[FEAT]: Extend the GoLLM task for equation styling to support branching ratios

[liunelson]

Is your feature request related to a problem? Please describe.

We often encounter "branching ratios" in disease forecasting. They are used to model infection into symptomatic/asymptomatic states and recovery into healed/death states. However, the ODE of the subject state (S in example 1) tends to be written in a simplified form that obscures the branching and SKEMA assumes that the branching terms in the outcome states (I, A in example 1) are completely

Example 1

S converts into either I or A with a branching ratio alpha

Input that is SKEMA-incompatible

\frac{d S}{d t} = -\beta S I
\frac{d I}{d t} = \beta \alpha S I
\frac{d A}{d t} = \beta (1 - \alpha) S I

Output that is SKEMA-compatible

\frac{d S}{d t} = -\beta \alpha S I - \beta (1 - \alpha) S I
\frac{d I}{d t} = \beta \alpha S I - \gamma I
\frac{d A}{d t} = \beta (1 - \alpha) S I

Example 2

S converts into A_0, A_1, A_2, ..., I_n states with branching ratio \alpha_0, \alpha_1, \alpha_2, ..., \alpha_n such that \alpha_0 + \alpha_1 + ... \alpha_n = 1.

Input that is SKEMA-incompatible

\frac{d S}{d t} = -\beta S I
\frac{d A_0}{d t} = \beta \alpha_0 S I
\frac{d A_1}{d t} = \beta \alpha_1 S I
\frac{d A_2}{d t} = \beta \alpha_2 S I
...
\frac{d A_n}{d t} = \beta \alpha_n S I

Output that is SKEMA-compatible

\frac{d S}{d t} = -\beta \alpha_0 S I - \beta \alpha_1 S I - \beta \alpha_2 S I - ... - \beta \alpha_n S I
\frac{d A_0}{d t} = \beta \alpha_0 S I
\frac{d A_1}{d t} = \beta \alpha_1 S I
\frac{d A_2}{d t} = \beta \alpha_2 S I
...
\frac{d A_n}{d t} = \beta \alpha_n S I