# Three Basic Epidemiological Models

• Herbert W. Hethcote
Chapter
Part of the Biomathematics book series (BIOMATHEMATICS, volume 18)

## Abstract

There are three basic types of deterministic models for infectious diseases which are spread by direct person-to-person contact in a population. Here these simplest models are formulated as initial value problems for systems of ordinary differential equations and are analysed mathematically. Theorems are stated regarding the asymptotic stability regions for the equilibrium points and phase plane portraits of solution paths are presented. Parameters are estimated for various diseases and are used to compare the vaccination levels necessary for herd immunity for these diseases. Although the three models presented are simple and their mathematical analyses are elementary, these models provide notation, concepts, intuition and foundation for considering more refined models. Some possible refinements are disease-related factors such as the infectious agent, mode of transmission, latent period, infectious period, susceptibility and resistance, but also social, cultural, Ecology by providing a sound intuitive understanding and complete proofs for the three most basic epidemiological models for microparasitic infections.

## Keywords

Equilibrium Point Epidemic Model Initial Value Problem Endemic Equilibrium Herd Immunity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Anderson, R.M. (1982) Directly transmitted viral and bacterial infections of man. In: Anderson, R.M. (ed.) Population Dynamics of Infectious Diseases. Theory and Applications. Chapman and Hall, New York, pp. 1–37.Google Scholar
2. Anderson, R.M., Jackson, H.C., May, R.M., Smith, A.D.M. (1981) Populations dynamics of fox rabies in Europe. Nature 289, 765–777.
3. Anderson, R.M., May, R.M. (1979) Population biology of infectious diseases I. Nature 280, 361–367
4. Anderson, R.M., May, R.M. (1981) The population dynamics of microparasites and their invertebrate hosts. Phil. Trans. Roy. Soc. London B291, 451–524.Google Scholar
5. Anderson, R.M., May, R.M. (1982) Directly transmitted infectious diseases: control by vaccination. Science 215, 1053–1060.
6. Anderson, R.M., May, R.M. (1983) Vaccination against rubella and measles: quantitative investigations of different policies. J. Hyg. Camb. 90, 259–325.
7. Anderson, R.M., May, R.M. (1985) Vaccination and herd immunity to infectious diseases. Nature 318, 323–329.
8. Bailey, N.T.J. (1975) The Mathematical Theory of Infectious Diseases, 2nd edn. Hafner, New York.
9. Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A., Liu, W.M. (1988) Cross-immunity in the dynamics of homogeneous and heterogeneous populations, In: T.G. Hallam, L. Gross, and S.A. Levin (eds.) Mathematical Ecology, World Scientific Publishing, Singapore, 303–316.Google Scholar
10. Centers for Disease Control (1971a) Infectious hepatitis-Kentucky. Morbidity and Mortality Weekly Report 20, 136–137Google Scholar
11. Centers for Disease Control (1971b) Measles-Dallas, Texas, Morbidity and Mortality Weekly Report 20, 191–192.Google Scholar
12. Centers for Disease Control (1981) Rubella-United States, 1978–1981. Morbidity and Mortality Weekly Report 30, 513–515.Google Scholar
13. Centers for Disease Control (1984) Measles in an immunized school-aged population-New Mexico. Morbidity and Mortality Weekly Report 34, 52–59.Google Scholar
14. Centers for Disease Control (1986a) Annual summary 1984: reported morbidity and mortality in the United States. Morbidity and Mortality Weekly Report 33 (54).Google Scholar
15. Centers for Disease Control (1986b) Rubella and congenital rubella syndrome-United States 1984–1985. Morbidity and Mortality Weekly Report 35, 129–135.Google Scholar
16. Centers for Disease Control (1987a) Measles-Dade County, Florida. Morbidity and Mortality Weekly Report 36, 45–48.Google Scholar
17. Centers for Disease Control (1987b) Enterically transmitted non-A, non-B hepatitis-East Africa, Morbidity and Mortality Weekly Report 36, 241–244.Google Scholar
18. Coddington, E.A., Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.
19. Coleman, C.S. (1978) Biological cycles and the fivefold way. In: Braun, M., Coleman, C.S., Drew, D.A. (eds.) Differential Equation Models. Springer, New York, pp. 251–278.Google Scholar
20. Dietz, K. (1975) Transmission and control of arbovirus diseases. In: Ludwig D. and Cooke, K.L (eds.) Epidemiology. SIMS 1974 Utah Conference Proceedings, SIAM, Philadelphia, pp. 104–121Google Scholar
21. Dietz, K. (1976) The incidence of infectious diseases under the influence of season fluctuations. In: Berger, J., Buhler, R., Repges, R., Tantu, P. (eds.) Mathematical Models in Medicine. Lecture Notes in Biomathematics, vol. 11. Springer, New York, pp. 1–15.Google Scholar
22. Evans, A.S. (1982) Viral Infections of Humans 2nd edn. Plenum Medical Book Company, New York.
23. Fenner, F. (1983) Biological control, as exemplified by smallpox eradication and myxomatosis. Proc. Roy. Soc. London B218, 259–285.Google Scholar
24. Fine, P.E.M., Clarkson, J.A. (1982) Measles in England and Wales I: An analysis of factors underlying seasonal patterns, and II: The impact of the measles vaccination programme on the distribution of immunity in the population. Int. J. Epid. 11, 5–14 and 15–24.
