**Martingale in Epidemic Theory**

Helmut Knolle

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A closed and homogeneously mixing
population is subdivided into tour classes, the susceptibles (S), the latent
infected (L), the infectious (I), and the immune or removed persons (R). It is
assumed that infectious persons may have contacts with any other member of the
population with equal probability. Any susceptible, once contacted by an
infective, becomes latent infected, then infective and at last immune or
removed from the infection process. In order to get a stochastic epidemic model
in discrete time it is assumed that the latent period has fixed length 1
whereas the infectious period reduces to a point. Thus, if a susceptible is
infected at time t, he (she) will be infectious at time t+1 and then removed.
An epidemic model of the SIR type in discrete time is a bivariate homogeneous
Markov chain (St, It), with specified transition probabilities, where St is the
number of susceptibles and It is the number of infectives at time t. Once I_t=
0, no further infection can occur, i.e. the epidemic is extinct T = mín {t : I_t = 0} is a stopping time and

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**Key words: **Final size of an epidemic, Harkov
chain, Martingale, Optional stopping theorem.

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