The solution of a recursive sequence arising from a combinatorial problem in botanical epidemiology |
| |
Authors: | Z AlSharawi A Burstein M Deadman A Umar |
| |
Institution: | 1. Department of Mathematics and Statistics , Sultan Qaboos University , P.O. Box 36, PC 123, Al-Khod , Sultanate of Oman alsha1zm@alsharawi.info;3. Department of Mathematics , Howard University , Washington , DC , 20059 , USA;4. Department of Crop Sciences , Sultan Qaboos University , P.O. Box 34, PC 123, Al-Khod , Sultanate of Oman;5. Department of Mathematics and Statistics , Sultan Qaboos University , P.O. Box 36, PC 123, Al-Khod , Sultanate of Oman |
| |
Abstract: | One of the central problems in botanical epidemiology is whether disease spreads within crops in a regular pattern or follows a random process. In this study, we consider a row of n plants in which m are infected. We then develop a rigorous mathematical approach to investigate the total number of ways to obtain k isolated individuals among m infected plants. We give a recurrence relation in three parameters that describes the problem, then we find a closed-form solution, and give two different approaches to tackle the proof. Finally, we find interesting formulae for the expectation and variance of the random variable that represents the number of infected and isolated plants. |
| |
Keywords: | spread of disease recurrence relation binomial coefficients hypergeometric function |
|
|