A few days ago it was uploaded on arXiv a preprint about radiactions from Fukushima reactors, Deciphering the measured ratios of Iodine-131 to Cesium-137 at the Fukushima reactors, by T. Matsui (University of Tokyo).
In the preprint, Matsui propose some simple theoretical calculations to evaluate the situation of TEPCO's reactors.
The physical basis is the radioactive decay,
\[N(t) = N_0 e^{-\lambda t}\]
where $N_0$ is the number of nucleus at time $t_0$ (in the beginning), $\lambda$ is the decay rate (inverse of life time $\tau$), $N (t)$ is the number of nucleus at time $t$.
To calculate the equation we start from the experimental law,
\[\frac{\text d N (t)}{\text d t} = - N(t) \lambda\]
For his calculation, Matsui used the following differential equation
\[\frac{\text d N_I (t)}{\text d t} = f_I N_0 \theta (t;t_i, t_f) - \lambda_I N_I\]
where $\theta (t;t_i, t_f) = 1$ for $t_i < t < t_f$ and $\theta (t;t_i, t_f) = 0$ otherwise, $N_0$ is the number of fission in time, $f_I$ is the fraction of I-131 produced in every fission, $\lambda_I$ the decay rate of I-131. Boundary conditions are: nuclear reactor had been in operation from $t_i$ to $t_f$; $N_I (t_i) = 0$.
After integration, and introducing the condition $\lambda_I (t_f - t_i) \gg 1$ (it means that the working time of the reactor is much longer than the half-life of I-131, that is 8 days), Matsui found:
