Skip to main content
PMC home page
Springer Nature - PMC COVID-19 Collection logoLink to Springer Nature - PMC COVID-19 Collection
. 2007 Aug 15;70(1):134–155. doi: 10.1007/s11538-007-9245-6

On the Role of Asymptomatic Infection in Transmission Dynamics of Infectious Diseases

Sze-Bi Hsu 1, Ying-Hen Hsieh 2,✉
PMCID: PMC7089371  PMID: 17701259

Abstract

We propose a compartmental disease transmission model with an asymptomatic (or subclinical) infective class to study the role of asymptomatic infection in the transmission dynamics of infectious diseases with asymptomatic infectives, e.g., influenza. Analytical results are obtained using the respective ratios of susceptible, exposed (incubating), and asymptomatic classes to the clinical symptomatic infective class. Conditions are given for bistability of equilibria to occur, where trajectories with distinct initial values could result in either a major outbreak where the disease spreads to the whole population or a lesser outbreak where some members of the population remain uninfected. This dynamic behavior did not arise in a SARS model without asymptomatic infective class studied by Hsu and Hsieh (SIAM J. Appl. Math. 66(2), 627–647, 2006). Hence, this illustrates that depending on the initial states, control of a disease outbreak with asymptomatic infections may involve more than simply reducing the reproduction number. Moreover, the presence of asymptomatic infections could result in either a positive or negative impact on the outbreak, depending on different sets of conditions on the parameters, as illustrated with numerical simulations. Biological interpretations of the analytical and numerical results are also given.

Keywords: Influenza, Asymptomatic infection, Basic reproduction number, Bistability, Threshold asymptomatic fraction

