Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Biological view of vaccination described by mathematical modellings: from rubella to dengue vaccines

1 Department de Matemática Aplicada, Universidade Estadual de Campinas, Campinas, SP, Brazil
2 Department of epidemiology, Faculty of Medicina São Leopoldo Mandic, Campinas, SP, Brazil

Special Issues: Mathematical Methods in the Biosciences

The only rubella vaccine available in North America is the RA27/3 strain (isolated from the kidney of a rubella-infected fetus and attenuated) licensed in 1979, which substituted HPV77/DE5 strain vaccine due to concerns about waning immunity. The first dengue vaccine (Dengvaxia CYDTDV) was first registered in Mexico in December, 2015, which is a live recombinant tetravalent dengue vaccine. Rubella vaccine was applied since 1969, but tetravalent dengue vaccine is being used in large scale nowadays. In the past, based on unavailable information regarded to rubella vaccine, mathematical models were used to design vaccination schemes in order to avoid congenital rubella syndrome (CRS). Currently, knowing that vaccine does not result in CRS, rubella vaccination is modelled as usual childhood infection. This experience of updated biological knowledge that influenced mathematical modellings of rubella vaccination is taken into account to reflect about the tetravalent dengue vaccine. We also address a discussion about the security of vaccination strategies.
  Article Metrics

Keywords rubella and dengue vaccines; vaccine failures; congenital rubella syndrome; severe dengue; antibody-dependent enhancement; average age at infection

Citation: Hyun Mo Yang, André Ricardo Ribas Freitas. Biological view of vaccination described by mathematical modellings: from rubella to dengue vaccines. Mathematical Biosciences and Engineering, 2019, 16(4): 3195-3214. doi: 10.3934/mbe.2019159


