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NS-2664-NS-2664-NS-2664 (LS-193,048) is an anxiolytic drug with a novel chemical structure, developed by the small pharmaceutical company NeuroSearch. It has similar effects to benzodiazepine drugs, but is structurally distinct and so is classed as a nonbenzodiazepine anxiolytic. NS-2664 is a potent but non-selective p... | milkshake721/2.1M-wiki-STEM |
Mean value theorem (divided differences)-Mean value theorem (divided differences)-In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. | milkshake721/2.1M-wiki-STEM |
Mean value theorem (divided differences)-Statement of the theorem-For any n + 1 pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point min max {x0,…,xn}) where the nth derivative of f equals n ! times the nth divided difference at these points: f[x0,…,x... | milkshake721/2.1M-wiki-STEM |
Mean value theorem (divided differences)-Proof-Let P be the Lagrange interpolation polynomial for f at x0, ..., xn.
Then it follows from the Newton form of P that the highest term of P is f[x0,…,xn](x−xn−1)…(x−x1)(x−x0) Let g be the remainder of the interpolation, defined by g=f−P . Then g has n+1 zeros: x0, ..... | milkshake721/2.1M-wiki-STEM |
Mean value theorem (divided differences)-Applications-The theorem can be used to generalise the Stolarsky mean to more than two variables. | milkshake721/2.1M-wiki-STEM |
Lek paradox-Lek paradox-The lek paradox is the conundrum of how additive or beneficial genetic variation is maintained in lek mating species in the face of consistent sexual selection based on female preferences. While many studies have attempted to explain how the lek paradox fits into Darwinian theory, the paradox re... | milkshake721/2.1M-wiki-STEM |
Lek paradox-Lek paradox-The basis of the lek paradox is continuous genetic variation in spite of strong female preference for certain traits. There are two conditions in which the lek paradox arises. The first is that males contribute only genes and the second is that female preference does not affect fecundity. Female... | milkshake721/2.1M-wiki-STEM |
Lek paradox-Lek paradox-In a lekking reproductive system, what male sexual characteristics can signal to females is limited, as the males provide no resources to females or parental care to their offspring. This implies that a female gains indirect benefits from her choice in the form of "good genes" for her offspring.... | milkshake721/2.1M-wiki-STEM |
Lek paradox-Lek paradox-Amotz Zahavi declared that male sexual characteristics only convey useful information to the females if these traits confer a handicap on the male. Otherwise, males could simply cheat: if the courtship displays have a neutral effect on survival, males could all perform equally and it would signi... | milkshake721/2.1M-wiki-STEM |
Lek paradox-Lek paradox-One potential resolution to the lek paradox is Rowe and Houle's theory of condition-dependent expression of male sexually selected traits. Similar to the handicap principle, Rowe and Houle argue that sexually selected traits depend on physical condition. Condition, in turn, summarizes a large nu... | milkshake721/2.1M-wiki-STEM |
Lek paradox-Lek paradox-In an alternate but non-exclusionary hypothesis, W. D. Hamilton and M. Zuk proposed that successful development of sexually selected traits signal resistance to parasites. Parasites can significantly stress their hosts so that they are unable to develop sexually selected traits as well as health... | milkshake721/2.1M-wiki-STEM |
Lek paradox-Lek paradox-One resolution to the lek paradox involves female preferences and how preference alone does not cause a drastic enough directional selection to diminish the genetic variance in fitness. Another conclusion is that the preferred trait is not naturally selected for or against and the trait is maint... | milkshake721/2.1M-wiki-STEM |
Dark Engine-Dark Engine-The Dark Engine was a game engine developed by Looking Glass Studios and was used from 1998 to 2000, mainly in the early Thief games. | milkshake721/2.1M-wiki-STEM |
Dark Engine-Features-The Dark Engine's renderer, originally created by Sean Barrett in 1995, supports graphics similar to that of the original Quake, with Unreal-like skybox effects and colored lighting introduced in Thief II. Due to the limited hardware of the time, the Dark Engine was not designed with scalability in... | milkshake721/2.1M-wiki-STEM |
Dark Engine-Features-The engine does not natively support advanced game scripting, with AI and object behavior being controlled by "Object Script Module" (.OSM) files, which are DLLs that are loaded at runtime. As such, new modules can be written and plugged into the level editor, DromEd, but are limited due to the sco... | milkshake721/2.1M-wiki-STEM |
Dark Engine-Features-For its time, the Dark Engine offered advanced AI and sound features, as well as a powerful object-oriented object system. | milkshake721/2.1M-wiki-STEM |
Dark Engine-Features-The designer has full control of sound propagation within the level, and the "artificial intelligence" of the non-player characters (NPCs) allows for three levels of awareness: vague acknowledgement caused by mild visual or auditive disturbances, which only prompts a startled bit of dialogue; defin... | milkshake721/2.1M-wiki-STEM |
Dark Engine-Source code-In 2009, a complete copy of the Dark Engine source code was discovered in the possession of an ex-Looking Glass Studios employee who was at the time continuing his work for Eidos Interactive. The code was a complete set of the engine's resources, and included the libraries needed to compile the ... | milkshake721/2.1M-wiki-STEM |
Dark Engine-Source code-In late April 2010, a user on the Dreamcast Talk forum disassembled the contents of a Dreamcast development kit he had purchased. The contents of the kit included, among other things, items pertaining to ports of Thief 2 and System Shock 2 to that system. By December 2010, it had been discovered... | milkshake721/2.1M-wiki-STEM |
Dark Engine-DromEd-DromEd is the level editor for the Dark Engine. It was originally used in the design of Thief: The Dark Project, but after a petition from the fan community it was released to the public, as were later versions. | milkshake721/2.1M-wiki-STEM |
Dark Engine-DromEd-There are four different versions of DromEd: for Thief: The Dark Project, for Thief Gold, for Thief II, and lastly for System Shock 2, commonly called "ShockEd." DromEd for Thief: The Dark Project and Thief Gold use the same version of the Dark Engine and therefore can open levels created for each ga... | milkshake721/2.1M-wiki-STEM |
Dark Engine-DromEd-The name of the level editor, DromEd, is a reference to the original project it was designed for — a game based on the Arthurian legend of Camelot — the Camel becoming Dromedary and thence Dromed. DromEd has been used by fans to create hundreds of fan missions for Thief and Thief II, and several miss... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Compartmental models in epidemiology-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, (Susceptible, Infectious, ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Compartmental models in epidemiology-The origin of such models is the early 20th century, with important works being that of Ross in 1916, Ross and Hudson in 1917, Kermack and McKendrick in 1927 and Kendall in 1956. The Reed-Frost model was also a significant and widely-overlooked a... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Compartmental models in epidemiology-Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. Such models can show how different publi... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The model consists of three compartments: S: The number of susceptible individuals. When a susceptible and an infectious individual come into "infectious cont... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-R for the number of removed (and immune) or deceased individuals. These are individuals who have been infected and have either recovered from the disease and entered the removed compartment, or died. It is assumed that the number of deaths is negligible with respect to... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Each member of the population typically progresses from susceptible to infectious to recovered. This can be shown as a flow diagram in which the boxes represent the different compartments and the arrows the transition between compartments, i.e. | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Transition rates For the full specification of the model, the arrows should be labeled with the transition rates between compartments. Between S and I, the transition rate is assumed to be d(S/N)/dt = -βSI/N2, where N is the total population, β is the average number of... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-For the special case in which there is no removal from the infectious compartment (γ=0), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected. | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-The SIR model without birth and death The dynamics of an epidemic, for example, the flu, are often much faster than the dynamics of birth and death, therefore, birth and death are often omitted in simple compartmental models. The SIR system without so-called vital dyna... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-This model was for the first time proposed by William Ogilvy Kermack and Anderson Gray McKendrick as a special case of what we now call Kermack–McKendrick theory, and followed work McKendrick had done with Ronald Ross.This system is non-linear, however it is possible t... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Secondly, we note that the dynamics of the infectious class depends on the following ratio: R0=βγ, the so-called basic reproduction number (also called basic reproduction ratio). This ratio is derived as the expected number of new infections (these new infections are s... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-There is also an effective reproduction number Re , which is similarly defined but in a population made up of both susceptible and infected individuals. The basic reproduction rate R0 quantifies the initial contagiousness of the disease, but the effective reproductio... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-This transcendental equation has a solution in terms of the Lambert W function, namely s∞=1−r∞=−R0−1W(−s0R0e−R0(1−r0)). | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-This shows that at the end of an epidemic that conforms to the simple assumptions of the SIR model, unless s0=0 , not all individuals of the population have been removed, so some must remain susceptible. A driving force leading to the end of an epidemic is a decline i... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-The role of both the basic reproduction number and the initial susceptibility are extremely important. In fact, upon rewriting the equation for infectious individuals as follows: dIdt=(R0SN−1)γI, it yields that if: R0⋅S(0)>N, then: dIdt(0)>0, i.e., there will be a prop... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-The force of infection Note that in the above model the function: F=βI, models the transition rate from the compartment of susceptible individuals to the compartment of infectious individuals, so that it is called the force of infection. However, for large classes of c... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Capasso and, afterwards, other authors have proposed nonlinear forces of infection to model more realistically the contagion process. | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Exact analytical solutions to the SIR model In 2014, Harko and coauthors derived an exact so-called analytical solution (involving an integral that can only be calculated numerically) to the SIR model. In the case without vital dynamics setup, for S(u)=S(t) , etc., it... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-A highly accurate analytic approximant of the SIR model as well as exact analytic expressions for the final values S∞ , I∞ , and R∞ were provided by Kröger and Schlickeiser, so that there is no need to perform a numerical integration to solve the SIR model (a simpli... