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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 partial agonist at GABAA receptors, although with little efficacy at the α1 subtype and more at α2 and α3. It has potent anticonvulsant effects in animal studies, but a relatively short duration of action, and produces little sedative effects or physical dependence.
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.
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,…,xn]=f(n)(ξ)n!. For n = 1, that is two function points, one obtains the simple mean value theorem.
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, ..., xn. By applying Rolle's theorem first to g , then to g′ , and so on until g(n−1) , we find that g(n) has a zero ξ . This means that 0=g(n)(ξ)=f(n)(ξ)−f[x0,…,xn]n! ,f[x0,…,xn]=f(n)(ξ)n!.
Mean value theorem (divided differences)	Applications	The theorem can be used to generalise the Stolarsky mean to more than two variables.
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 remains. Persistent female choice for particular male trait values should erode genetic diversity in male traits and thereby remove the benefits of choice, yet choice persists. This paradox can be somewhat alleviated by the occurrence of mutations introducing potential differences, as well as the possibility that traits of interest have more or less favorable recessive alleles.
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 choice should lead to directional runaway selection, resulting in a greater prevalence for the selected traits. Stronger selection should lead to impaired survival, as it decreases genetic variance and ensures that more offspring have similar traits. However, lekking species do not exhibit runaway selection.
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. Hypothetically, in choosing a male that excels at courtship displays, females gain genes for their offspring that increase their survival or reproductive fitness.
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 signify nothing to the females. But if the courtship display is somehow deleterious to the male’s survival—such as increased predator risk or time and energy expenditure—it becomes a test by which females can assess male quality. Under the handicap principle, males who excel at the courtship displays prove that they are of better quality and genotype, as they have already withstood the costs to having these traits. Resolutions have been formed to explain why strong female mate choice does not lead to runaway selection. The handicap principle describes how costly male ornaments provide females with information about the male’s inheritable fitness. The handicap principle may be a resolution to the lek paradox, for if females select for the condition of male ornaments, then their offspring have better fitness.
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 number of genetic loci, including those involved in metabolism, muscular mass, nutrition, etc. Rowe and Houle claim that condition dependence maintains genetic variation in the face of persistent female choice, as the male trait is correlated with abundant genetic variation in condition. This is the genic capture hypothesis, which describes how a significant amount of the genome is involved in shaping the traits that are sexually selected. There are two criteria in the genic capture hypothesis: the first is that sexually selected traits are dependent upon condition and the second is that general condition is attributable to high genetic variance.Genetic variation in condition-dependent traits may be further maintained through mutations and environmental effects. Genotypes may be more effective in developing condition dependent sexual characteristics in different environments, while mutations may be deleterious in one environment and advantageous in another. Thus genetic variance remains in populations through gene flow across environments or generation overlap. According to the genic capture hypothesis, female selection does not deplete the genetic variance, as sexual selection operates on condition dependence traits, thereby accumulating genetic variance within the selected for trait. Therefore, females are actually selecting for high genetic variance.
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 healthy males. According to this theory, a male who vigorously displays demonstrates that he has parasite-resistant genes to the females. In support of this theory, Hamilton and Zuk found that male sexual ornaments were significantly correlated with levels of incidence of six blood diseases in North American passerine bird species. The Hamilton and Zuk model addresses the lek paradox, arguing that the cycles of co-adaptation between host and parasite resist a stable equilibrium point. Hosts continue to evolve resistance to parasites and parasites continue to bypass resistant mechanisms, continuously generating genetic variation. The genic capture and parasite resistance hypotheses could logically co-occur in the same population.
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 maintained because it implies increased attractiveness to the male. Thus, there may be no paradox.
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.
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 mind, and can therefore only display 1024 terrain polygons onscreen at once, as well as various other limits on objects and lights. In terms of textures, the game supports palletized PCX and TGA textures, in powers of two up to 256x256. Textures are grouped in "families" which share the same palette. There is a maximum of 216 textures and independent palettes, excluding 8 animated water textures.
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 scope of the functions made available by the core engine. In order to overcome this, editors must resort to complicated Rube Goldberg machine-like effects using a combination of its other systems.
Dark Engine	Features	For its time, the Dark Engine offered advanced AI and sound features, as well as a powerful object-oriented object system.
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; definite acknowledgement caused by significant visual or auditive disturbances, which causes the NPC to enter "search mode", and definite acquisition (triggered by visual on the fully lit player, or face-first contact with a player regardless of the light level), prompting a direct attack.