  • 1. S. A. Plotkin, The history of rubella and rubella vaccination leading to elimination, Clin. Infect. Dis., 43 (2006), S164–168.
  • 2. G. Mandell, J. Bennett and R. Dolin, Mandell, Douglas and Bennett's Principles and Practices of Infectious Diseases, 6th edition, Elsevier Inc., Philadelphia, 2005.
  • 3. G. Strickland, Hunter's Tropical Medicine and Emerging Infectious Diseases, 8th edition, W.S. Saunders Co., Philadelphia, 2000.
  • 4. E. Massad, R.S. Azevedo Neto, M.N. Burattini, et al., Assessing the efficacy of s mixed vaccination strategy against rubella In São Paulo, Brazil, Int. J. Epidemiol., 24 (1995), 842–850.
  • 5. H. M. Yang, Modelling vaccination strategy against directly transmitted diseases using a series of pulses, J. Biol. Syst., 6 (1998), 187–212.
  • 6. X. Song, Y. Jiang and H. Wei, Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays, Appl. Math. Comput., 214 (2009), 381–390.
  • 7. M. De la Sen, R. P. Agarwal, A. Ibeas, et al., On a generalized time-varying SEIR epidemic model with mixed point and Distributed Time-Varying Delays and Combined Regular and Impulsive vaccination controls, Adv. Differ. Equ., 2010 (2010), 281612.
  • 8. M. De la Sen, R. P. Agarwal, R. Nistal1, et al., A switched multicontroller for an SEIADR epidemic model with monitored equilibrium points and supervised transients and vaccination costs, Adv. Differ. Equ., 2018 (2018), 390.
  • 9. H. Yang, Epidemiologia Matemática: Estudo dos Efeitos da Vacina¸cão em Doen¸cas de Transmissão Direta, Edunicamp & Fapesp, Campinas, 2001.
  • 10. L. Esteva and H. M. Yang, Assessing the effects of temperature and dengue virus load on dengue transmission, J. Biol. Syst., 23 (2015), 527–554.
  • 11. J. Arino, M. Connell and P. Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260–276.
  • 12. S. Chengjun and Y. Wei, Global results for an SIRS model with vaccination and isolation, Nonlinear Anal. Real World Appl., 11 (2010), 4223–4237.
  • 13. J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36 (1998), 227–248.
  • 14. H. Jing and Z. Deming, Global stability and periodicity on SIS epidemic models with backward bifurcation, Comput. Math. Appl., 50 (2005), 1271–1290.
  • 15. C. M. Kribs-Zaleta and J. X. Velasco-Hernández, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183–201.
  • 16. J. Li, Z. Ma and Y. Zhou, Global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria, Acta Math. Sci., 26B (2006), 83–93.
  • 17. M. A. Safi and A. B. Gumel, Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine, Comput. Mathe. Appl., 61 (2011), 3044–3070.
  • 18. O. Sharomi, C. N. Podder, A. B. Gumel, et al., Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Math. Biosci., 210 (2007), 436–463.
  • 19. L. Freitas, Vacina¸cão de Doen¸cas Infecciosas de Transmissão Direta: Quantificando Condi¸cões de Controle Considerando Portadores, Ph.D thesis, State University at Campinas City (Brazil), 2018.
  • 20. H. M. Yang, Modeling directly transmitted infections in a routinely vaccinated population – the force of infection described by Volterra integral equation, Appl. Math. Comput., 122 (2001), 27– 58.
  • 21. H. M. Yang and S. M. Raimundo, Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis, Theor. Biol. Med. Model., 7 (2010), 41.
  • 22. R. Anderson and R. May, Infectious Diseases of Human: Dynamics and Control, Oxford University Press, Oxford, New York, Tokyo, 1991.
  • 23. O. Zgorniak-Nowosielska, B. Zawillinska and S. Szostek, Rubella infection during pregnancy in the 1985-86 epidemic: follow-up after seven years, Eur. J. Epidemiol., 12 (1996), 303–308.
  • 24. S.W. Bart, H. C. Steller, S. R. Preblud, et al., Fetal risk associated with rubella vaccine: an update, Rev. Infect. Dis., 7 (1985), S95–102.
  • 25. C. Castillo-Solórzano, S. E. Feef, M. Morice, et al., Rubella vaccination of unknowingly pregnant women during mass campaigns for rubella and congenital rubella syndrome elimination, the Americas 2001-2008, J. Infect. Dis., 204 (2011), S713–717.
  • 26. E. Massad, M. N. Burattini, R. S. Azevedo Neto, et al., A model-based design of a vaccination strategy against rubella in a non-immunized community of São Paulo State, Brazil, Epidemiol. Infect., 112 (1994), 579–594.
  • 27. H. M. Yang, Directly transmitted infections modeling considering age-structured contact rate, Math. Comput. Model., 29 (1999), 39–48.
  • 28. H. M. Yang, Directly transmitted infections modeling considering age-structured contact rate – epidemiological analysis, Math. Comput. Model., 29 (1999), 11–30.
  • 29. R. S. Azevedo Neto, A. S. B. Silveira, D. J. Nokes, et al., Rubella seroepidemiology in a nonimmunized population of São Paulo State, Brazil, Epidemiol. Infect., 113 (1994), 161–173.
  • 30. X. Badilla, A. Morice, M. L. Avila-Aguero, et al., Fetal risk associated with rubella vaccination during pregnancy, Pediatr. Infect. Dis. J., 26 (2007), 830–835.
  • 31. A. M. Ergenoglu, A. O. Yeniel, N. Yildirim, et al., Rubella vaccination during the periconception period or in pregnancy and perinatal and fetal outcomes, Turk J. Pediatr., 54 (2012), 230–233.
  • 32. B. J. Freij, M. A. South and J. L. Sever, Maternal rubella and the congenital rubella syndrome, Clin. Perinatol., 15 (1988), 247–257.
  • 33. L. Minussi, R. Mohrdieck, M. Bercini, et al., Prospective evaluation of pregnant women vaccinated against rubella in southern Brazil, Reprod. Toxicol., 25 (2008), 120–123.
  • 34. M. H. Namae, M. Ziaee and N. Nasseh, Congenital rubella syndrome in infants of women vaccinated during or just before pregnancy with measles-rubella vaccine, Indian J. Med. Res., 127 (2008), 551–554.
  • 35. R. Nasiri, J. Yoseffi, M. Khaejedaloe, et al., Congenital rubella syndrome after rubella vaccination in 1-4 weeks periconceptional period, Indian J. Pediatr., 76 (2009), 279–282.
  • 36. H. K. Sato, A. T. Sanajotta, J. C. Moraes, et al., Rubella vaccination of unknowingly pregnant women: the São Paulo experience, 2001, J. Infect. Dis., 204 (2011), S734–744.
  • 37. G. R. da Silva e Sá, L. A. Camacho, M. S. Stavola, et al., Pregnancy outcomes following rubella vaccination: a prospective study in the state of Rio de Janeiro, Brazil, 2001-2002, J. Infect. Dis., 204 (2011), S722–728.
  • 38. R. C. Soares, M. M. Siqueira, C. M. Toscano, et al., Follow-up study of unknowingly pregnant women vaccinated against rubella in Brazil, 2001-2002, J. Infect. Dis., 204 (2011), S729–736.
  • 39. P. A. Tookey, G. Jones, B. H. Miller, et al., Rubella vaccination in pregnant, CDR (Lond. Engl. Rev.), 1 (1991), R86–88.
  • 40. D. Gubler, Dengue and dengue hemorrhagic fever, Clin. Microb. Rev., 11 (1998), 480–496.
  • 41. S. B. Halstead, S. Mahalingam, M. A. Marovich, et al., Intrinsic antibody-dependent enhanced of microbial infection in macrophages: disease regulation by immune complexies, Lancet Infect. Dis., 10 (2010), 712–722.
  • 42. D. M. Morens, Antibody-dependent enhancement of infection and the pathogenesis of viral disease, Clin. Infect. Dis., 19 (1994), 500–512.
  • 43. WHO, Immunization, vaccines and biologicals, 2017. Available from: http://www.who.int/ immunization/research/development/dengue_vaccines/en/.
  • 44. L. Coudeville, N. Baurin and E. Vergu, Estimation of parameters related to vaccine efficacy and dengue transmission from two large phase III studies, Vaccine, 34 (2016), 6417–6425.
  • 45. S. Gailhardou, A. Skipetrova, G. H. Dayan, et al., Safety overview of a recombinant live-attenuated tetravalent dengue vaccine: pooled analysis of data from 18 clinical trials, Plos NTD, 10 (2016), 1–25.
  • 46. L. J. Scott, Tetravalent dengue vaccine: a review in the prevention of dengue disease, Drugs, 76 (2016), 1301–1312.
  • 47. L. Villar, G. H. Dayan, J. L. Arredondo-Garcia, et al., Efficacy of a tetravalent dengue vaccine in children in Latin America, N. Engl. J. Med., 372 (2015), 113–123.
  • 48. S. B. Halstead and P. K. Russell, Protective and immunological behavior of chimeric yellow fever dengue vaccine, Vaccine, 34 (2016), 1643–1647.
  • 49. S. R. S. Hadinegoro, J. L. Arredondo-Garcia, B. Guy, et al., Answer to the review from Healstead and Russell "Protective and immunological behavior of chimeric yellow fever dengue vaccine", Vaccine, 34 (2016), 4273–4274.
  • 50. Sanofi updates information on dengue vaccine, Nov. 29, 2017, Sanofi, 2017. Available from: http://mediaroom.sanofi.com/sanofi-updates-information-on-dengue-vaccine/.
  • 51. Philippines gripped by dengue vaccine fears, Feb. 3, 2018, BBC, 2017. Available from: http: //www.bbc.com/news/world-asia-42929255.
  • 52. L. Billings, I. B. Schwartz, L. B. Shaw, et al., Instabilities in multiserotype disease model with antibody-dependent enhancement, J. Theoret. Biol., 246 (2007), 18–27.
  • 53. L. Esteva and C. Vargas, Coexistence of different serotypes of dengue virus, J. Math. Biol., 46 (2003), 31–47.
  • 54. L. Coudeville and G. P. Garnett, Transmission dynamics of the four dengue serotypes in Southern Vietnam and the potential impact of vaccination, PlosOne, 7(12) (2012), e51244.
  • 55. H. M. Yang, M. L. Macoris, K. C. Galvani, et al., Follow up estimation of Aedes aegypti entomological parameters and mathematical modellings, BioSystems, 103 (2011), 360–371.
  • 56. H. M. Yang, M. L. Macoris, K. C. Galvani, et al., Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue, Epidemiol. Infect., 137 (2009), 1188–1202.
  • 57. N. Chotiwan, B. G. Andre, I. Sanchez-Vargas, et al., Dynamic remodeling of lipids coincides with dengue virus replication in the midgut of Aedes aegypti mosquitoes, PLOS Pathog., 14 (2018), e1006853.
  • 58. H. M. Yang, J. L. Boldrini, A. C. Fassoni, et al., Fitting the incidence data from the City of Campinas, Brazil, based on dengue transmission modellings considering time-dependent entomological parameters, PLOS One, 11 (2016), 1–41.
  • 59. Vigilˆancia Sanitária, MG, Boletim epidemiológico de monitoramento dos casos de dengue, febre chikungunya e febre zika, Semana epidemiológica, 1 (2016), 01.
  • 60. M. C. Gomez and H. M. Yang, A simple mathematical model to describe antibody-dependent enhancement in heterologous secondary infection in dengue, Math. Med. Biol.: A Journ. IMA, (2016), DOI: 10.1093/imammb/dqy016.
  • 61. WHO, Dengue vaccine: WHO position paper – July 2016, Weekly epidemiological record, 30 (2016), 349–364.
  • 62. A. Ong, M. Sandar, M. I. Chen, et al., Fatal dengue hemorrhagic fever in adults during a dengue epidemic in Singapore, Int. J. Infect. Disease., 11 (2007), 263–267.
  • 63. H. M. Yang and C. P. Ferreira, Assessing the effects of vector control on dengue transmission, Appl. Math. Comput., 198 (2008), 401–413.
  • 64. H. M. Yang, The basic reproduction number obtained from Jacobian and next generation matrices – A case study of dengue transmission modelling, BioSystems, 126 (2014), 52–75.
  • 65. H. M. Yang and D. Greenhalgh, Proof of conjecture in: The basic reproduction number obtained from Jacobian and next generation matrices – A case study of dengue transmission modelling, Appl. Math. Comput., 265 (2015), 103–107.


This article has been cited by

  • 1. Manuel De la Sen, On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls, Symmetry, 2019, 11, 5, 712, 10.3390/sym11050712

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved