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I am attempting to find the maximum likelihood estimate for disease prevalence in trapped mice by using data on the probability of being trapped each year and the number of mice actually trapped that year.

Imagine a group of juvenile mice. There is a number of disease-free mice, $S$, and a number of infected mice $I$. I trap these mice, wherein each disease-free mouse has a binomial probability $p_S$ of being captured and each infected mouse has a $p_I$ change of being captured. I capture a known number of $s_1$ disease-free mice and $i_1$ infected mice, that are now removed from the population.

After these captures, an unknown number of mice transition from being disease-free to infected with a probability $B$. Once this transition occurs, I trap them again, and the disease-free and infected mice have the same probabilities of being captured as before (i.e., $p_S$ and $p_I$, respectively). I capture and remove a known number $s_2$ and $i_2$ of disease-free and infected mice.

I want to write out the likelihood of observing particular values of $s_1, i_1, s_2$, and $i_2$ given certain values of $S, I, B, p_S$, and $p_I$. I know how to write out the binomial likelihood is individuals remain in the same cohort through time:

$$L(s_1,s_2|S,p_S)={S \choose s_1,s_2}(p_S)^{s_1}((1-p_S)p_S)^{s_2}((1-p_S)(1-p_S))^{S-s_1-s_2}$$

My issue is what to do when an unknown number of individuals leaves a cohort and the same number enters the other cohort. Essentially, the number of infected individuals changes, so $I-i_1$ is not the number of infected mice available to be captured the second time around because an unknown number of new mice moved to this cohort from the disease-free one with probability $B$.

ZombiePlan37
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  • please change notation to more readable – marco11 May 11 '15 at 14:47
  • Due to the lack of feedback from this and similar forums, I ended up asking some friends in the stats department.

    The solution is to assume that a number of individuals n migrated from the susceptible pool into the infected pool. Then you can write out a joint binary likelihood for the probability of observing s1, s2, i1, i2, and n individuals. However, n is unknown, so you marginalize over it, essentially adding up the joint likelihood for every feasible integer value of n.

     – ZombiePlan37 May 13 '15 at 17:33

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