So in this final section, we'll bring all these concepts together to see what they imply for control in public health practice. First, let's think about R-naught. Remember I said earlier when the reproductive number falls below one, the number of cases will decrease. So in the epidemic curve I'm showing here again on the right, when R-naught was below one, the epidemic started to recede. Because of this, any control measures that cause R to be less than one will ultimately be successful. That is, it might take some time, but eventually the epidemic will die out because it's not replenishing itself since R is less than one. If R is less than one before an epidemic starts into population, that is before the disease is introduced into a population, the disease can't take off and the population is effectively protected from that disease because that initial index case that comes into the population won't cause on average another full case. This is often called herd immunity. This idea that if you have enough immunity and population or get R less than one in a population, introduction of a disease can't cause it to take off. So this concept of herd immunity is important in planning for disease control. In one of the places that's most important is this idea of the critical vaccination threshold. So that is the proportion of the population that we need to be successfully vaccinated to bring R below one. So a little bit of simple algebra can be used to find a formula for this critical vaccination threshold. We find it to be that V, the number of people who need to be successfully vaccinated, and it's important that it's successful, it's equal to 1 minus 1 over the basic reproductive number. As I said, this V is referred to as the critical vaccination threshold. So as in real-world example, if the R-naught for measles is 11-19, we need to vaccinate somewhere between 91 percent to 94 percent of the population to prevent measles outbreaks when it's introduced in the population. While vaccination is the prime example of this idea that if we can bring R below one, we can protect a population from the disease, similar logic can be used for any intervention. So for instance, if we were treating people fast enough to have the number of infections they caused, it would have the reproductive number, and if that brought R below one, then this population would be protected by that treatment. Now let's think a little bit about disease natural history and control. Remember, the incubation period tells us when people develop symptoms. So that tells us when cases in the course of their infection can be detected by passive measures or symptom-based surveillance. The latent period tells us when people will be infectious, so that tells us when they can actually start causing new cases. So the relationship between these values has implications for the success of different intervention strategies, particularly quarantine and symptom-based control strategies. So first, let's talk about quarantine. Quarantine is the segregation or isolation of people potentially infected with a disease until we're confident they're not infected. So this is taking, say somebody who had been exposed to somebody with Ebola and having them stay in a room where they're not in contact with anyone else and not potentially infecting anyone else until we're confident that they're not going to develop Ebola symptoms and they're not infected. It's important to note this is different from isolation of those who are already infected where we have somebody who's already developed symptoms of say Ebola, and we're keeping them out of the general population. So the correct length of this quarantine, because we're waiting to see if people develop symptoms is based on the incubation period. We can think a little bit about why this is true. When I'm isolated, not just I'm I not going to infect anyone else, no one else can infect me, so I must have been infected before that period of isolation happens. So as time goes on and I don't develop symptoms, become more and more confident that I'm actually not infected with that disease, I can be safely allowed to circulate in the normal population. That length is of course the incubation period. If we think about the shape of the incubation period as illustrated here on the right and it's long right tail, we want to get way down on that tail if we want to be really confident. So think about it in this way that you have some confidence, some surety, you want to have that the person you've quarantine is not actually infected and going to develop symptoms after they leave isolation. So we want to wait till maybe that probability is five percent. So we're 95 percent confident that they won't develop symptoms after leaving isolation or maybe we'll wait even longer, and we want to wait to a 99 percent confident they're not infected because they haven't developed symptoms yet. So that time of waiting is based on the incubation period. Quarantine is not viewed as positively in this day and age as it used to be, because we generally don't feel like it's okay to take people who may or may not be sick and have no symptoms and force them to be isolated from the rest of the population, though it's still used occasionally. A less intrusive alternative to quarantine is active monitoring, and this was a strategy that was used in the 2014-'15 Ebola outbreak. So the idea of active monitoring is that we'd have you stay at home. You can still circulate in the population, but not do it too much, and to test yourself from symptoms and report back regularly. In the Ebola outbreak, state health departments under the guidance from the CDC were sending people home with thermometers and those people were supposed to take their temperatures twice a day or more often, and then check in with the public health officers and tell them what their temperature was. The idea being if they started showing an elevated temperature or a fever, that they might be infected with Ebola, and then they should have more extreme isolation procedures. It should be noted for this to work, it depends on the latent period being longer than the incubation period. So that strategy of quarantine we discussed in the last slide, that works even if the latent period is less, because we isolate people from their moment of being exposed. So even if they become infectious before they develop symptoms, they're isolated and they're not going to infect other people. In the active monitoring strategy, people are allowed to have at least some circulation or go out into the population at least some while they don't have symptoms. As long as they can't infect people before they have symptoms, that is the latent period is longer than the incubation period, then that's fine. Fortunately, that was the case for Ebola, so active monitoring was a good strategy. On the right here, we illustrate an analysis that I actually did with some colleagues on how the probability of symptoms after active monitoring based on both the risk of being infected and how long that active monitoring process happened. So here we show that if active monitoring is only a single incubation period, or one times the median incubation period, that there might be a one in 10,000 chance of people at high risk of an infection developing symptoms after or at that active monitoring period. As we go on by two median incubation periods after that period, it becomes one in 100,000 for Ebola and less for other diseases. By three times the median incubation period, it becomes one in a million for Ebola and even less for other diseases. So these numbers even for very short active monitoring seem low, but it should be remembered that's A in part because the chances of being infected with Ebola are low, and B, it should be remembered that the impact of somebody actually going into the community with Ebola could be in the millions of dollars even for controlling just that one case which we know from a couple of instances where it actually occurred. So if secondary cases occur, it could be catastrophic at unfathomable levels. A more general class of control measures that both quarantine and active monitoring fall under are case-based control measures. Case-based control measures are any control strategy based on intervening on symptomatic cases, so people who are actually sick. These include treatment of people who are sick, isolation of sick people as we discussed, and things like symptom screening at the borders to stop international transmission of diseases. The effectiveness of these measures depends on the percentage of infections caused by symptomatic cases. So that is if the latent period is much, much less than the incubation period, say for a disease like HIV, these methods are going to be completely ineffective because most infections are going to be caused by people who are not yet symptomatic. Whereas if the latent period is longer than the incubation period, these measures can be very effective at controlling the disease because we can catch people who have symptoms and prevent them from causing additional transmission. To summarize some key points from this section. If the reproductive number is brought below one an epidemic will die out. Because we know that the reproductive number is a function of susceptibility and the basic reproductive number, this rule combined with knowing the basic reproductive number tells us the percent of the population we need to successfully vaccinate, protect the population from epidemics through herd immunity. The incubation period can be used to help determine the length of quarantine or active monitoring. The relationship between the incubation period and the latent period tells us if case-based interventions will work. So for instance, if the incubation period is much longer like with HIV, they will be doomed to fail, while if only symptomatic cases can transmit, such as with smallpox, they may be quite effect.