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-While Kendall considered the so-called all-time SIR model where the initial conditions S(0) , I(0) , and R(0) are coupled through the above relations, Kermack and McKendrick proposed to study the more general semi-time case, for which S(0) and I(0) are both arbitr... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Numerical solutions to the SIR model with approximations Numerical solutions to the SIR model can be found in the literature. An example is using the model to analyze COVID-19 spreading data. Three reproduction numbers can be pulled out from the data analyzed with nume... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Rt does not tell us whether or not the spreading will speed up or slow down in the latter stages when the fraction of susceptible people in the community has dropped significantly after recovery or vaccination. Re corrects this dilution effect by multiplying the fract... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Using the differential equations of the SIR model and converting them to numerical discrete forms, one can set up the recursive equations and calculate the S, I, and R populations with any given initial conditions but accumulate errors over a long calculation time from... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Among these three reproduction numbers, R0 is very useful to judge the control pressure, e.g., a large value meaning the disease will spread very fast and is very difficult to control. Rt is most useful in predicting future trends, for example, if we know the social ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-In this case, we can derive a basic reproduction number: R0=βμ+γ, which has threshold properties. In fact, independently from biologically meaningful initial values, one can show that: lim DFE =(Λμ,0,0) lim EE =(γ+μβ,μβ(R0−1),γβ(R0−1)). | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-The point EE is called the Endemic Equilibrium (the disease is not totally eradicated and remains in the population). With heuristic arguments, one may show that R0 may be read as the average number of infections caused by a single infectious subject in a wholly susce... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-The SIR model In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments: susceptible, S(t) ; infected, I(t) ; and recovered, R(t) . The compartments used for this model consist of three class... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-I(t) denotes the individuals of the population who have been infected with the disease and are capable of spreading the disease to those in the susceptible category. | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-R(t) is the compartment used for the individuals of the population who have been infected and then removed from the disease, either due to immunization or due to death. Those in this category are not able to be infected again or to transmit the infection to others.The ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-dSdt=−βSIN dIdt=βSIN−γI dRdt=γI Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of a and an e... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Steady-state solutions The expected duration of susceptibility will be min (TL∣TS)] where TL reflects the time alive (life expectancy) and TS reflects the time in the susceptible state before becoming infected, which can be simplified to: min (TL∣TS)]=∫0∞e−(μ+δ)xdx=... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The SIR model-Other compartmental models There are many modifications of the SIR model, including those that include births and deaths, where upon recovery there is no immunity (SIS model), where immunity lasts only for a short period of time (SIRS), where there is a latent period o... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-The SIS model Some infections, for example, those from the common cold and influenza, do not confer any long-lasting immunity. Such infections may give temporary resistance but do not give long-term immunity upon recovery from infection, and individ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-It is possible to find an analytical solution to this model (by making a transformation of variables: I=y−1 and substituting this into the mean-field equations), such that the basic reproduction rate is greater than unity. The solution is given as ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-As a special case, one obtains the usual logistic function by assuming γ=0 . This can be also considered in the SIR model with R=0 , i.e. no removal will take place. That is the SI model. The differential equation system using S=N−I thus reduces ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-The SIRV model The Susceptible-Infectious-Recovered-Vaccinated model is an extended SIR model that accounts for vaccination of the susceptible population. This model uses the following system of differential equations: dSdt=−β(t)ISN−v(t)S,dIdt=β(t)I... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-The MSIR model For many infections, including measles, babies are not born into the susceptible compartment but are immune to the disease for the first few months of life due to protection from maternal antibodies (passed across the placenta and add... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-To indicate this mathematically, an additional compartment is added, M(t). This results in the following differential equations: dMdt=Λ−δM−μMdSdt=δM−βSIN−μSdIdt=βSIN−γI−μIdRdt=γI−μR Carrier state Some people who have had an infectious disease such a... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-The SEIR model For many important infections, there is a significant latency period during which individuals have been infected but are not yet infectious themselves. During this period the individual is in compartment E (for exposed).
