Bulletin of Mathematical Biology

, Volume 70, Issue 1, pp 134–155 | Cite as

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

  • Sze-Bi Hsu
  • Ying-Hen HsiehEmail author
Original Article


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.


Influenza Asymptomatic infection Basic reproduction number Bistability Threshold asymptomatic fraction 


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

Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  1. 1.Department of MathematicsNational Tsing-Hua UniversityHsinchuTaiwan
  2. 2.Department of Public Health and Biostatistics CenterChina Medical UniversityTaichungTaiwan

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