of multiple this, 24A873F8

Lamar performing in 2012.\cdot S=1.}\ R_{0}\cdot S=1.| S can be rewritten as (1 - p) because p is the proportion of the population that is immune and p + S equals one. Then, the equation can be rearranged to place p by itself as follows:| {\displaystyle \ R_{0}\cdot (1-p)=1,}\ R_{0}\cdot (1-p)=1, > {\displaystyle 1-p={\frac {1}{R_{0}}},}1-p={\frac {1}{R_{0}}}, > {\displaystyle p_{c}=1-{\frac {1}{R_{0}}}.}p_{c}=1-{\frac {1}{R_{0}}}.| With p being by itself on the left side of the equation, it can now be written as pc to represent the critical proportion of the population needed to become immune to stop the transmission of disease, or the herd immunity threshold.9 R0 functions as a measure of contagiousness, so low R0 values are associated with lower HITs, whereas higher R0s result in higher HITs.3043 For example, the HIT for a disease with an R0 of 2 is theoretically only 50%, whereas with disease with an R0 of 10 the theoretical HIT is 90%.30 These calculations assume that the entire population is susceptible, meaning no individuals are immune to the disease. In reality, varying proportions of the population are immune to any given disease at any given time.9 To account for this, the effective reproductive number Re, also written as Rt, or the average number of infections caused at time t, can found by multiplying R0 by the fraction of the population that is still susceptible. When Re is reduced to and sustained below 1, the number of cases occurring in the population gradually decreases until the disease has been eliminated.93055 If a population is immune to a disease in excess of that disease's HIT, the number of cases reduces at a faster rate, outbreaks are even less likely to happen, and outbreaks that occur are smaller than they would be otherwise.19 If Re increases to above 1, then the disease is neither in a steady state nor decreasing in incidence but is actively spreading through the population and infecting a larger number of people than usual.4455| A second assumption in these calculations is that populations are homogeneous, or well-mixed, meaning that every individual comes into contact with every other individual, when in reality populations are better described as social networks as individuals tend to cluster together, remaining in relatively close contact with a limited number of other individuals. In these networks, transmission only occurs between those who are geographically or physically close to one another.14344 The shape and size of a network is likely to alter a disease's HIT, making incidence either more or less common.3043 In heterogeneous populations, R0 is now considered to be a measure of the number of cases generated by a "typical" infectious person, which depends on how individuals within a network interact with each other.1 Interactions within networks are more common than between networks, in which case the most highly connected networks transmit disease more easily, resulting in a higher R0 and a higher HIT than would be required in a less connected network.144 In networks that either opt not to become immune or are not immunized sufficiently, diseases may persist despite existing in better-immunized networks.44| Boosts| Further information: Artificial induction of immunity| Vaccination| Main articles: Vaccination and Vaccine| The primary way to boost levels of immunity in a population is through vaccination.156 Vaccination is originally based on the observation that milkmaids exposed to cowpox were immune to smallpox, so the practice of inoculating people with the cowpox virus began as a way to prevent smallpox.42 Well-developed vaccines provide protection in a far safer way than natural infections, as vaccines generally do not cause the diseases they protect against and severe adverse effects are significantly less common than complications from natural infections.5758 The immune system does not distinguish between natural infections and vaccines, forming an active response to both, so immunity induced via vaccination is similar to what would have occurred from contracting and recovering from the disease.59 To achieve herd immunity through vaccination, vaccine manufacturers aim to produce vaccines with low failure rates and policy makers aim to encourage their use.56 After the successful introduction and widespread use of a vaccine, sharp declines in the incidence of diseases it protects against can be observed, necessarily decreasing the number of hospitalizations and deaths caused by such diseases.606162| Assuming a vaccine is 100% effective, then the equation used for calculating the herd immunity threshold can be used for calculating the vaccination level needed to eliminate a disease, written as Vc.1 Vaccines are usually imperfect however, so the effectiveness, E, of a vaccine must be accounted for:| {\displaystyle V_{c}={\frac {1-{\frac {1}{R_{0}}}}{E}}.}V_{c}={\frac {1-{\frac {1}{R_{0}}}}{E}}.| >From this equation, it can be observed that if E is less than (1 ? 1/R0), then it is impossible to eliminate a disease, even if the entire population is vaccinated.1 Similarly, waning vaccine-induced immunity, as occurs with acellular pertussis vaccines, requires higher levels of booster vaccination to sustain herd immunity.121 If a disease has ceased to be endemic to a population, then natural infections no longer contribute to a reduction in the fraction of the population that is susceptible. Only vaccination contributes to this reduction.9 The relation between vaccine coverage and effectiveness and disease incidence can be shown by subtracting the product of the effectiveness of a vaccine and the proportion of the population that is vaccinated, pv, from the herd immunity threshold equation as follows:| Measles vaccine coverage and reported measles cases in Eastern Mediterranean countries. As coverage increased, the number of cases decreased.