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: