On a combinatorial problem in botanical epidemiology

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• 作者：Toufik Mansour^a ; ^{tmansour@univ.haifa.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Matthias Schork^b ; ^{mschork@member.ams.org" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Spread of disease ; Lattice model ; Recurrence relation
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 March 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：131-150
• 全文大小：488 K

文摘

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004102&_mathId=si8.gif&_user=111111111&_pii=S0166218X15004102&_rdoc=1&_issn=0166218X&md5=228008c1b1507579b400fc9e5a818a9c" title="Click to view the MathML source">ℓclass="mathContainer hidden">class="mathCode">$\ell$ isolated infected individuals among class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004102&_mathId=si9.gif&_user=111111111&_pii=S0166218X15004102&_rdoc=1&_issn=0166218X&md5=12f484b0af56cde6cd8e1639e5a82fba" title="Click to view the MathML source">mclass="mathContainer hidden">class="mathCode">$m$ infected plants in a row of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004102&_mathId=si10.gif&_user=111111111&_pii=S0166218X15004102&_rdoc=1&_issn=0166218X&md5=e5ded2df2892c4b1c4b1dd57509072a1" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode">$n$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15004102&_mathId=si10.gif&_user=111111111&_pii=S0166218X15004102&_rdoc=1&_issn=0166218X&md5=e5ded2df2892c4b1c4b1dd57509072a1" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode">$n$ 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.