Immune Response to a Variable Pathogen: A Stochastic Model with Two Interlocked Darwinian Entities

Figure 9

Computer result of a pathogen 𝑃 (with antigen 𝐴) coupled through averages to an immune system consisting of an effector 𝐸 (producing antigen receptor 𝑎) and memory 𝑀. Case where pathogen is eliminated. Glimpses at generation (a) 𝑛=13, (b) 𝑛=84 and (c) 𝑛=200. Upper left (triangle graph for 0 and scales 0.15 for 𝑊 and 32 for 𝑁): probability 𝑊^𝑆_𝑛(𝑁_𝑃) of finding 0≤𝑁_𝑃≤𝑁_max individuals of the pathogen 𝑃 (green). Lower left (triangle graph): probability 𝑊^𝑆_𝑛(𝑁_𝐸,𝑁_𝑀) of finding 𝑁_𝐸 individuals of the immune effector 𝐸 (red) and 𝑁_𝑀 individuals of the immune memory 𝑀 (blue), (0≤𝑁_𝐸+𝑁_𝑀≤𝑁_max). Upper right: extinction probability 𝑊(0) as a function of generations 𝑛 of pathogen (green), immune effector, and immune memory (violet). Lower right: average of pathogen 𝑃 (green), of immune effector 𝐸 (red), and of immune memory 𝑀 (blue) as a function of generations 𝑛. Parameter values: maximal total number 𝑁_max=32; 𝛼_𝑃=0.1, 𝛼_𝐸=0.1, and 𝛼_𝑀=0.1. For average 𝐸<𝑇_𝐸=1.5 (below threshold value, immune effector 𝐸 inactive): 𝜌_𝑃=6, 𝛽_𝑃=0.96. For average 𝐸>𝑇_𝐸=1.5 (above threshold value, immune effector 𝐸 active): 𝜌_𝑃=2, 𝛽_𝑃=0.65. For average 𝑃>𝑇_𝑃=0.5 (below threshold value, pathogen 𝑃 not seen by immune system): 𝜌_𝐸=2, 𝜌_𝑀=2, 𝜀_𝐸𝑀=0.50, 𝜀_𝑀𝐸=0.01, 𝛽_𝐸=0.90, and 𝛽_𝑀=0.93. For average 𝑃>𝑇_𝑃=0.5 (above threshold value, pathogen 𝑃 seen by immune system): 𝜌_𝐸=6, 𝜌_𝑀=2, 𝜀_𝐸𝑀=0.50, 𝜀_𝑀𝐸=0.50, 𝛽_𝐸=0.95, and 𝛽_𝑀=0.93.