Epidemiological Models with Non-Exponentially Distributed Disease Stages and Applications to Disease Control

Abstract

SEIR epidemiological models with the inclusion of quarantine and isolation are used to study the control and intervention of infectious diseases. A simple ordinary differential equation (ODE) model that assumes exponential distribution for the latent and infectious stages is shown to be inadequate for assessing disease control strategies. By assuming arbitrarily distributed disease stages, a general integral equation model is developed, of which the simple ODE model is a special case. Analysis of the general model shows that the qualitative disease dynamics are determined by the reproductive number \(\mathcal R_c\), which is a function of control measures. The integral equation model is shown to reduce to an ODE model when the disease stages are assumed to have a gamma distribution, which is more realistic than the exponential distribution. Outcomes of these models are compared regarding the effectiveness of various intervention policies. Numerical simulations suggest that models that assume exponential and non-exponential stage distribution assumptions can produce inconsistent predictions.

References

  1. Anderson, R.M., May, R.M., 1979. Prevalence of schistosome infections within molluscan populations: Observed patterns and theoretical predictions. Parasitology 79, 63–94.

    Article  Google Scholar 

  2. Chowell, G., Fenimorea, P., Castillo-Garsowc, M., Castillo-Chavez, C., 2003. SARS outbreaks in Ontario, Hong Kong and Singapore: The role of diagnosis and isolation as a control mechanism. J. Theo. Bio. 224, 1–8.

    Article  Google Scholar 

  3. Feng, Z., Huang, W., Castillo-Chavez, C., 2001. On the role of variable latent periods in mathematical models for tuberculosis. J. Dyn. Diff. Eqs. 13, 435–452.

    Article  MathSciNet  Google Scholar 

  4. Feng, Z., Thieme, H.R., 2000a. Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: General theory. SIAM J. Appl. Math. 61(3), 803–833.

    MATH  Article  MathSciNet  Google Scholar 

  5. Feng, Z., Thieme, H.R., 2000b. Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages II: Fast disease dynamics and permanent recovery. SIAM J. Appl. Math. 61(3), 983–1012.

    MATH  Article  MathSciNet  Google Scholar 

  6. Hethcote, H., Tudor, D., 1980. Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47.

    MATH  Article  MathSciNet  Google Scholar 

  7. Hethcote, H., Stech, H.W., van den Driessche, P., 1981. Nonlinear oscillations in epidemic models. SIAM J. Appl. Math. 40(1), 1–9.

    MATH  Article  MathSciNet  Google Scholar 

  8. Kermack, W.O., McKendrick, A.G., 1927. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. A 115, 700–721.

    Article  Google Scholar 

  9. Kermack, W.O., McKendrick, A.G., 1932. Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc. Roy. Soc. A 138, 55–83.

    MATH  Article  Google Scholar 

  10. Kermack, W.O., McKendrick, A.G., 1933. Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity. Proc. Roy. Soc. A 141, 94–122.

    MATH  Article  Google Scholar 

  11. Lipsitch, M., Cohen, T., Copper, B., et al., 2003. Transmission dynamics and control of severe acute respiratory syndrome. Science 300, 1966–1970.

    Article  Google Scholar 

  12. Lloyd, A., 2001a. Realistic distributions of infectious periods in epidemic models. Theor. Pop. Biol. 60, 59–71.

    Article  Google Scholar 

  13. Lloyd, A., 2001b. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. Lond. B. 268, 985–993.

    Article  Google Scholar 

  14. MacDonald, N., 1978. Time Lags in Biological Models. Springer-Verlag, New York.

    Google Scholar 

  15. McLean, A.R., Robert, M.M., Pattison, J., Weiss, R.A., 2005. A Case Study in Emerging Infections. Springer-Verlag, Oxford University Press, New York.

    Google Scholar 

  16. Plant, R.E., Wilson, L.T., 1986. Models for age-structured populations with distributed maturation rates. J. Math. Biol. 23, 247–262.

    MATH  Article  MathSciNet  Google Scholar 

  17. Riley, S., Fraser, C., Donnelly, C.A., Ghani, A.C., et al., 2003. Transmission dynamics of the etiological agent of SARS in Hong Kong—impact of public health interventions. Science 300, 1961–1966.

    Article  Google Scholar 

  18. Ross, R., 1911. The Prevention of Malaria. John Murray, London.

    Google Scholar 

  19. Taylor, H.M., Karlin, S., 1998. An Introduction to Stochastic Modeling, 3rd edition. Academic Press, San Diego.

    Google Scholar 

  20. Thieme, H., 2003. Mathematics in Population Biology. Princeton University Press, Princeton.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zhilan Feng.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Feng, Z., Xu, D. & Zhao, H. Epidemiological Models with Non-Exponentially Distributed Disease Stages and Applications to Disease Control. Bull. Math. Biol. 69, 1511–1536 (2007). https://doi.org/10.1007/s11538-006-9174-9

Download citation

Keywords

  • Epidemiological model
  • Distributed disease stage
  • Integral equation
  • Disease control strategies