25. Guckenheimer, J., Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York.Google Scholar
26. Hamer, W.H. (1906) Epidemic disease in England. Lancet 1, 733–739.Google Scholar
27. Hethcote, H.W. (1973) Asymptotic behavior in a deterministic epidemic model. Bull. Math. Biology 35, 607–614.
28. Hethcote, H.W. (1974) Asymptotic behavior and stability in epidemic models. In: van den Driessche P. (ed.) Mathematical Problems in Biology. Lecture Notes in Biomathematics, vol. 2. Springer, Berlin Heidelberg New York, pp. 83–92.Google Scholar
29. Hethcote, H.W. (1976) Qualitative analysis for communicable disease models. Math. Biosci. 28,335–356.
30. Hethcote, H.W. (1978) An immunization model for a heterogeneous population. Theor. Prop. Biol. 14, 338–349.
31. Hethcote, H.W. (1983) Measles and rubella in the United States. Am. J. Epidemiol. III, 2–13.Google Scholar
32. Hethcote, H.W. (1988) Optimal ages of vaccination for measles. Math. Biosci. 89, 29–52.
33. Hethcote, H.W. (1989) Rubella. In: Levin, S.A., Hallam, T.G., Gross, L. (eds.) Applied Mathematical Ecology. Biomathematics, vol. 18. Springer, Berlin, Heidelberg, New York.Google Scholar
34. Hethcote, H.W., Levin, S.A. (1988) Periodicity in epidemiological models. In: Levin, S.A., Hallam, T.G. Gross, L. (eds.) Applied Mathematical Ecology. Biomathematics, vol. 18. Springer, Berlin, Heidelberg, New York, 193–211.Google Scholar
35. Hethcote, H.W., Stech, H.W., van den Driessche, P. (1981a) Nonlinear oscillations in epidemic models. SIAM J. Appl. Math. 40, 1–9.
36. Hethcote, H.W. Stech, H.W., van den Driessche, P. (1981b) Stability analysis for models of diseases without immunity. J. Math. Biology 13, 185–198.
37. Hethcote, H.W., Stech, H.W., van den Driessche, P. (1981c) Periodicity and stability in epidemic models: a survey. In: Busen berg, S. and Cooke, K.L. (eds.) Differential Equations and Applications in Ecology, Epidemics and Populations Problems. Academic Press, New York, pp. 65–82.Google Scholar
38. Hethcote, H.W., Tudor, D.W. (1980) Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37–47.
39. Hethcote, H.W., Van Ark, J.W. (1987) Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation and immunization programs. Math. Biosci. 84, 85–118.
40. Hethcote, H.W., Yorke, J.A. (1984) Gonorrhea Transmission Dynamics and Control. Lecture Notes in Biomathematics, vol. 56, Springer, Berlin Heidelberg New York.Google Scholar
42. Jordan, D.W., Smith, P. (1977) Nonlinear Ordinary Differential Equations. Oxford University Press, Oxford.
43. Kermack, W.O., McKendrick, A.G. (1927) A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. A115, 700–721.Google Scholar
44. London, W.A., Yorke, J.A. (1973) Recurrent outbreaks of measles, chickenpox and mumps. I. Am. J. Epid. 98, 453–468.Google Scholar
45. Longini, I.M., Jr. (1986) The generalized discrete-time epidemic model with immunity: a synthesis. Math. Biosci. 82, 19–41.
46. Longini, I.M., Jr., Ackerman, E., Elveback, L.R. (1978) An optimization model for influenza A epidemics. Math. Biosci. 38, 141–157
47. May, R.N. (1986) Population biology of microparasitic infections, In: Hallam T.G. and Levin, S.A. (eds.) Mathematical Ecology. Biomathematics, vol. 17. Springer, Berlin, Heidelberg, New York, pp. 405–442.Google Scholar
48. May, R.M., Anderson, R.M. (1979) Population biology of infectious diseases II. Nature 280, 455–461.
49. May, R.M., Anderson, R.M. (1984a) Spatial heterogeneity and the design of immunization programs. Math. Biosci. 72, 83–111.
50. May, R.M., Anderson, R.M. (1984b) Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes. IMA J. of Math. App. Med. Biol. 1, 233–266.
51. Miller, R.K., Michel, A.N. (1982) Ordinary Differential Equations. Academic Press, New York.
52. Mollison, D. (1977) Spatial contact models for ecological and epidemic spread. J.R. Statist. Soc. Ser. B39, 283–326.
53. Mollison, D., Kuulasmaa, K. (1985) Spatial epidemic models: theory and simulations. In: Bacon, P.J. (ed.) Population Dynamics of Rabies in Wildlife. Academic Press, London, pp. 291–309Google Scholar
54. Radcliffe, J., Rass, L. (1986) The asymptotic speed of propagation of the deterministic nonreducible H-type epidemic. J. Math. Biol. 23, 341–359.
55. Ross, R. (1911). The Prevention of Malaria, 2nd edn. Murray, London.Google Scholar
56. Rvachev, L.A., Longini, I.M. Jr. (1985) A mathematical model for the global spread ofinfluenza. Math. Biosci. 75, 3–22.
57. Schenzle, D. (1984) An age structured model of pre and post-vaccination measles transmission.IMA J. Math. Appl. Biol. Med. 1, 169–191.
58. Soper, H.E. (1929) Interpretation of periodicity in disease prevalence. J.R. Statist. Soc. 92, 34—73.Google Scholar
59. World Health Organization (1980) The Global Eradication of Smallpox. Final report, WHO, GenevaGoogle Scholar
60. Yorke, J.A., London, W.P. (1973) Recurrent outbreaks of measles, chickenpox and mumps II. Am. J. Epid. 98, 469–482.Google Scholar
61. Yorke, J.A., Nathanson, N. Pianigiani, G. Martin, J. (1979) Seasonality and the requirementsfor prepetuation and eradication of viruese In population. Am.J. Emplidemiol. 109, 103–123Google Scholar

## Authors and Affiliations

• Herbert W. Hethcote

There are no affiliations available