrett C. L.  Commentary on Ferguson, et al., "Impact of Non-pharmaceutical Interventions (NPIs) to Reduce {COVID}-19 Mortality and Healthcare Demand".  Bulletin of Mathematical Biology. 82, 52 (2020).
  5. Park M., Cook A. R., Lim J. T., Sun Y., Dickens B. L.  A Systematic Review of COVID-19 Epidemiology Based on Current Evidence.  Journal of Clinical Medicine. 9 (4), 967 (2020).
  6. Bai Z., Gong Y., Tian X., Cao Y., Liu W., Li J.  The Rapid Assessment and Early Warning Models for COVID-19.  Virologica Sinica. 35 (3), 272–279 (2020).
  7. Cheng Z. J., Shan J.  2019 Novel coronavirus: where we are and  what we know.  Infection. 48 (2), 155–163 (2020).
  8. Anirudh A.  Mathematical modeling and the transmission dynamics in predicting the Covid-19 – What next in combating the pandemic.  Infectious Disease Modelling. 5, 366–374 (2020).
  9. Carletti T., Fanelli D., Piazza F.  COVID-19: The unreasonable effectiveness of simple models.  Chaos, Solitons & Fractals: X. 5, 100034 (2020).
  10. Nazarimehr F., Pham V. T., Kapitaniak T.  Prediction of bifurcations by varying critical parameters of COVID-19.  Nonlinear Dynamics. 101, 1681–1692 (2020).
  11. Contreras S., Villavicencio H. A., Medina-Ortiz D., Biron-Lattes J. P., Olivera-Nappa Á.  A multi-group SEIRA model for the spread of COVID-19 among heterogeneous populations.  Chaos, Solitons & Fractals. 136, 109925 (2020).
  12. Guerrero-Nancuante C., Manríquez R. P.  An epidemiological forecast of COVID-19 in Chile based on the generalized {SEIR} model and the concept of recovered.  Medwave. 20 (4), e7898–e7898 (2020).
  13. Rawson T., Brewer T., Veltcheva D., Huntingford C., Bonsall M. B.  How and When to End the COVID-19 Lockdown: An Optimization Approach.  Frontiers in Public Health. 8, 262 (2020).
  14. Tang Y., Wang S.  Mathematic modeling of COVID-19 in the United States.  Emerging Microbes & Infections. 9 (1), 827–829 (2020).
  15. Tang Y., Serdan T. D. A., Masi L. N., Tang S., Gorjao R., Hirabara S. M.  Epidemiology of COVID-19 in Brazil: using a mathematical model to estimate the outbreak peak and temporal evolution.  Emerging Microbes & Infections. 9 (1), 1453–1456 (2020).
  16. Gatto M., Bertuzzo E., Mari L., Miccoli S., Carraro L., Casagrandi R., Rinaldo A.  Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures.  Proceedings of the National Academy of Sciences. 117 (19), 10484–10491 (2020).
  17. Götz T., Heidrich P.  Early stage COVID-19 disease dynamics in Germany: models and parameter identification.  Journal of Mathematics in Industry. 10 (1), 20 (2020).
  18. Liu Z., Magal P., Seydi O., Webb G.  A COVID-19 epidemic model with latency period.  Infectious Disease Modelling. 5, 323–337 (2020).
  19. Chen J., Fu M. C., Zhang W., Zheng J.  Predictive Modeling for Epidemic Outbreaks: A New Approach and COVID-19 Case Study.  Asia-Pacific Journal of Operational Research. 37 (03), 2050028 (2020).
  20. Sun J., He W. T., Wang L., Lai A., Ji X., Zhai X., Li G., Suchard M. A., Tian J., Zhou J., Veit M., Su S.  COVID-19: Epidemiology, Evolution, and Cross-Disciplinary Perspectives.  Trends in Molecular Medicine. 26 (5), 483–495 (2020).
  21. Sun T., Weng D.  Estimating the effects of asymptomatic and imported patients on COVID-19 epidemic using mathematical modeling.  Journal of Medical Virology. 92, 1995–2003 (2020).
