The paper considers models of dynamics of infectious disease in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive. If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur. A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.

Pimsner introduced the C*-algebra O X generated by a Hilbert bimodule X over a C*-algebra A. We look for additional conditions that X should satisfy in order to study the simplicity and, more generally, the ideal structure of O X when X is finite projective. We introduce two conditions,``(I)-freeness'' and``(II )-freeness,'' stronger than the former, in analogy with J. the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of``Cuntz Krieger bimodules.'' If X satisfies this condition the C*-algebra O X does not depend on the choice of the generators when A is faithfully represented. As a consequence, if X is (I )-free and A is X-simple, then O X is simple. In the case of Cuntz Krieger algebras O A , X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then O X is p.i.; if A is nonnuclear then O X is nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore if X is (II )-free, we determine the ideal structure of O X . 1998 Academic Press article no. FU983306 295

We introduce the notion of finite right (or left) numerical index on a C Ã -bimodule A X B with a bi-Hilbertian structure, based on a Pimsner-Popa-type inequality. The right index of X can be constructed in the centre of the enveloping von Neumann algebra of A . The bimodule X is called of finite right index if the right index lies in the multiplier algebra of A: In this case the Jones basic construction enjoys nice properties. The C Ã -algebra of bimodule mappings with a right adjoint is a continuous field of finite dimensional C Ã -algebras over a compact Hausdorff space, whose fiber dimensions are bounded above by the index. If A is unital, the right index belongs to A if and only if X is finitely generated as a right module. A finite index bimodule is a bi-Hilbertian C Ã -bimodule which is at the same time of finite right and left index.Bi-Hilbertian, finite index C Ã -bimodules, when regarded as objects of the tensor 2-C Ã -category of right Hilbertian C Ã -bimodules, are precisely those objects with a conjugate in the same category, in the sense of Longo and Roberts. r

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Abstract. In this paper we introduce the notion of Hilbert C * -bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert C * -bimodules and show that tensor products of minimal bimodules are also minimal. For an A-A bimodule which is compatible with a trace on a unital C * -algebra A, its dimension (square root of Jones index) depends only on its KK-class. Finally, we show that the dimension map transforms the Kasparov products in KK(A, A) to the product of positive real numbers, and determine the subring of KK(A, A) generated by the Hilbert C * -bimodules for a C * -algebra generated by Jones projections. IntroductionStrong Morita equivalence for C * -algebras A and B was introduced by M. A. Rieffel ([43], [44]) by the existence of an imprimitivity bimodule X, which is a left Hilbert A-module as well as a right Hilbert B-module with full C * -algebra valued inner products A , and , B such that A x, y z = x y, z B . If σ-unital C * -algebras A and B are strongly Morita equivalent, then K * (A) and K * (B) are isomorphic by the imprimitivity bimodule, and they are stably isomorphic by a result of L. Brown, P. Green and M. A. Rieffel ([2]).The purpose of this paper is to study a relation between the index theory invented by V. F. R. Jones ([23]) and K-theory for C * -algebras (cf. [1]). There exists a Ktheoretical obstruction even for the inclusion of simple C * -algebras of index two. Jones introduced an index for a subfactor N of a type, which can be identified with the elements of K 0 (N ). The index of a subfactor N ⊂ M is analyzed by the bimodule N M M . The important point to note here is that Jones index is also regarded as an element of the Kasparov group "KK(N, N )" ([26], [15]). We studied the inclusion of C * -algebras, introducing the index for C * -subalgebras in [47]. In this paper, we study the C * -index theory from the viewpoint of bimodules. We introduce the notion of a Hilbert A-B bimodule A X B by replacing the associativity condition A x, y z = x y, z B in Rieffel's imprimitivity bimodule by Pimsner-Popa type inequalities [41]. We are forced to postulate that Hilbert C * -bimodules have two-sided inner products in order to make the category of them have the notion of conjugation, which plays an important role in subfactor theory. We remark that considering two-sided inner products corresponds to the orientation of tangles [24].A