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The Kermack-McKendrick Model of Epidemics and Its Extensions

Jan Hurt
Organization: Charles University of Prague

2006 Wolfram Technology Conference
Conference location

Champaign IL

In the classical deterministic model of epidemics proposed by Kermack and McKendrick in 1927 it is assumed that the population of size a, constant in time a, consists of three parts: a, the number of susceptibles (exposed to the infection), a, the number of infectives, and a, the number of removals (recovered, dead, or isolated), a. The model is governed by the system of ODEs. Positive constants a and a represent the infection and recovery rates, respectively. Thus the transition a is realized with rate a, a with rate γ, consequently a with rate a. In the first part the numerical solution of the above system by NDSolve will be discussed. In the second part selected intervention models will be studied. Intervention takes the form of vaccination at time where the epidemic spreads to some unacceptable or prescribed level and may be modeled by changing a. Procedures, utilizing NDSolve again, will be given. The third part will be devoted to a stochastic generalization in that the population size is a stochastic process taking into account immigration and/or emigration, for example, the population size a is governed by the stochastic differential equation. The processes a, a, and a behave on average like in the deterministic case. Various approaches will be analyzed and a simulation model will be developed. All the methods will be numerically demonstrated on influenza and other diseases epidemics.

*Science > Medicine

Kermack-McKendrick, epidemics, disease
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