## St exp[at t2544

In this equation, ai is a measure of the cost of deviations from the optimal response. The smaller this value, the less important are deviations from the optimal mix of T-helper cells. If we assume that the n different pathogens or parasites affect survival independently, then the probability of surviving all of them is n

and our objective is to maximize S(t) by choosing t.

Exercise 5.12 (E)

Show that the optimal mixture of Thj cells is t: tiOLi t* = Mn--(5.46)

This equation tells us two important things about the immune response. First, when a number of pathogens or parasites attack a host, the optimal response will be a mixture of the individual optimal responses. Second, this mixture will be weighted by the consequences of non-optimal response to each disease. Note that if one of the ai is very large, then the optimal value of t will be very close to the response of Th1 cells for that parasite or pathogen.

Here is an interesting extension of Graham's work, due to Steve Munch. There are some diseases that will kill a host, even if the host mounts the appropriately optimal response. Thus, we could generalize Eq. (5.44) to Si(t) = ßiexp[—ai(t — t)2] where the parameter ßi, with 0 <ßi < 1, measures the probability of surviving the ith pathogen or parasite when the optimal response is applied.

Show that the optimal response when confronted by n different pathogens or parasites is still given by Eq. (5.46). How is this result to be interpreted? Does this suggest that we should change the model in some way?

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