

Part 6: Herd Immunity
Each strain of flu is a disease that confers future immunity on its sufferers. For such a disease, if almost everyone has had it, then those who have not had it are protected from getting it  there are not enough susceptibles left in the population to allow an epidemic to get under way. This group protection is called herd immunity.
In Part 4 you experimented with the relative sizes of b and k, and you found that, if b is small enough relative to k, then no epidemic can develop. In the language of Part 5, if the contact number c = b/k is small enough_{}, then there will be no epidemic. But another way to prevent an epidemic is to reduce the initial susceptible population artificially by inoculation.
The point of inoculation is to create herd immunity by stimulating in as many people as possible the antibodies that confer immunity  but without actually giving those people the disease. Thus inoculation creates a direct path from the susceptible group to the recovered group without passing through the infected group (see the diagram below). And a largescale inoculation program to head off an impending epidemic does this rapidly enough to lower the initial susceptible population to a safe level  safe enough that if a trace level of infection enters the population, a few people may get sick, but no epidemic will develop.
So, what fraction of the population must be inoculated to obtain herd immunity? Or, put another way, how small must s_{0} be to insure that an epidemic cannot get started? It depends on the contact number.
(b) Does di/dt negative for a particular value of t imply that no new individuals are getting sick?
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