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Model of Immune Response with Time-Dependent Immune Reactivity
Model of Immune Response with Time-Dependent Immune Reactivity
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
v'=(β-γ)v
c'=α(t)v(t-τ)f(t-τ)-μ(c-1)
f'=μρ(t)(c-f)-ηvf
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.
f(t≤0)=c((t≤0)=1
v(t≤0)=
-5
10
α(t)
α(t)=1+sin
a
c
2t
T
ρ(t)=1
v
c
f
β
γ
μ
η
a
c
t
τ
T
f(t)
v(t)