Previous |  Up |  Next


SIR epidemic models; vaccination; differential equation
This paper proposes a deterministic model for the spread of an epidemic. We extend the classical Kermack–McKendrick model, so that a more general contact rate is chosen and a vaccination added. The model is governed by a differential equation (DE) for the time dynamics of the susceptibles, infectives and removals subpopulation. We present some conditions on the existence and uniqueness of a solution to the nonlinear DE. The existence of limits and uniqueness of maximum of infected individuals are also discussed. In the final part, simulations, numerical results and comparisons of the different vaccination strategies are presented.
[1] Amann H.: Ordinary Differential Equations: An Introduction to Nonlinear Analysis. Walter de Gruyter, Berlin – New York 1990 MR 1071170 | Zbl 0708.34002
[2] Bailey N. T. J.: The Mathematical Theory of Epidemics. Hafner Publishing Company, New York 1957 MR 0095085
[3] Daley D. J., Gani J.: Epidemic Modelling: An Introduction. Cambridge University Press, Cambridge 1999 MR 1688203 | Zbl 0964.92035
[4] Greenwood P., Gordillo L. F., Marion A. S., Martin-Löf A.: Bimodal Epidemic Side Distributions for Near-Critical SIR with Vaccination. In preparation
[5] Kalas J., Pospíšil Z.: Spojité modely v biologii (Continuous Models in Biology). Masaryk University, Brno 2001
[6] Kermack W. O., McKendrick A. G.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. London A 155 (1927), 700–721
[7] Štěpán J., Hlubinka D.: Kermack–McKendrick epidemic model revisited. Kybernetika 43 (2007), 395–414 MR 2377919 | Zbl 1137.37338
[8] Štěpán J.: Private communicatio.
[9] Wai-Yuan T., Hulin W.: Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention. World Scientific, Singapore 2005 MR 2169300
Partner of
EuDML logo