# A Mathematical Model

### José A.M. Borghans, Rob J. de Boer, Eli Sercarz & Vipin Kumar

T-cell vaccination (TCV) is a method to induce resistance to autoimmune diseases by priming the immune system with autoreactive T cells. This priming evokes an anti-idiotypic regulatory T-cell response to the receptors on the autoreactive T cells. Hence resistance is induced. To prevent the inoculated autoreactive cells from inducing autoimmunity, cells are given in a sub-pathogenic dose or in an attenuated form. We develop a mathematical model to study how the interactions between autoreactive T cells, self epitopes, and regulatory cells can explain TCV. The model is based on detailed data on experimental autoimmune encephalomyelitis, but can be generalized to other autoimmune diseases. We show that all of the phenomena collectively described as TCV occur quite naturally in systems where autoreactive T cells can be controlled by anti-idiotypic regulatory T cells. The essential assumption that we make is that TCV generally involves self epitopes for which T-cell tolerance is incomplete. The model predicts a qualitative difference between the two vaccination methods: vaccination with normal autoreactive cells should give rise to a steady state of long lasting protection, whereas vaccination with attenuated cells should only confer transient resistance. Moreover, the model shows how autoimmune relapses can occur naturally without the involvement of T cells arising due to determinant spreading.

J. Immunol. (1998) 161, 1087-1093.

 Model Figure: Schematic representation of the simplified model. The autoreactive cells A recruit a regulatory population R which consists of both CD4+ and CD8+ cells. The regulatory cells R inhibit the autoreactive cells A. This results in a negative feedback loop between A and R. The autoreactive cells proliferate upon stimulation by presented self epitopes Sa. Since the autoreactive cells stimulate the presentation of self epitopes on MHC molecules, for example by IFN-gamma production, there is a positive feedback loop between A and Sa.

 Results Figure: Model experiments. The large panels show the model behavior in conventional time plots; the insets show the same behavior in the state space. In the state space we have plotted the number of regulatory cells R as a function of the number of autoreactive cells A. The black squares denote the two stable steady states (i.e. the attractors) of the system. The thick line in the figure is the separatrix. The thin lines represent the sizes of both clones at subsequent moments in time. Panel (a): A large dose of autoreactive cells (A = 100) given in the normal state N (see dashed line) causes a vigorous autoreactive response which is interpreted as autoimmunity. Eventually the vaccinated state is approached, leaving the animal healthy and resistant to autoimmunity. Panel (b): If the same large dose of autoreactive cells (A = 100) is given in the vaccinated state V (see dashed line), the regulatory cells are able to control the autoreactive response. There is no autoimmune disease and the system returns to the vaccinated state. Panel (c): A small dose of autoreactive cells (A = 0.5) given in the normal state N leads to a switch to the vaccinated state V while no autoimmune disease is induced. Panel (d): Attenuated autoreactive cells or regulatory cells (R = 10) given in the normal state N (see the vertical line) are able to confer transient protection. If a previously pathogenic dose of live autoreactive cells (A = 100) is given when the concentration of regulatory cells is still large (see the horizontal line), the system switches to the vaccinated state while no autoimmunity is induced.