Terminal distributions along a 'knight's line' for a stochastic epidemic

The model under consideration relates to the spread of a disease in a finite population where removal of infectives is allowed to occur. It is equivalent to a restricted random walk over a discrete grid of points interior to a trapezium. We consider an artificial barrier on the grid at which the epidemic ceases. The barrier is a semi-diagonal resembling the path of the knight in chess and allows some simplification of the analysis. In particular, we investigate the probability distribution along this barrier when time is infinite. Recursive relations are obtained for these terminal distributions and are used to develop an exact numerical method for their computation. In certain cases, a slight modification of the binomial distribution provides a good approximation to the actual distribution. A direct recursive method for calculating moments of the distribution is given and we show how the whole distribution may be recovered from these moments.