I haul this problem over from Pielou, 1965 (The spread of disease in patchily-infected forest stands. Forest Science, 11(1):18–26).
Diseased plants in a plant population are often found to be patchily dispersed, patches of diseased plants being intermixed with gaps of healthy plants. Within each patch a certain proportion of the plants will be diseased, and another proportion healthy. The disease can spread in two ways:
- Increase in the proportion $\pi$ of the total area of the stand occupied by patches, and
- Increase in the proportion $v$ of within-patch trees that are diseased.
To estimate the parameters $\pi$ and $v$ take one or several transects through the study area, noting the linear sequence of diseased and healthy trees. An example might be
DDD HHH D HHH DDD H
Each sequence of one or more of the same letter is termed a run, and the runs for this particular sequence are as shown by ordering of letters separated by space. The two end letters of the sequences are ignored. The mean length of each letter is determined. The mean run length of diseased trees $\bar{d}$ is defined as:
$$ \bar{d} = \frac{1}{1 - av} $$
The mean run length of diseased trees (for the above sequence) is $\bar{d} = (2 + 1 + 3)/3 = 2$.
To determine $\pi$ and $v$ it is necessary to make two assumptions:
- If a tree is in a patch, the probability that it is diseased is independent of its locality or the condition of the trees next to it. More succinctly, diseased and healthy trees within a patch are not segregated.
- The patches and gaps are randomly mingled
In modeling of epidemiology of plant stands, the probability $h(r)$ of a run of $r$ healthy trees is defined as $\frac{bvy + a^2uv}{1 - av}$ for $r = 1$ and for $r = 2$ is $\frac{bvxy + a^3u^2v + 2abuvy}{1 = av}$, where $a$, $b$, $u$, $v$, $x$ and $y$ are the parameters.
- $y$ is the probability that a tree in a gap is succeeded by a tree in a patch in the linear transect,
- $v$ is the proportion of within-patch trees that are diseased
- $a$ is the probability that a patch tree is succeeded by another patch tree in the linear transect
- $b = 1 - a$, $x = 1 - y$ and $u = 1 - v$
and the probability of encountering a run of $r$ diseased trees is $d(r) = (av)^{r-1} (1 - av)$
Of several ideas, I do not understand 1. Why $d$ has to be parametrized by $a$ and $v$ and, 2. Although, the expressions are already stated in the Pielou, 1965, how are those exact probability functions derived for higher values (for example, 3 and 4) of $r$ ?
