jacobians

How is the Jacobian Matrix Constructed?

Constructing a Jacobian matrix involves the following steps:

1. Model Formulation: Develop a system of differential equations representing the disease dynamics. For example, the SIR model consists of equations for Susceptible (S), Infected (I), and Recovered (R) populations.

2. Partial Derivatives: Compute the first-order partial derivatives of each equation with respect to each variable in the system. These derivatives form the elements of the Jacobian matrix.

3. Matrix Assembly: Arrange these partial derivatives into a matrix. For an SIR model, the Jacobian might look like this:

\[
J = \begin{pmatrix}
\frac{\partial f_1}{\partial S} & \frac{\partial f_1}{\partial I} & \frac{\partial f_1}{\partial R} \\
\frac{\partial f_2}{\partial S} & \frac{\partial f_2}{\partial I} & \frac{\partial f_2}{\partial R} \\
\frac{\partial f_3}{\partial S} & \frac{\partial f_3}{\partial I} & \frac{\partial f_3}{\partial R}
\end{pmatrix}
\]