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 code. Fans of the Thief and System Shock series subsequently petitioned the publisher to consider releasing the code.
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 by the user and subsequently the greater Looking Glass Studios fan community that a compact disc included with the kit - the contents of which had been uploaded to the Internet - included a second copy of the Dark Engine source, minus the libraries needed to compile the code.In September 2012, a significant unofficial update to the Dark Engine was published anonymously in a French forum, most probably based on the leaked Dreamcast source code. This unofficial patch extended the limits of the engine, introduced support for recent graphics and sound hardware, as well as better support for newer versions of Windows.
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.
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 game, although Thief Gold levels may refer to in-game objects that are not found in Thief. Thief II uses a revised version of the Dark Engine, and therefore it is difficult to open levels created for Thief with DromEd for Thief II. ShockEd is not compatible with any Dark Engine games aside from System Shock 2. However, basic level geometry can be moved between editors using a geometry export feature called "multibrush". System Shock 2 levels can be loaded by DromEd 2 with some work.
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 missions for System Shock 2.
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, or Recovered). 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.
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 ancestor of modern epidemiological modelling approaches.The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze.
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 public health interventions may affect the outcome of the epidemic, e.g., what the most efficient technique is for issuing a limited number of vaccines in a given population.
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 contact", the susceptible individual contracts the disease and transitions to the infectious compartment. I: The number of infectious individuals. These are individuals who have been infected and are capable of infecting susceptible individuals.
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 the total population. This compartment may also be called "recovered" or "resistant".This model is reasonably predictive for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps and rubella. It has also been used outside of epidemiology, for example in modeling the spread of song popularity , political influence , rumors and gun ownership. These variables (S, I, and R) represent the number of people in each compartment at a particular time. To represent that the number of susceptible, infectious and removed individuals may vary over time (even if the total population size remains constant), we make the precise numbers a function of t (time): S(t), I(t) and R(t). For a specific disease in a specific population, these functions may be worked out in order to predict possible outbreaks and bring them under control.As implied by the variable function of t, the model is dynamic in that the numbers in each compartment may fluctuate over time. The importance of this dynamic aspect is most obvious in an endemic disease with a short infectious period, such as measles in the UK prior to the introduction of a vaccine in 1968. Such diseases tend to occur in cycles of outbreaks due to the variation in number of susceptibles (S(t)) over time. During an epidemic, the number of susceptible individuals falls rapidly as more of them are infected and thus enter the infectious and removed compartments. The disease cannot break out again until the number of susceptibles has built back up, e.g. as a result of offspring being born into the susceptible compartment.
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.
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 contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject, and SI/N2 is the fraction of those contacts between an infectious and susceptible individual which result in the susceptible person becoming infected. (This is mathematically similar to the law of mass action in chemistry in which random collisions between molecules result in a chemical reaction and the fractional rate is proportional to the concentration of the two reactants).Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI. This is equivalent to assuming that the probability of an infectious individual recovering in any time interval dt is simply γdt. If an individual is infectious for an average time period D, then γ = 1/D. This is also equivalent to the assumption that the length of time spent by an individual in the infectious state is a random variable with an exponential distribution. The "classical" SIR model may be modified by using more complex and realistic distributions for the I-R transition rate (e.g. the Erlang distribution).
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.
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 dynamics (birth and death, sometimes called demography) described above can be expressed by the following system of ordinary differential equations: {dSdt=−βISN,dIdt=βISN−γI,dRdt=γI, where S is the stock of susceptible population, I is the stock of infected, R is the stock of removed population (either by death or recovery), and N is the sum of these three.
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 to derive its analytic solution in implicit form. Firstly note that from: dSdt+dIdt+dRdt=0, it follows that: constant =N, expressing in mathematical terms the constancy of population N . Note that the above relationship implies that one need only study the equation for two of the three variables.
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 sometimes called secondary infections) from a single infection in a population where all subjects are susceptible. This idea can probably be more readily seen if we say that the typical time between contacts is Tc=β−1 , and the typical time until removal is Tr=γ−1 . From here it follows that, on average, the number of contacts by an infectious individual with others before the infectious has been removed is: Tr/Tc.