| {\displaystyle (1-{\frac {1}{R_{0}}})-(E\times p_{v}).}(1-{\frac {1}{R_{0}}})-(E\times p_{v}).| It can be observed from this equation that, ceteris paribus, any increase in either vaccine coverage or vaccine effectiveness, including any increase in excess of a disease's HIT, further reduces the number of cases of a disease.9 The rate of decline in cases depends on a disease's R0, with diseases with lower R0 values experiencing sharper declines.9 Vaccines usually have at least one contraindication for a specific population for medical reasons, but if both effectiveness and coverage are high enough herd immunity can protect these individuals.151720 Vaccine effectiveness is often, but not always, adversely affected by passive immunity,6364 so additional doses are recommended for some vaccines while others are not administered until after an individual has lost his or her passive immunity.1620| Passive immunity| Main article: Passive immunity| Individual immunity can also be gained passively, in which antibodies to a pathogen are transferred from one individual to another. This can occur naturally, whereby maternal antibodies, primarily immunoglobulin G antibodies, are transferred across the placenta and in colostrum to fetuses and newborns,6566 or artificially, by which antibodies from the serum or plasma of an immune individual are injected into a susceptible person.5967 Protection generated from passive immunity is immediate but wanes over the course of weeks to months, so any contribution to herd immunity is temporary.55968 For diseases that are especially severe among fetuses and newborns, such as influenza and tetanus, pregnant women may be immunized in order to transfer antibodies to the child.156970 In the same way, high-risk groups that are either more likely to experience infection or are more likely to develop complications from infection may receive antibody preparations to prevent these infections or to reduce the severity of symptoms.67| Cost–benefit analysis| Herd immunity is often accounted for when conducting cost–benefit analyses of vaccination programs. It is regarded as a positive externality of high levels of immunity, producing an additional benefit of disease reduction that would not occur had no herd immunity been generated in the population.7172 Therefore, herd immunity's inclusion in cost–benefit analyses results in more favorable cost-effectiveness or cost–benefit ratios and an increase in the number of disease cases averted by vaccination.72 Study designs done to estimate herd immunity's benefit include recording disease incidence in households in which a member was vaccinated, randomizing a population in a single geographic area to be vaccinated or not, and observing disease incidence before and after a vaccination program is introduced.73 From these, it can be observed that disease incidence may decrease to a level beyond what can be predicted from direct protection alone, indicating that herd immunity contributed to the reduction.73 When serotype replacement is accounted for, it reduces the predicted benefits of vaccination.72| History| Measles cases in the United States before and after mass vaccination against measles began.| The term herd immunity was first used in 1923 to refer to an entire population's immunity, in reference to research examining disease mortality in mouse populations with varying degrees of immunity.74 Herd immunity was first recognized as a naturally occurring phenomenon in the 1930s when A. W. Hedrich published research on the epidemiology of measles in Baltimore and took notice that after many children had become immune to measles, the number of new infections temporarily decreased, including among susceptible children.8 In spite of this knowledge, efforts to control and eliminate measles were unsuccessful until mass vaccination using the measles vaccine began in the 1960s.8 Mass vaccination, discussions of disease eradication, and cost–benefit analyses of vaccination subsequently prompted more widespread use of the term herd immunity.1 In the 1970s, the theorem used to calculated a disease's herd immunity threshold was developed.1 During the smallpox eradication campaign in the 1960s and 1970s, the practice of ring vaccination, of which herd immunity is integral to, began as a way to immunize every person in a "ring" around an infected individual to prevent outbreaks from spreading.75| Since the adoption of mass and ring vaccination, complexities and challenges to herd immunity have arisen.156 Modeling of the spread of infectious disease originally made a number of assumptions, namely that entire populations are susceptible and well-mixed, which do not exist in reality, so more precise equations have been developed.1 In recent decades, it has been recognized that the dominant strain of a microorganism in circulation may change due to herd immunity, either because of herd immunity acting as an evolutionary pressure or because herd immunity against one strain allowed another already-existing strain to spread.3238 Emerging or ongoing vaccine controversies and various reasons for opposing vaccination have reduced or eliminated herd immunity in certain communities, allowing preventable diseases to persist in or return to these communities.101112| See also| Premunity| References| Fine, P.; Eames, K.; Heymann, D. 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