  22. Chatterjee S., Sarkar A., Chatterjee S., Karmakar M., Paul R.  Studying the progress of COVID-19 outbreak in India using SIRD model.  Indian Journal of Physics  (2020).
  23. Kucharski A. J., Russell T. W., Diamond C., Liu Y., Edmunds J., Funk S., Eggo R. M., Sun F., Jit M., Munday J. D., Davies N., Gimma A., van Zandvoort K.,  Gibbs H., Hellewell J., Jarvis C. I., Clifford S., Quilty B. J., Bosse N. I.,  Abbott S.,  Klepac P.,  Flasche S.  Early dynamics of transmission and control of COVID-19: a mathematical modeling study.  The Lancet Infectious Diseases. 20 (5), 553–558 (2020).
  24. Rahimi F., Abadi A. T. B.  Practical Strategies Against the Novel Coronavirus and COVID-19 – the Imminent Global Threat.  Archives of Medical Research. 51 (3), 280–281 (2020).
  25. Linka K., Peirlinck M., Costabal F. S., Kuhl E.  Outbreak dynamics of COVID-19 in Europe and the effect of travel restrictions.  Computer Methods in Biomechanics and Biomedical Engineering. 23 (11), 710–717 (2020).
  26. Mushayabasa S., Ngarakana-Gwasira E. T., Mushanyu J.  On the role of governmental action and individual reaction on COVID-19 dynamics in South Africa: A mathematical modeling study.  Informatics in Medicine Unlocked. 20, 100387 (2020).
  27. Davies N. G., Kucharski A. J., Eggo R. M., Gimma A., Edmunds W. J., Jombart T., O\'{}Reilly K., Endo A., Hellewell J., Nightingale E. S., Quilty B. J.,  Jarvis C. I., Russell T. W., Klepac P., Bosse N. I., Funk S., Abbott S., Medley G. F.,  Gibbs H.,  Pearson C. A. B., Flasche S.,  Jit M.,  Clifford S.,  Prem K., Diamond C.,  Emery J.,  Deol A. K.,  Procter S. R., van Zandvoort K., Sun Y. F., Munday J. D., Rosello A.,  Auzenbergs M.,  Knight G., Houben R. M. G. J.,  Liu Y.  Effects of non-pharmaceutical interventions on COVID-19 cases, deaths, and demand for hospital services in the UK: a modeling study.  The Lancet Public Health. 5 (7), e375–e385 (2020).
  28. Choi S., Ki M.  Estimating the reproductive number and the outbreak size of COVID-19 in Korea.  Epidemiology and Health. 42, e2020011 (2020).
  29. Peirlinck M., Linka K., Costabal F. S., Kuhl E.  Outbreak dynamics of COVID-19 in China and the United States.  Biomechanics and Modeling in Mechanobiology. 19, 2179–2193 (2020).
  30. Zhang J., Litvinova M., Wang W., Wang Y., Deng X., Chen X., Li M., Zheng W., Yi L., Chen X., Wu Q., Liang Y., Wang X., Yang J., Sun K., Longini I. M.,  Halloran M. E., Wu P., Cowling B. J., Merler S., Viboud C., Vespignani A., Ajelli M., Yu H.  Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: a descriptive and modeling study.  The Lancet Infectious Diseases. 20 (7), 793–802 (2020).
  31. Chowdhury R., Heng K., Shawon M. S. R., Goh G., Okonofua D., Ochoa-Rosales C., Gonzalez-Jaramillo V., Bhuiya A., Reidpath D., Prathapan S., Shahzad S., Althaus C. L., Gonzalez-Jaramillo N., Franco O. H.  Dynamic interventions to control COVID-19 pandemic: a multivariate prediction modeling study comparing 16 worldwide countries.  European Journal of Epidemiology. 35 (5), 389–399 (2020).
  32. Block P., Hoffman M., Raabe I. J., Dowd J. B., Rahal C., Kashyap R., Mills M. C.  Social network-based distancing strategies to flatten the COVID-19 curve in a post-lockdown world.  Nature Human Behaviour. 4 (6), 588–596 (2020).
