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.
Anderson, R.M., May, R.M., 1979. Prevalence of schistosome infections within molluscan populations: Observed patterns and theoretical predictions. Parasitology 79, 63–94.
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.
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.
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.
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.
Hethcote, H., Tudor, D., 1980. Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47.
Hethcote, H., Stech, H.W., van den Driessche, P., 1981. Nonlinear oscillations in epidemic models. SIAM J. Appl. Math. 40(1), 1–9.
Kermack, W.O., McKendrick, A.G., 1927. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. A 115, 700–721.
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.
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.
Lipsitch, M., Cohen, T., Copper, B., et al., 2003. Transmission dynamics and control of severe acute respiratory syndrome. Science 300, 1966–1970.
Lloyd, A., 2001a. Realistic distributions of infectious periods in epidemic models. Theor. Pop. Biol. 60, 59–71.
Lloyd, A., 2001b. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. Lond. B. 268, 985–993.
MacDonald, N., 1978. Time Lags in Biological Models. Springer-Verlag, New York.
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.
Plant, R.E., Wilson, L.T., 1986. Models for age-structured populations with distributed maturation rates. J. Math. Biol. 23, 247–262.
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.
Ross, R., 1911. The Prevention of Malaria. John Murray, London.
Taylor, H.M., Karlin, S., 1998. An Introduction to Stochastic Modeling, 3rd edition. Academic Press, San Diego.
Thieme, H., 2003. Mathematics in Population Biology. Princeton University Press, Princeton.
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
- Epidemiological model
- Distributed disease stage
- Integral equation
- Disease control strategies