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

Figure 7

Dynamical stochastic process for initial form 𝐴 and mutant 𝐵. One initial ancestor of form 𝐴. Three cardinal examples (a) 𝑛=48, (b) 𝑛=164, and (c) 𝑛=600. Upper left (triangle graph for 0 and scales 0.15 for 𝑊 and 32 for 𝑛): probability distribution 𝑊^𝑀_𝑛(𝜈_𝐴,𝜈_𝐵) of finding 𝜈_𝐴 individuals of initial form 𝐴 (red) and 𝜈_𝐵 individuals of mutant 𝐵 (blue) after multiplication phase 𝑀 of generation 𝑛 (9), (10). Lower left (triangle graph): probability distribution 𝑊^𝑆_𝑛(𝑁_𝐴,𝑁_𝐵) of finding 𝑁_𝐴 individuals of initial form 𝐴 (red) and 𝑁_𝐵 individuals of mutant 𝐵 (blue) after selection phase 𝑆 of generation 𝑛 (12). Upper right: total extinction probability 𝑊(0) of both initial form (𝐴) and mutant (𝐵) together (violet) along generation 𝑛. Lower right: average number of individuals 𝜈 and 𝑁 of initial form 𝐴 (red) and mutant 𝐵 (blue) with their standard deviations (black) along generation 𝑛. Parameters: maximal total number 𝑁_max=32, copy probability 𝛼_𝐴=0.1 and 𝛼_𝐵=0.1, multiplication factor 𝜌_𝐴=3 of initial form 𝐴, multiplication factor 𝜌_𝐵=6 of mutant 𝐵, mutation probability 𝜀_𝐴𝐵=0.01 from initial form 𝐴 to mutant 𝐵, mutation probability 𝜀_𝐵𝐴=0.001 from mutant 𝐵 to initial form 𝐴, surviving probability 𝛽_𝐴=0.95 of initial form 𝐴, surviving probability 𝛽_𝐵=0.95 of mutant 𝐵. Deterministic model (green in lower right, (13)): 𝑅_𝑎=0.128, 𝑅_𝑏=0.14, and 𝐾=16.