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

Figure 12

Computer result of a varying pathogen (𝑃_𝐴 with antigen 𝐴 changing into 𝑃_𝐵 with antigen 𝐵) coupled through averages to an immune system against antigen 𝐴 consisting of an effector 𝐸_𝑎 and memory 𝑀_𝑎 and against antigen 𝐵 consisting of an effector 𝐸_𝑏 and memory Mb. Partial change of pathogen 𝑃_𝐴 to pathogen 𝑃_𝐵 escaping immune effector 𝐸_𝑎 and with delayed immune response of effector 𝐸_𝑏. Glimpses at generation (a) 𝑛=22, (b) 𝑛=100. Upper left (triangle graph for 0 and scales 0.15 for 𝑊 and 32 for 𝑁): probability 𝑊^𝑆_𝑛(𝑁_𝑃𝐴,𝑁_𝑃𝐵) of finding 𝑁_𝑃𝐴 individuals of the pathogen 𝑃_𝐴 with antigen 𝐴 and of finding 𝑁_𝑃𝐵 individuals of the pathogen 𝑃_𝐵 with antigen 𝐵 (green) (0≤𝑁_𝑃𝐴+𝑁_𝑃𝐵≤𝑁_max). Middle left (triangle graph): probability 𝑊^𝑆_𝑛(𝑁_𝐸𝑎,𝑁_𝑀𝑎) of finding 𝑁_𝐸𝑎 individuals of the immune effector (red) and 𝑁_𝑀𝑎 individuals of immune memory (blue) against antigen 𝐴 (0≤𝑁_𝐸𝑎+𝑁_𝑀𝑎≤𝑁_max). Lower left (triangle graph): probability 𝑊^𝑆_𝑛(𝑁_𝐸𝑏,𝑁_𝑀𝑏) of finding 𝑁_𝐸𝑏 individuals of the immune effector (red) and 𝑁_𝑀𝑏 individuals of immune memory (blue) against antigen 𝐵 (0≤𝑁_𝐸𝑏+𝑁_𝑀𝑏≤𝑁_max). Upper right: extinction probability 𝑊(0) as a function of generations 𝑛 of pathogen (green), immune effector & memory against antigen 𝐴 and immune effector & memory against antigen 𝐵 (both violet). Middle and lower right: average of pathogen 𝑃 (green), of immune effector 𝐸 (red) and of immune memory 𝑀 (blue) against antigen 𝐴 (middle right) and against antigen 𝐵 (lower right) as a function of generations 𝑛. Parameter values others than Figure 2: average 𝑇_𝑃𝐴=1.5 and average 𝑇_𝑃𝐵=1.5, respectively (threshold value to switch immune system); 𝜀_{𝑃𝐴𝑃𝐵}=0.01, 𝜀_{𝑃𝐵𝑃𝐴}=0.001 (pathogen variability).