On a combinatorial problem in botanical epidemiology

In botanical epidemiology, one of the major problems is to study the spread of diseases within crops. Several approaches to study the patterns and temporal evolution of diseases have been discussed in the literature, e.g., statistical techniques or variogram analysis. Recently, AlSharawi, Burstein, Deadman and Umar determined the total number of ways to have ℓ isolated infected individuals among m infected plants in a row of n plants. They discussed this in a straightforward combinatorial fashion and considered several associated points, like expectation value and variance of the number of isolated infected plants. In the present paper, we derive their results with the help of generating function techniques and use this method to extend the discussion to plants arranged in a circle as well as in two rows. The dependence of the expected number of isolated infected plants on the proportion of infected plants is considered for large n in all these cases as well as in the case of an arbitrary number of rows and a simple asymptotic behavior is found. Connections to several combinatorial sequences are established.