Assuming that... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-M→S→E→I→R→S Variable contact rates It is well known that the probability of getting a disease is not constant in time. As a pandemic progresses, reactions to the pandemic may change the contact rates which are assumed constant in the simpler models.... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-In addition, Some diseases are seasonal, such as the common cold viruses, which are more prevalent during winter. With childhood diseases, such as measles, mumps, and rubella, there is a strong correlation with the school calendar, so that during th... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-Thus, our model becomes dSdt=μN−μS−β(t)INSdIdt=β(t)INS−(γ+μ)I (the dynamics of recovered easily follows from R=N−S−I ), i.e. a nonlinear set of differential equations with periodically varying parameters. It is well known that this class of dynamic... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-This allowed to give a contribution to explain the poly-annual (typically biennial) epidemic outbreaks of some infectious diseases as interplay between the period of the contact rate oscillations and the pseudo-period of the damped oscillations near... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-SIR model with diffusion Spatiotemporal compartmental models describe not the total number, but the density of susceptible/infective/recovered persons. Consequently, they also allow to model the distribution of infected persons in space. In most cas... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-Interacting Subpopulation SEIR Model As social contacts, disease severity and lethality, as well as the efficacy of prophylactic measures may differ substantially between interacting subpopulations, e.g., the elderly versus the young, separate SEIR ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-SIR Model on Networks The SIR model has been studied on networks of various kinds in order to model a more realistic form of connection than the homogeneous mixing condition which is usually required. A simple model for epidemics on networks in whic... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-SIRSS model - combination of SIR with modelling of social stress Dynamics of epidemics depend on how people's behavior changes in time. For example, at the beginning of the epidemic, people are ignorant and careless, then, after the outbreak of epid... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-The simplest SIR-social stress (SIRSS) model is organised as follows. The susceptible individuals (S) can be split in three subgroups by the types of behavior: ignorant or unaware of the epidemic (Sign), rationally resistant (Sres), and exhausted (S... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Variations on the basic SIR model-The differences between countries are concentrated in two kinetic constants: the rate of mobilization and the rate of exhaustion calculated for COVID-19 epidemic in 13 countries. These constants for this epidemic in all countries can be extracted by... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Modelling vaccination-The SIR model can be modified to model vaccination. Typically these introduce an additional compartment to the SIR model, V , for vaccinated individuals. Below are some examples. | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Modelling vaccination-Vaccinating newborns In presence of a communicable diseases, one of the main tasks is that of eradicating it via prevention measures and, if possible, via the establishment of a mass vaccination program. Consider a disease for which the newborn are vaccinated (... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Modelling vaccination-In other words, if P<P∗=1−1R0 the vaccination program is not successful in eradicating the disease, on the contrary, it will remain endemic, although at lower levels than the case of absence of vaccinations. This means that the mathematical model suggests that ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Modelling vaccination-Vaccination and information Modern societies are facing the challenge of "rational" exemption, i.e. the family's decision to not vaccinate children as a consequence of a "rational" comparison between the perceived risk from infection and that from getting damag... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Modelling vaccination-In such a case the eradication condition becomes: P(0)≥P∗, i.e. the baseline vaccination rate should be greater than the "mandatory vaccination" threshold, which, in case of exemption, cannot hold. Thus, "rational" exemption might be myopic since it is based on... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Modelling vaccination-Vaccination of non-newborns In case there also are vaccinations of non newborns at a rate ρ the equation for the susceptible and vaccinated subject has to be modified as follows: dSdt=μN(1−P)−μS−ρS−βINSdVdt=μNP+ρS−μV leading to the following eradication conditi... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The influence of age: age-structured models-Age has a deep influence on the disease spread rate in a population, especially the contact rate. This rate summarizes the effectiveness of contacts between susceptible and infectious subjects. Taking into account the ages of the epidemic ... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The influence of age: age-structured models-Complexity is added by the initial conditions for newborns (i.e. for a=0), that are straightforward for infectious and removed: i(t,0)=r(t,0)=0 but that are nonlocal for the density of susceptible newborns: s(t,0)=∫0aM(φs(a)s(a,t)+φi(a)i(a... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The influence of age: age-structured models-Moreover, defining now the density of the total population n(t,a)=s(t,a)+i(t,a)+r(t,a) one obtains: ∂tn(t,a)+∂an(t,a)=−μ(a)n(a,t) In the simplest case of equal fertilities in the three epidemic classes, we have that in order to have demog... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-The influence of age: age-structured models-A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator. | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Other considerations within compartmental epidemic models-Vertical transmission In the case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This transmission of the disease down from the mother is referred to a... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Other considerations within compartmental epidemic models-Vector transmission Diseases transmitted from human to human indirectly, i.e. malaria spread by way of mosquitoes, are transmitted through a vector. In these cases, the infection transfers from human to insect and an epidemic... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Deterministic versus stochastic epidemic models-It is important to stress that the deterministic models presented here are valid only in case of sufficiently large populations, and as such should be used cautiously.To be more precise, these models are only valid in the thermodynamic... | milkshake721/2.1M-wiki-STEM |
Compartmental models in epidemiology-Deterministic versus stochastic epidemic models-One of the possible extensions of mean-field models considers the spreading of epidemics on a network based on percolation theory concepts. Stochastic epidemic models have been studied on different networks and more recently applied to... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-Boltzmann distribution-In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperat... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-Boltzmann distribution-The term system here has a wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore the Boltzmann distribution can be used to solve a wide variety of problems. The dis... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-Boltzmann distribution-The ratio of probabilities of two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference: exp (εj−εikT) The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of th... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-The distribution-The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state's energy and temperature of the system to which the distribution is applied. It is given as where: exp() is the exponential function, pi is the proba... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-The distribution-The partition function can be calculated if we know the energies of the states accessible to the system of interest. For atoms the partition function values can be found in the NIST Atomic Spectra Database.The distribution shows that states with lower energy will always have a hi... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-The distribution-The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over bound states accessible to them. If we have a system consisting of many particles, the probability of a particle being in state i is practically the probability th... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-The distribution-pi=NiN where Ni is the number of particles in state i and N is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives ... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-The distribution-The softmax function commonly used in machine learning is related to the Boltzmann distribution: softmax [−ε1kT,…,−εMkT] | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-Generalized Boltzmann distribution-Distribution of the form Pr exp [∑η=1nXηxη(ω)kBT−E(ω)kBT] is called generalized Boltzmann distribution by some authors.The Boltzmann distribution is a special case of the generalized Boltzmann distribution. The generalized Boltzmann distribution is used in stat... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-Generalized Boltzmann distribution-It is the only distribution that is mathematically consistent with the fundamental thermodynamic relation where state functions are described by ensemble average. | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-In statistical mechanics-The Boltzmann distribution appears in statistical mechanics when considering closed systems of fixed composition that are in thermal equilibrium (equilibrium with respect to energy exchange). The most general case is the probability distribution for the canonical ensemble... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-In statistical mechanics-Statistical frequencies of subsystems' states (in a non-interacting collection) When the system of interest is a collection of many non-interacting copies of a smaller subsystem, it is sometimes useful to find the statistical frequency of a given subsystem state, among th... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-In statistical mechanics-Maxwell–Boltzmann statistics of classical gases (systems of non-interacting particles) In particle systems, many particles share the same space and regularly change places with each other; the single-particle state space they occupy is a shared space. Maxwell–Boltzmann st... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-In statistical mechanics-If the subsystems within a collection do interact with each other, then the expected frequencies of subsystem states no longer follow a Boltzmann distribution, and even may not have an analytical solution. The canonical ensemble can however still be applied to the collect... | milkshake721/2.1M-wiki-STEM |
Boltzmann distribution-In statistical mechanics-With quantum gases of non-interacting particles in equilibrium, the number of particles found in a given single-particle state does not follow Maxwell–Boltzmann statistics, and there is no simple closed form expression for quantum gases in the canonical ensemble. In the g... | milkshake721/2.1M-wiki-STEM |
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