The immunization of any individual, by developing antibodies to either natural infection, or to a vaccine. Results in direct protection, for that individual, from subsequent infection. It also results, in the indirect protection, to all the individuals, that they might have infected. Thus, to assess the full potential impact of vaccination. One must consider, both the direct benefit of vaccination to the individual. And, the indirect benefits of vaccination to the population. Consider the vaccination of infants. A strategy that has been employed, to limit many potentially dangerous human infections. Like polio, measles, mumps, rubella, and whooping cough. We can then ask, what will be the impact, at the population scale, of immunizing some fraction, P, of all infants. Where P, is the product of the proportion of infants vaccinated, and the vaccine efficacy. Or, the probability that a vaccinated infant will successfully develop immunity. Recall, that infection removes susceptible individuals, at a rate, related to the basic reproductive number, R-naught. And susceptibles are added to the population, through births. Thus, immunizing a proportion, P, of births, reduces the rate, at which the susceptible population is replenished. It can be shown then, that the new reproductive rate. For an infection, in a population, where P percent of individuals are immunized, is R effective. Which is 1 minus P times R-naught, in the absence of vaccination. Recall again, that R effective, is the expected number of new infections, due to a single infection. And an epidemic, is only expected to establish in a population, if this number is greater than one. So, if R effective is less than one, then we would expect that infection would not be able to invade. Solving this relationship for P, we can calculate, the critical immunization fraction needed, to ensure that R effective is less than one. Which gives us P, greater than one, minus one, over R-naught. Which says, that, so long as the proportion vaccinated, is greater than this quantity, an epidemic will not be able to spread. This fairly simple equation suggests a very powerful result for public health. That, we can effectively prevent an infection from invading a population. By immunizing, less than, 100% of all births. This phenomenon is referred to as herd immunity. We can see immediately, that as R-naught increases. The proportion of children born, that we would need to vaccinate, also increases. And, that proportion rises rapidly, at first. So, for a pathogen like smallpox, with an R-naught of approximately five. We might only need to immunize 80% of children. But, for pathogens like mumps or chicken pox, with R-naught nearer to ten. We would need to immunize 90%. And, for pathogens like whooping cough and measles, with R-naught near 20. We would need to immunize, close to 95% of all children born each year. Recall, that the herd immunity threshold, is the percentage that must be immunized, not just vaccinated. Pediatric vaccination, with pertussis or measles vaccine. Generally, stimulates immunity in less that 90% of infants, due to interactions with maternal immunity. So, infant based vaccination strategies alone. Are not expected to be effective to achieve eradication of pathogens, with such high transmissibility. So, while, herd immunity is an encouraging phenomenon. Indicating that eradication may be possible, even with less than perfect vaccination coverage. it also highlights the inherent challenges to eradication. Even if the critical vaccination level necessary, for herd immunity, is difficult to achieve. The indirect effects of vaccination will still have positive benefits. The equilibrium prevalence of infection. That is, the average proportion of the population that is infected, at any given time. Is expected to decrease linearly, with the vaccination coverage, even when coverage is below the critical level for herd immunity. When the prevalence of infection, and the proportion of the population unvaccinated, drop to low levels. It is increasingly likely, that transmission will fail to occur, due to random chance. Thus, even if vaccination alone cannot achieve elimination. It increases the chance of local elimination, due to stochastic dynamics. It's important to note, that in this event, if vaccine coverage is below the level necessary to achieve herd immunity. And infection fades out, locally, due to stochastic dynamics, the population is susceptible to reinvasion of infection. Further, as time passes, if vaccination stays below the critical coverage level. The proportion of the population, that is susceptible to infection, will continue to increase. Leaving the population, increasingly, susceptible to new outbreaks. Thus, the combination of indirect protection, and stochastic dynamics, can be useful to achieve disease elimination goals. But, the local elimination of infection, must be viewed with caution. As the absence of infection does not, necessarily, translate into the absence of, subsequent, outbreak risk.