  33. López L., Rodó X.  The end of social confinement and COVID-19 re-emergence risk.  Nature Human Behaviour. 4 (7), 746–755 (2020).
  34. Organization W. H. "Immunity passports" in the context of COVID-19  (24 April 2020).
  35. Edridge A. W. D., Kaczorowska J., Hoste A. C. R., Bakker M., Klein M., Loens K., Jebbink M. F., Matser A., Kinsella C. M., Rueda P., Ieven M., Goossens H., Prins M., Sastre P., Deijs M., van~der Hoek L.  Seasonal coronavirus protective immunity is short-lasting.  Nature Medicine. 26 (11), 1691–1693 (2020).
  36. Ibarrondo F. J., Fulcher J. A., Goodman-Meza D., Elliott J., Hofmann C., Hausner M. A., Ferbas K. G., Tobin N. H., Aldrovandi G. M., Yang O. O.  Rapid Decay of Anti-SARS-CoV-2 Antibodies in Persons with Mild Covid-19.  New England Journal of Medicine. 383 (11), 1085–1087 (2020).
  37. COVID research updates: A coronavirus vaccine shows lasting benefit. Nature (2021).
  38. Tillett R. L., Sevinsky J. R., Hartley P. D., Kerwin H., Crawford N., Gorzalski A., Laverdure C.,  Verma S. C., Rossetto C. C., Jackson D.,   Farrell M. J., Hooser S. V., Pandori M.  Genomic evidence for reinfection with SARS-CoV-2: a case study.  The Lancet Infectious Diseases. 21 (1), 52–58 (2021).
  39. Vrieze J.  More people are getting COVID-19 twice, suggesting immunity wanes quickly in some.  Science  (2020).
  40. Haseltine W. A.  Covid-19 Reinfection Is Possible And Should Inform Pandemic Priorities Moving Forward  2020 (accessed 20 Novemeber, 2020).
  41. Mallapaty S.  COVID mink analysis shows mutations are not dangerous – yet.  Nature. 587 (7834), 340–341 (2020).
  42. Hou Y. J., Chiba S., Halfmann P., Ehre C., Kuroda M., Dinnon K. H., Leist S. R., Sch\"{a}fer A., Nakajima N., Takahashi K., Lee R. E., Mascenik T. M.,  Graham R., Edwards C. E., Tse L. V., Okuda K.,  Markmann A. J., Bartelt L., de~Silva A., Margolis D. M., Boucher R. C., Randell S. H., Suzuki T., Gralinski L. E., Kawaoka Y., Baric R. S.  SARS-CoV-2 D614G variant exhibits efficient replication ex vivo and transmission in vivo.  Science. 370 (6523), 1464–1468 (2020).
  43. Goodman B.  Study: New Mutation Sped Up Spread of Coronavirus 2020 (accessed 13 Novemeber, 2020).
  44. Terry M.  SARS-CoV-2, the COVID-19 Virus, is Mutating, But So Far,  Slowly  2020 (accessed 10 December 2020).
  45. Organization W. H. SARS-CoV-2 Variants  (31 December 2020).
  46. Korn G., Korn T.  Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review.  Dover Civil and Mechanical Engineering, Dover Publications (2013).
  47. Pongkitivanichkul C., Samart D., Tangphati T., Koomhin P., Pimton P., Dam-O P., Payaka A., Channuie P.  Estimating the size of COVID-19 epidemic outbreak.  Physica Scripta. 95 (8), 085206 (2020).
  48. Keeling M. J., Rohani P.  Modeling infectious diseases in humans and animals.  Princeton University Press, Princeton (2008).
  49. Piccolomini E. L., Zama F.  Monitoring Italian COVID-19 spread by a forced EIRD model.  PLOS ONE. 15 (8), e0237417 (2020).
Mathematical Modeling and Computing, Vol. 8, No. 2, pp. 282–303 (2021)