The uniqueness follows from the fact, that the corresponding homogeneous problem has only the

trivial solution. Hence the proof.

As our problem is linear, this question could be reformulated equivalently as follows: When the corresponding homogeneous problem has only

trivial solution, i.e., when

Since 0 < [alpha] < 2k + 1, we observe from Lemma 1 that (17) has only a

trivial solution when t = 0.

It follows that (22) has only

trivial solution u [equivalent to] 0.

In this section, based on Tian's work [20], we give a sufficient condition for the

trivial solution of Problem (1) to be asymptotically stable.

To prove the uniqueness, it is sufficient to show that the homogenous boundary value problem (1.2a)-(1.2b) has only

trivial solution. Assume on the contrary that y(t) [not equivalent to] 0 is a solution of the homogenous boundary value problem (1.2a)-(1.2b).

Note that the global stability of the

trivial solution u = 0 of (20) is equivalent to the global stability of the infection-free equilibrium [P.sub.0] = ([S.sup.*], 0, 0).

Next, we shall discuss the stability of the

trivial solution of the stochastic differential equation (8).

In the same way, one can calculate the same series solution around a = [infinity], which results in the

trivial solution; all the coefficients are identically zero in this limit.

The

trivial solution of system (1) is said to be mean-square exponentially input-to-state stable if for every [mathematical expression not reproducible] and u(t) [member of] [l.sub.[infinity]] there exist scalars [[alpha].sub.0] > 0, [[beta].sub.0] > 0, and [[gamma].sub.0] > 0 such that the following inequality holds:

Specifically, first the method involves yielding the large amplitude stable limit cycle, then decreasing speed with small stepsize to find smaller amplitude stable limit cycle, and decreasing speed stepwise until converging to the

trivial solution (the

trivial solution here is zero solution (as shown in Figure 3), which is different from periodical solution (as shown in Figure 11)).

In order to obtain no

trivial solution, it is necessary to propose that the determinant of (16) is zero, so