Observe that if all roots of the equation det (z[alpha](z)I - A) = 0 are inside the unit circle, then system (5) (or equivalently, (6)) is

asymptotically stable, see for instance [9].

If [R.sub.0] [less than or equal to] 1, then the infection-free equilibrium [E.sub.0] is globally

asymptotically stable and it becomes unstable if [R.sub.0] > 1.

The equilibrium points [x.sup.*] are locally

asymptotically stable if all the eigenvalues [[lambda].sub.j] (j = 1,2, ..., n) of the Jacobian matrix A = [partial derivative]f/[partial derivative]x evaluated at the equilibrium points [x.sup.*] satisfy the following condition:

Hence, the low-criminality equilibrium [P.sub.l] = ([S.sub.0], [D.sub.0], [C.sub.1], [S.sub.1], [D.sub.1], 0,0,0) is locally

asymptotically stable if [R.sup.*.sub.1] < 1, where [R.sup.*.sub.1], defined as Criminality Reproduction Number (CRN), is given by

Further, (i) if [R.sub.0] < 1, the disease-free equilibrium [P.sub.02]([K.sub.p], 0) is locally

asymptotically stable and (ii) if [R.sub.0] > 1, the disease-free equilibrium [P.sub.02]([K.sub.p], 0) is unstable and the endemic equilibrium [P.sup.*]([S.sup.*.sub.p], [I.sup.*.sub.p]) exists and is locally

asymptotically stable.

Then system (1) is

asymptotically stable. Combined with the above two cases, as long as the linear matrix inequality (11) is satisfied, system (1) is

asymptotically stable.

The system [mathematical expression not reproducible] is

asymptotically stable if [absolute value of arg(eig(A))] > [alpha][pi]/2 where 0 < [alpha] < 2, and eig(A) are the eigenvalues of matrix A.

Based on the Routh-Hurwitz criterion and the discussion in [31], it follows that the positive equilibrium [E.sub.*] is locally

asymptotically stable if the following condition holds: ([H.sub.1]):

The positive equilibrium point [E.sup.*]([x.sup.*], [y.sup.*]) is locally

asymptotically stable, if (I) holds.

Thus, according to the Routh-Hurwithz theorem, we know that if conditions ([H.sub.11]) [a.sub.10]- > 0, [a.sub.13] > 0, and [a.sub.1]3 [a.sub.12] > [a.sub.11] hold, then viral equilibrium [E.sub.*]([S.sub.*])[L.sub.*],[A.sub.*],[R.sub.*]) of system (2) without delay is locally

asymptotically stable.

They showed using LaSalle's invariance principle that the disease-free equilibrium, [P.sub.0], is globally

asymptotically stable in T if [R.sub.0] [less than or equal to] 1 and unstable if [R.sub.0] > 1.

(a) [E.sub.1] is locally

asymptotically stable if [A.sub.1] > 1.