Inducible defense against pathogens and parasites: optimal choice among multiple options

Defense against pathogen, parasites and herbivores is often enhanced after their invasion into the host's body. Sometimes different options are adopted depending on the identity and the quantity of the pathogen, exemplified by the switch between Th1 and Th2 systems in mammalian immunity. In this paper, we study the optimal defense of the host when two alternative responses are available, which differ in the effectiveness of suppressing the growth of pathogen (parasite, or herbivore), the damage to the host caused by the defense response, and the magnitude of time delay before the defense response becomes fully effective. The optimal defense is the one that minimizes the sum of the damages caused by the pathogen and the cost due to defense activities. The damage by pathogens increases in proportion to the time integral of the pathogen abundance, and the cost is proportional to the defense activity. We can prove that a single globally optimal combination of defense options always exists and there is no other local optimum. Depending on the parameters, the optimal is to adopt only the early response, only the late response, or both responses. The defense response with a shorter time delay is more heavily used when the pathogen grows fast, the initial pathogen abundance is large, and the difference in time delay is long. We also study the host's optimal choice between constitutive and inducible defenses. In the constitutive defense, the response to pathogen attack works without delay, but it causes the cost even when the pathogen attack does not occur. We discuss mammalian immunity and the plant chemical defense from the model's viewpoint.