Question: 3.4 Consider the microbial infection data of Example 1.2. A possible model for the progress of the infection is that a single infecting bacterium living
3.4 Consider the microbial infection data of Example 1.2. A possible model for the progress of the infection is that a single infecting bacterium living according to a stochastic linear birth-and-death process (see Cox and Miller, 1965, p.156) with respective birth and death rates λ and μ, with λ>μ, produces the symptoms of a sore throat if and only if the resulting colony grows to size n. Then the time, T , to the onset of symptoms is governed by:
pr(T ≤ t) ≈ (1 − α)βn−1(1 − μ
λ)
−1
(Morgan and Watts, 1980), where
α = μ{1 − e(μ−λ)t
}{λ − μe(μ−λ)t
}−1,
β = λ{1 − e(μ−λ)t
}{λ − μe(μ−λ)t
}−1.
Verify that the likelihood is given by:
L(λ, μ, n) =
13 i=1
{pr(T ≤ i) − pr(T ≤ i − 1)}ri , where ri is the number of individuals in the ith class interval, and try to obtain the maximum-likelihood estimates of λ, μ and n.
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