References

  1. Arino J., Brauer F., van den Driessche P., Watmough J., Wu J. Simple models for containment of a pandemic. J. R. Soc. Interface. 2006;3:453–457. doi: 10.1098/rsif.2006.0112. [DOI] [PMC free article] [PubMed] [Google Scholar]
  2. Bell D.M., World Health Organization Writing Group Nonpharmaceutical interventions for pandemic influenza, international measures. Emerg. Infect. Dis. 2006;12(1):81–87. doi: 10.3201/eid1201.051370. [DOI] [PMC free article] [PubMed] [Google Scholar]
  3. Chan P.K. Outbreak of avian influenza A(H5N1) virus infection in Hong Kong in 1997. Clin. Infect. Dis. 2002;34(Suppl. 2):S58–S64. doi: 10.1086/338820. [DOI] [PubMed] [Google Scholar]
  4. Coppell W.A. Stability and Asymptotic Behavior of Solutions of Differential Equations. Boston: Heath; 1965. [Google Scholar]
  5. Diekmann O., Heesterbeek J.A.P. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. New York: Wiley; 2000. [Google Scholar]
  6. Eccles R. Asymptomatic spread of flu is not proved. Br. Med. J. 2005;331(7525):1145. doi: 10.1136/bmj.331.7525.1145. [DOI] [PMC free article] [PubMed] [Google Scholar]
  7. Ferguson N.M., Cummings D.A., Cauchemez S., Fraser C., Riley S., Meeyai A., Iamsirithaworn S., Burke D.S. Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature. 2005;437:209–214. doi: 10.1038/nature04017. [DOI] [PubMed] [Google Scholar]
  8. Ferguson N.M., Cummings D.A., Fraser C., Cajka J.C., Cooley P.C., Burke D.S. Strategies for mitigating an influenza pandemic. Proc. Natl. Acad. Sci. U.S.A. 2006;103(15):5935–5940. doi: 10.1073/pnas.0601266103. [DOI] [PMC free article] [PubMed] [Google Scholar]
  9. Germann T.C., Kadau K., Longini I.M., Jr., Macken C.A. Mitigation strategies for pandemic influenza in the United States. Proc. Natl. Acad. Sci. U.S.A. 2006;103:5935–5940. doi: 10.1073/pnas.0601266103. [DOI] [PMC free article] [PubMed] [Google Scholar]
  10. Graat J.M., Schouten E.G., Heijnen M.L., Kok F.J., Pallast E.G., de Greeff S.C., Dorigo-Zetsma J.W. A prospective, community-based study on virologic assessment among elderly people with and without symptoms of acute respiratory infection. J. Clin. Epidemiol. 2003;56(12):1218–1223. doi: 10.1016/S0895-4356(03)00171-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
  11. Hsieh Y.-H., van den Driessche P., Wang L. A multi-patch model for spatial spread of disease: impact of travel between patches. Bull. Math. Biol. 2007;69(4):1355–1375. doi: 10.1007/s11538-006-9169-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
  12. Hsu S.B., Hsieh Y.-H. Modeling intervention measures and public response during SARS outbreak. SIAM J. Appl. Math. 2006;66(2):627–647. doi: 10.1137/040615547. [DOI] [Google Scholar]
  13. Longini I.M., Halloran M.E., Nizam A., Yang Y. Containing pandemic influenza with antiviral agents. Am. J. Epidemiol. 2004;159:623–633. doi: 10.1093/aje/kwh092. [DOI] [PubMed] [Google Scholar]
  14. Longini I.M., Nizam A., Xu S., Ungchusak K., Hanshaoworakul W., Cummings D.A., et al. Containing pandemic influenza at the source. Science. 2005;309:1083–1087. doi: 10.1126/science.1115717. [DOI] [PubMed] [Google Scholar]
  15. Monto A.S., Gunn R.A., Bandyk M.G., King C.L. Prevention of Russian influenza by amantadine. J. Am. Med. Assoc. 1979;241:1003–1007. doi: 10.1001/jama.241.10.1003. [DOI] [PubMed] [Google Scholar]
  16. Nafta I., Turcanu A.G., Braun I., Companetz W., Simionescu A., Birt E., Florea V. Administration of amantadine for the prevention of Hong Kong influenza. Bull. World Health. Organ. 1970;42:423–427. [PMC free article] [PubMed] [Google Scholar]
  17. Oker-Blom N., Hovi T., Leinikki P., Palosuo T., Pettersson R., Suni J. Protection of man from natural infection with influenza A2 Hong Kong virus by amantadine: a controlled field trial. Br. Med. J. 1970;3:676–678. doi: 10.1136/bmj.3.5724.676. [DOI] [PMC free article] [PubMed] [Google Scholar]
  18. Pettersson R.F., Hellstrom P.E., Penttinen K., Pyhala R., Tokola O., Vartio T., Visakorpi R. Evaluation of amantadine in the prophylaxis of influenza A (H1N1) virus infection: a controlled field trial among young adults and high-risk patients. J. Infect. Dis. 1980;142:377–383. doi: 10.1093/infdis/142.3.377. [DOI] [PubMed] [Google Scholar]
  19. Quarles J.M., Couch R.B., Cate T.R., Goswick C.B. Comparison of amantadine and rimantadine for prevention of type A (Russian) influenza. Antiviral. Res. 1981;1:149–155. doi: 10.1016/0166-3542(81)90003-6. [DOI] [PubMed] [Google Scholar]
  20. Stilianakis N.I., Perelson A.S., Hayden F.G. Emergence of drug resistance during an influenza epidemic: insights from a mathematical model. J. Infect. Dis. 1998;177(4):863–873. doi: 10.1086/515246. [DOI] [PubMed] [Google Scholar]
  21. Webster R.G. Wet markets—a continuing source of severe acute respiratory syndrome and influenza? Lancet. 2004;363(9404):234–236. doi: 10.1016/S0140-6736(03)15329-9. [DOI] [PMC free article] [PubMed] [Google Scholar]

Articles from Bulletin of Mathematical Biology are provided here courtesy of Nature Publishing Group

RESOURCES