Model of Immune Response with Time-Dependent Immune Reactivity

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The probability of getting a disease is related to the efficiency of the immune system, which can change with the seasons of the year. This Demonstration shows the solution of a model of the immune system that has periodic changes in the immune reactivity due to changes in the environment.


The model consists of three delay ordinary differential equations




with initial history functions and ; is the immune reactivity (the immune response of the infected individual): , and is the antibody production rate per plasma cell due to the presence of antigens. Here , , and represent antigen, plasma cells, and antibody concentrations; is the antigen reproduction rate; is the probability of an antigen-antibody encounter; is the reciprocal of the plasma cell lifetime; is the number of antibodies necessary to suppress one antigen; and is a constant. Values of these parameters are taken from the reference. Time is , is the time delay necessary for the formation of plasma cells and antibodies, and is the length of the season. Large ratios of antibody to antigen concentrations, , correspond to a strong immune system in which reactions are fast and in which the organism has strong resistance; on the other hand, small values of this ratio imply immunodeficiency.


Contributed by: Clay Gruesbeck (May 2013)
Open content licensed under CC BY-NC-SA




[1] M. Bodnar and U. Fory\:015b, "A Model of Immune System with Time-Dependent Immune Reactivity," Nonlinear Analysis, 70(2), 2009 pp. 1049–1058. doi:10.1016/

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