where the endemic fixed point is now located at the origin. The Jacobian of Eqs. ͑5a͒-͑5d͒ is computed to zeroth order in and is evaluated at the origin. Ignoring the components, the Jacobian has only two linearly independent eigenvectors. Therefore, the Jacobian is not diagonalizable. However, it is possible to transform Eqs. ͑5a͒-͑5c͒ to a block diagonal form with the eigenvalue structure that is needed to use the center manifold theory. We use a transformation matrix P consisting of the two linearly independent eigenvectors of the Jacobian along with a third vector chosen to be linearly independent. There are many choices for this third vector; our choice is predicated on keeping the vector as simple as possible. This transformation matrix is given as