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 reproduction number Re is a time-dependent rate. By dividing the first differential equation by the third, separating the variables and integrating we get S(t)=S(0)e−R0(R(t)−R(0))/N, where S(0) and R(0) are the initial numbers of, respectively, susceptible and removed subjects. Writing s0=S(0)/N for the initial proportion of susceptible individuals, and s∞=S(∞)/N and r∞=R(∞)/N for the proportion of susceptible and removed individuals respectively in the limit t→∞, one has s∞=1−r∞=s0e−R0(r∞−r0) (note that the infectious compartment empties in this limit).
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)).
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 in the number of infectious individuals. The epidemic does not typically end because of a complete lack of susceptible individuals.
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 proper epidemic outbreak with an increase of the number of the infectious (which can reach a considerable fraction of the population). On the contrary, if R0⋅S(0)<N, then dIdt(0)<0, i.e., independently from the initial size of the susceptible population the disease can never cause a proper epidemic outbreak. As a consequence, it is clear that both the basic reproduction number and the initial susceptibility are extremely important.
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 communicable diseases it is more realistic to consider a force of infection that does not depend on the absolute number of infectious subjects, but on their fraction (with respect to the total constant population N ): F=βIN.
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.
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 corresponds to the following time parametrization S(u)=S(0)u I(u)=N−R(u)−S(u) ln ⁡(u) for t=Nβ∫u1du∗u∗I(u∗),ρ=γNβ, with initial conditions (S(1),I(1),R(1))=(S(0),N−R(0)−S(0),R(0)),uT<u<1, where uT satisfies I(uT)=0 . By the transcendental equation for R∞ above, it follows that uT=e−(R∞−R(0))/ρ(=S∞/S(0) , if S(0)≠0) and I∞=0 An equivalent so-called analytical solution (involving an integral that can only be calculated numerically) found by Miller yields S(t)=S(0)e−ξ(t)I(t)=N−S(t)−R(t)R(t)=R(0)+ρξ(t)ξ(t)=βN∫0tI(t∗)dt∗ Here ξ(t) can be interpreted as the expected number of transmissions an individual has received by time t . The two solutions are related by e−ξ(t)=u Effectively the same result can be found in the original work by Kermack and McKendrick.These solutions may be easily understood by noting that all of the terms on the right-hand sides of the original differential equations are proportional to I . The equations may thus be divided through by I , and the time rescaled so that the differential operator on the left-hand side becomes simply d/dτ , where dτ=Idt , i.e. τ=∫Idt . The differential equations are now all linear, and the third equation, of the form dR/dτ= const., shows that τ and R (and ξ above) are simply linearly related.
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 simplified example practice on COVID-19 numerical simulation using Microsoft Excel can be found here ), to obtain its parameters from existing data, or to predict the future dynamics of an epidemics modeled by the SIR model. The approximant involves the Lambert W function which is part of all basic data visualization software such as Microsoft Excel, MATLAB, and Mathematica.
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 arbitrary. This latter version, denoted as semi-time SIR model, makes predictions only for future times t>0 . An analytic approximant and exact expressions for the final values are available for the semi-time SIR model as well.
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 numerical approximation, the basic reproduction number: R0=β0γ0 the real-time reproduction number: Rt=βtγt and the real-time effective reproduction number: Re=βtSγtN R0 represents the speed of reproduction rate at the beginning of the spreading when all populations are assumed susceptible, e.g. if 0.4 day−1 and 0.2 day−1 meaning one infectious person on average infects 0.4 susceptible people per day and recovers in 1/0.2=5 days. Thus when this person recovered, there are two people still infectious directly got from this person and R0=2 , i.e. the number of infectious people doubled in one cycle of 5 days. The data simulated by the model with R0=2 or real data fitted will yield a doubling of the number of infectious people faster than 5 days because the two infected people are infecting people. From the SIR model, we can tell that β is determined by the nature of the disease and also a function of the interactive frequency between the infectious person I with the susceptible people S and also the intensity/duration of the interaction like how close they interact for how long and whether or not they both wear masks, thus, it changes over time when the average behavior of the carriers and susceptible people changes. The model use SI to represent these factors but it indeed is referenced to the initial stage when no action is taken to prevent the spread and all population is susceptible, thus all changes are absorbed by the change of β . γ is usually more stable over time assuming when the infectious person shows symptoms, she/he will seek medical attention or be self-isolated. So if we find Rt changes, most probably the behaviors of people in the community have changed from their normal patterns before the outbreak, or the disease has mutated to a new form. Costive massive detection and isolation of susceptible close contacts have effects on reducing 1/γ but whose efficiencies are under debate. This debate is largely on the uncertainty of the number of days reduced from after infectious or detectable whichever comes first to before a symptom shows up for an infected susceptible person. If the person is infectious after symptoms show up, or detection only works for a person with symptoms, then these prevention methods are not necessary, and self-isolation and/or medical attention is the best way to cut the 1/γ values. The typical onset of the COVID-19 infectious period is in the order of one day from the symptoms showing up, making massive detection with typical frequency in a few days useless.
