Then, calculating the respective derivatives, it is understood that the

Jacobian matrix [J.sub.1] is the zero matrix.

and, for the equilibrium [P.sup.+](x, y, z) = (+ [square root of b], + [square root of b], 0), the

Jacobian matrix of (6) at points [P.sup.+] is obtained as

Since J = J + O(h), we conclude that the

Jacobian matrix J = [[partial derivative][[PHI].sub.n]/[partial derivative]U] in (3.8) is nonsingular for all sufficiently small h and for [x.sub.n] in a small neighbourhood of the exact solution of problem (1.1).

For the equilibrium point [P.sub.1] = (0,0, 0) the

Jacobian matrix is given by

The Frechet derivative matrix in (4) is a

Jacobian matrix with large dimensionality.

The key point here is that the diagonal elements of the simplified diagonal

Jacobian matrix are identical to those of the flux

Jacobian matrix in the time-domain solution algorithm.

The

Jacobian matrix of model (3) around the disease-free equilibrium, [E.sub.0], is given by

where [mathematical expression not reproducible] is the overall

Jacobian matrix of the 2-RPU&2-SPS SPM.

We evaluate the

Jacobian matrix (41) at the endemic equilibrium to obtain

where the values of each element in the

Jacobian matrix can be expressed as follows:

At the disease-free equilibrium [D.sub.o] the corresponding

Jacobian matrix [J.sub.o]([zeta]) of system (3) is computed as follows:

The elementary

Jacobian matrix is obtained as follows: