The pulse vaccination strategy is a method used to eradicate an epidemic by repeatedly vaccinating a group at risk, over a defined age range, until the spread of the pathogen has been stopped. It is most commonly used during measles and polio epidemics to quickly stop the spread and contain the outbreak. [1] [2]
Where T= time units is a constant fraction p of susceptible subjects vaccinated in a relatively short time. This yields the differential equations for the susceptible and vaccinated subjects as[ citation needed ]
Further, by setting I = 0, one obtains that the dynamics of the susceptible subjects is given by: [3]
and that the eradication condition is: [4]
Poliomyelitis, commonly shortened to polio, is an infectious disease caused by the poliovirus. Approximately 75% of cases are asymptomatic; mild symptoms which can occur include sore throat and fever; in a proportion of cases more severe symptoms develop such as headache, neck stiffness, and paresthesia. These symptoms usually pass within one or two weeks. A less common symptom is permanent paralysis, and possible death in extreme cases. Years after recovery, post-polio syndrome may occur, with a slow development of muscle weakness similar to that which the person had during the initial infection.
Vaccination is the administration of a vaccine to help the immune system develop immunity from a disease. Vaccines contain a microorganism or virus in a weakened, live or killed state, or proteins or toxins from the organism. In stimulating the body's adaptive immunity, they help prevent sickness from an infectious disease. When a sufficiently large percentage of a population has been vaccinated, herd immunity results. Herd immunity protects those who may be immunocompromised and cannot get a vaccine because even a weakened version would harm them. The effectiveness of vaccination has been widely studied and verified. Vaccination is the most effective method of preventing infectious diseases; widespread immunity due to vaccination is largely responsible for the worldwide eradication of smallpox and the elimination of diseases such as polio and tetanus from much of the world. However, some diseases, such as measles outbreaks in America, have seen rising cases due to relatively low vaccination rates in the 2010s – attributed, in part, to vaccine hesitancy. According to the World Health Organization, vaccination prevents 3.5–5 million deaths per year.
Herd immunity is a form of indirect protection that applies only to contagious diseases. It occurs when a sufficient percentage of a population has become immune to an infection, whether through previous infections or vaccination, thereby reducing the likelihood of infection for individuals who lack immunity.
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries of physical space.
Polio vaccines are vaccines used to prevent poliomyelitis (polio). Two types are used: an inactivated poliovirus given by injection (IPV) and a weakened poliovirus given by mouth (OPV). The World Health Organization (WHO) recommends all children be fully vaccinated against polio. The two vaccines have eliminated polio from most of the world, and reduced the number of cases reported each year from an estimated 350,000 in 1988 to 33 in 2018.
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.
In mathematics, Itô's lemma or Itô's formula is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.
The DPT vaccine or DTP vaccine is a class of combination vaccines against three infectious diseases in humans: diphtheria, pertussis, and tetanus. The vaccine components include diphtheria and tetanus toxoids and either killed whole cells of the bacterium that causes pertussis or pertussis antigens. The term toxoid refers to vaccines which use an inactivated toxin produced by the pathogen which they are targeted against to generate an immune response. In this way, the toxoid vaccine generates an immune response which is targeted against the toxin which is produced by the pathogen and causes disease, rather than a vaccine which is targeted against the pathogen itself. The whole cells or antigens will be depicted as either "DTwP" or "DTaP", where the lower-case "w" indicates whole-cell inactivated pertussis and the lower-case "a" stands for "acellular". In comparison to alternative vaccine types, such as live attenuated vaccines, the DTP vaccine does not contain any live pathogen, but rather uses inactivated toxoid to generate an immune response; therefore, there is not a risk of use in populations that are immune compromised since there is not any known risk of causing the disease itself. As a result, the DTP vaccine is considered a safe vaccine to use in anyone and it generates a much more targeted immune response specific for the pathogen of interest.
In epidemiology, the basic reproduction number, or basic reproductive number, denoted , of an infection is the expected number of cases directly generated by one case in a population where all individuals are susceptible to infection. The definition assumes that no other individuals are infected or immunized. Some definitions, such as that of the Australian Department of Health, add the absence of "any deliberate intervention in disease transmission". The basic reproduction number is not necessarily the same as the effective reproduction number , which is the number of cases generated in the current state of a population, which does not have to be the uninfected state. is a dimensionless number and not a time rate, which would have units of time−1, or units of time like doubling time.
Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health and plant health interventions. Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programs. The modelling can help decide which intervention(s) to avoid and which to trial, or can predict future growth patterns, etc.
Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R,. People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.
Pulse Polio is an immunisation campaign established by the government of India to eliminate poliomyelitis (polio) in India by vaccinating all children under the age of five years against the polio virus. The project fights polio through a large-scale, pulse vaccination programme and monitoring for poliomyelitis cases.
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.
Vaccination and religion have interrelations of varying kinds. No major religion prohibits vaccinations, and some consider it an obligation because of the potential to save lives. However, some people cite religious adherence as a basis for opting to forego vaccinating themselves or their children. Many such objections are pretextual: in Australia, anti-vaccinationists founded the Church of Conscious Living, a "fake church", leading to religious exemptions being removed in that country, and one US pastor was reported to offer vaccine exemptions in exchange for online membership of his church.
Polio eradication, the permanent global cessation of circulation of the poliovirus and hence elimination of the poliomyelitis (polio) it causes, is the aim of a multinational public health effort begun in 1988, led by the World Health Organization (WHO), the United Nations Children's Fund (UNICEF) and the Rotary Foundation. These organizations, along with the U.S. Centers for Disease Control and Prevention (CDC) and The Gates Foundation, have spearheaded the campaign through the Global Polio Eradication Initiative (GPEI). Successful eradication of infectious diseases has been achieved twice before, with smallpox in humans and rinderpest in ruminants.
A vaccination policy is a health policy adopted in order to prevent the spread of infectious disease. These policies are generally put into place by State or local governments, but may also be set by private facilities, such as workplaces or schools. Many policies have been developed and implemented since vaccines were first made widely available.
Mass vaccination is a public policy effort to vaccinate a large number of people, possibly the entire population of the world or of a country or region, within a short period of time. This policy may be directed during a pandemic, when there is a localized outbreak or scare of a disease for which a vaccine exists, or when a new vaccine is invented.
Targeted immunization strategies are approaches designed to increase the immunization level of populations and decrease the chances of epidemic outbreaks. Though often in regards to use in healthcare practices and the administration of vaccines to prevent biological epidemic outbreaks, these strategies refer in general to immunization schemes in complex networks, biological, social or artificial in nature. Identification of at-risk groups and individuals with higher odds of spreading the disease often plays an important role in these strategies, since targeted immunization in high-risk groups is necessary for effective eradication efforts and has a higher return on investment than immunizing larger but lower-risk groups.
Misinformation related to immunization and the use of vaccines circulates in mass media and social media in spite of the fact that there is no serious hesitancy or debate within mainstream medical and scientific circles about the benefits of vaccination. Unsubstantiated safety concerns related to vaccines are often presented on the internet as being scientific information. A high proportion of internet sources on the topic are "inaccurate on the whole" which can lead people searching for information to form "significant misconceptions about vaccines".
