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].

They obtain detailed asymptotics of that singularity formation, and show in a precise way that mean curvature flow solutions become

asymptotically rotationally symmetric near a neckpinch singularity.

In this paper, we would like to consider only the two-subdomain decomposition described above, which allows us to obtain

asymptotically accurate formulas of the best transmission parameters.

Almost periodic and

asymptotically almost periodic solutions of differential equations in Banach spaces have been considered by many authors so far (for the basic information on the subject, we refer the reader to the monographs [1-10]).

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.

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.

On the other hand, the concept of the

asymptotically almost periodicity was introduced into the research field by French mathematician Frechet [17,18].

Then the existence, uniqueness, and the

asymptotically orbit stability of order-one periodic solution (OOPS) of system (7) are proved in Section 3.

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 point of system (1) is said to be globally

asymptotically stable if it is locally stable in sense of Lyapunov and globally attractive.

This solution has the same thermodynamic characteristics as the black hole solution in

asymptotically flat space-time; that is, the black hole entropy is equal to a quarter of the event horizon area, while the corresponding thermodynamics quantity satisfies the law of thermodynamics of black hole.