Martingale in Epidemic Theory

Helmut Knolle



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 S0- ST, is the final size of the epidemic, i.e. the number of persons who became infected during the epidemic. Lefevre & Picard (1989) have shown that a certain function of  St, It and an integer-valued parameter r is a martingale, where 1 ≤ r ≤ n = S0. Applying the optional stopping theorem, they obtain a triangular system of n linear equations from which the distribution of the final size of the epidemic can be calculated.


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


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