Elimination of danger (e.g. pathogens) is an important activity for organisms. From this viewpoint, it is clear that the optimization analysis is a valid method to study immune system.

There are different patterns of defense observed in immune system. Higher vertebrates have both innate immunity and acquired immunity, whilst lower animals don't. Some proteins for defense are stored beforehand, whilst others are produced after infection. It is likely that some advantages lay under the realized patterns of defense. The organisms should prevent pathogen growth, with saving defense cost (e.g. tissue injury). I present mathematical models to clarify the optimal defense pattern using the above criterion. There are 3 parts in my talk. First, I present models focusing on difference between two defense options, such as delay, cost, effectiveness, and uncertainty of information available. If defense proteins are produced via gene expression after infection, there is a serious risk where pathogen abundance increases quickly until the level of defense protein becomes enough. In contrast, if defense proteins are produced and stored before infection, there is no disadvantage mentioned above. However, storing holds the other disadvantage where stored proteins can be wasteful, because the information available before infection is limited. Taking the above into consideration, I discuss the optimal defense strategy. Second, I present more detailed models clarifying dynamic defense strategy. I discuss the pattern of acquired immune response. Immune system realizes "bang-bang control", to produce effector and memory cells economically. Lastly, I present the model, which we have just started, on JAK-STAT pathway. The last study is based on Yamada et al (2003).