Mathematical model of immune processes

In the model the time lags of the antibody production and immune memory formation are taken into account explicitly. The antibody-antigen reaction is supposed to be very fast. The cases of a reproducing antigen as well as that of a non-reproducting antigen are considered. The conditions of the infinite increase of the antigen quantity and of the antigen elimination are obtained. For the rapidly reproducing antigen the latter condition includes the requirement for the time lag of the immune response to be not too short or not too long. In the case of the poorly catabolized non-reproducing antigen the cyclic appearance of the antibody producing cells due to the immune memory is described in the frame-work of the model.The mathematical structure of the model is similar to that of the Volterra-Lotka jequations. The only difference is the presence of the time lags in the non-linear terms. The time lags lead to the instability of the stationary state. In the prolonged reaction the antigen quantity may perform several oscillations before the elimination of the antigen.