Abstract
Kendall's (1956) approach to the ‘general’ epidemic is generalized by dropping the assumptions of constant infectivity and random recovery or death of ill individuals. A great deal of attention is paid to the biological background and the heuristics of the model formulation. Some new results are: (l) the derivation of Kermack's and McKendrick's integral equation from what seems to be the most general set of assumptions in section 2.2, (2) the use of Kermack's and McKendrick's final value equation to arrive at a finite time version of the threshold theorem for the general case, comparable to that for the case of only one Markovian state of illness in section 2.5, (3) the analysis of the behaviour of the solutions of the integral equation when the starting infection approaches zero in section 2.7, (4) the derivation of the probability structure of a general branching process, after conditioning on extinction in section 3.6, (5) the statement of the generalized versions of Kendall's ideas in the form of precise limit conjectures in section 4, (6) the derivation of a closed expression for the limit epidemic resulting from (3) in appendix 4.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bailey, N. T. J. (1975). The mathematical theory of infectious diseases. - London, Griffin, xvi+412 pp.
Barbour, A. D. (1974). On a functional central limit theorem for Markov population processes. - Adv. appl. Prob. 6, p. 21–39.
Barbour, A. D. (1975a). A note on the maximum size of a closed epidemic. - J. R. statist. Soc. B 37, p. 459–460.
Barbour, A. D. (1975b). The duration of the closed stochastic epidemic. - Biometrika 62, p. 477–482.
Barlow, R. E. & F. Proschran (1975). Statistical theory of reliability and life testing. Vol. I. Probability models. - New York, Holt, Rinehart & Winston, xiii + 290 pp.
Bellman, R. & K. L. Cooke (1963). Differential difference equations. - New York, Acad. Press, xvi + 462 pp.
Cooke, K. L. (1967). Functional differential equations: some models and perturbation problems. - In: J. K. Hale & J. P. La Salle, eds., Differential equations and dynamical systems, p. 167–183. - New York, Acad. Press, xvii + 544 pp.
Daniels, H. E. (1961). Mixtures of geometric distributions. - J. R. statist. Soc. B 23, p. 409–413.
Daniels, H. E. (1974). The maximum size of a closed epidemic. - Adv. appl. Prob. 6, p. 607–621.
Diekmann, O. (1977). Limiting behaviour in an epidemic model. - Nonlinear Analysis, Theory, Methods and Applications 1, p. 459–470.
Doney, R. A. (1972). A limit theorem for a class of supercritical branching processes. - J. appl. Prob. 9, p. 707–724.
Feller, W. (1968). An introduction to probability theory and its applications. vol. I, 3rd ed. - New York, Wiley, xviii + 509 pp.
Feller, W. (1971). An introduction to probability theory and its applications. vol. II, 2nd ed. - New York, Wiley, xxiv + 669 pp.
Grasman, J. & B. J. Matkowsky (1976). Evolution of an epidemic with threshold. - In prep.
Haight, F. A. (1961). An analogue to the Borel-Tanner distribution. - Biometrika 48, p. 167–173.
Haight, F. A. & M. A. Breuer (1960). The Borel-Tanner distribution. - Biometrika 47, p. 143–150.
Hethcote, H. (1970). Note on determining the limiting susceptible population in an epidemic model. - Math. Biosciences 9, p. 161–163.
Hoppensteadt, F. (1975). Mathematical theories of populations: demographics, genetics and epidemics. - SIAM, vii + 72 pp.
Hoppensteadt, F. & P. Waltmann (1970). A problem in the theory of epidemics. I. - Math. Biosciences 9, p. 71–91.
Hoppensteadt, F. & P. Waltmann (1971). A problem in the theory of epidemics. II. - Math. Biosciences 12, p. 133–145.
Jagers, P. (1975). Branching processes with biological applications. - London, Wiley, xiii + 268 pp.
Kaplan, N. (1975). A note on the supercritical generalized age-dependent branching process. - J. appl. Prob. 12, p. 341–345.
Karlin, S. & H. M. Taylor (1975). A first course in stochastic processes. 2nd ed. - New York, Academic Press, xvi + 557 pp.
Kendall, D. G. (1949). Stochastic processes and population growth. - J. R. statist. Soc. B 11, p. 230–264.
Kendall, D. G. (1956). Deterministic and stochastic epidemics in closed populations. - Proc. 3rd Berkeley Symp. Math. Stat. IV, p. 149–165.
Kermack, W. O. & A. G. McKendrick (1927). A contribution to the mathematical theory of epidemics. - Proc. Royal Soc. London Ser. A, 115, p. 700–721.
Keyfitz, N. (1968). Introduction to the mathematics of population. - Reading Mass. Addison-Wesley, xiv + 450 pp.
Kryscio, R. J. (1975). The transition probabilities of the general stochastic epidemic model. - J. appl. Prob. 11, p. 415–424.
Kurtz, T. G. (1971). Limit theorems for sequences of jump Markov processes approximating ordinary differential equations. - J. appl. Prob. 8, p. 344–356.
Landau, H. G. (1952). On some problems of random nets. - Bull. Math. Biophys. 14, p. 203–212.
Landau, H. G. & A. Rapoport (1953). Contribution to the mathematical theory of contagion and spread of information: I. Spread through a thoroughly mixed population. - Bull. math. Biophys. 15, p. 173–183.
Lauwerier, H. A. (1973). College biomathematica. - Universiteit van Amsterdam, 82 pp.
Lauwerier, H. A. (1976). Mathematische modellen voor epidemische processen. - Handout at the 12th Dutch Mathematical Congress.
Lotka, A. J. (1956). Elements of mathematical biology. - New York, Dover, xxx + 465 pp. - Formerly: Elements of physical biology (1925). - Baltimore, Williams & Wilkins.
Ludwig, D. (1974). Stochastic population theories. - Lecture Notes in Biomathematics 3. - Berlin, Springer Verlag, vi + 108 pp.
Ludwig, D. (1975a). Final size distribution for epidemics. - Math. Biosciences 23, p. 33–46.
Ludwig, D. (1975b). Qualitative behaviour of stochastic epidemics. - Math. Biosciences 23, 47–73.
McKendrick, A. G. (1914). Studies on the theory of continuous probabilities with special reference to its bearing on natural phenomena of a progressive nature. - Proc. Lond. math. Soc. (2) 13, p. 401–416.
McKendrick, A. G. (1926). Applications of mathematics to medical problems. - Proc. Edinburgh math. Soc. 44, p. 98–130.
Mollison, D. (1977). Spatial contact models for ecological and epidemic spread. -J. R. statist. Soc. B, 39, p. 283–326.
Otter, R. (1949). The multiplicative process. - Am. Math. Statist. 20, p. 206–224.
Reddingius, J. (1971). Notes on the mathematical theory of epidemics. - Acta biotheoretica 10, p. 125–157.
Waltmann, P. (1974). Deterministic threshold models in the theory of epidemics. - Lecture Notes in Biomathematics 1. - Berlin, Springer-Verlag, ii + 101 pp.
Wang, F. J. S. (1975). Limit theorems for age and density dependent stochastic population models. - J. Math. Biology 2, p. 373–400.
Waugh, W. A. O'N. (1958). Conditioned Markov processes. - Biometrika 45, p. 241–249.
Wilson, L. O. (1972). An epidemic model involving a threshold. - Math. Biosciences 15, p. 109–121.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Metz, J.A.J. The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections. Acta Biotheor 27, 75–123 (1978). https://doi.org/10.1007/BF00048405
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00048405