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 fraction of the susceptible population over the total population. It corrects the effective/transmissible interaction between an infectious person and the rest of the community when many of the interaction is immune in the middle to late stages of the disease spreading. Thus, when Re>1 , we will see an exponential-like outbreak; when Re=1 , a steady state reached and no number of infectious people changes over time; and when Re<1 , the disease decays and fades away over time.
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 the reference point. Sometimes a convergence test is needed to estimate the errors. Given a set of initial conditions and the disease-spreading data, one can also fit the data with the SIR model and pull out the three reproduction numbers when the errors are usually negligible due to the short time step from the reference point. Any point of the time can be used as the initial condition to predict the future after it using this numerical model with assumption of time-evolved parameters such as population, Rt , and γ . However, away from this reference point, errors will accumulate over time thus convergence test is needed to find an optimal time step for more accurate results.
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 interactions have reduced 50% frequently from that before the outbreak and the interaction intensities among people are the same, then we can set 0.5 R0 . If social distancing and masks add another 50% cut in infection efficiency, we can set 0.25 R0 . Re will perfectly correlate with the waves of the spreading and whenever Re>1 , the spreading accelerates, and when Re<1 , the spreading slows down thus useful to set a prediction on the short term trends. Also, it can be used to directly calculate the threshold population of vaccination/immunization for the herd immunity stage by setting Rt=R0 The SIR model with vital dynamics and constant population Consider a population characterized by a death rate μ and birth rate Λ , and where a communicable disease is spreading. The model with mass-action transmission is: dSdt=Λ−μS−βISNdIdt=βISN−γI−μIdRdt=γI−μR for which the disease-free equilibrium (DFE) is: (S(t),I(t),R(t))=(Λμ,0,0).
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)).
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 susceptible population, the above relationship biologically means that if this number is less than or equal to one the disease goes extinct, whereas if this number is greater than one the disease will remain permanently endemic in the population.
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 classes: S(t) is used to represent the individuals not yet infected with the disease at time t, or those susceptible to the disease of the population.
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.
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 flow of this model may be considered as follows: S→I→R Using a fixed population, N=S(t)+I(t)+R(t) in the three functions resolves that the value N should remain constant within the simulation, if a simulation is used to solve the SIR model. Alternatively, the analytic approximant can be used without performing a simulation. The model is started with values of S(t=0) , I(t=0) and R(t=0) . These are the number of people in the susceptible, infected and removed categories at time equals zero. If the SIR model is assumed to hold at all times, these initial conditions are not independent. Subsequently, the flow model updates the three variables for every time point with set values for β and γ . The simulation first updates the infected from the susceptible and then the removed category is updated from the infected category for the next time point (t=1). This describes the flow persons between the three categories. During an epidemic the susceptible category is not shifted with this model, β changes over the course of the epidemic and so does γ . These variables determine the length of the epidemic and would have to be updated with each cycle.
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 equal fraction b of people that an individual makes contact with per unit time. Then, let β be the multiplication of a and b . This is the transmission probability times the contact rate. Besides, an infected individual makes contact with b persons per unit time whereas only a fraction, S/N of them are susceptible. Thus, we have every infective can infect abS=βS susceptible persons, and therefore, the whole number of susceptibles infected by infectives per unit time is βSI . For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, a number equal to the fraction γ (which represents the mean recovery/death rate, or 1/γ the mean infective period) of infectives are leaving this class per unit time to enter the removed class. These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned. Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.
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=1μ+δ, such that the number of susceptible persons is the number entering the susceptible compartment μN times the duration of susceptibility: S=μNμ+λ. Analogously, the steady-state number of infected persons is the number entering the infected state from the susceptible state (number susceptible, times rate of infection) λ=βIN, times the duration of infectiousness 1μ+v :I=μNμ+λλ1μ+v.
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 of the disease where the person is not infectious (SEIS and SEIR), and where infants can be born with immunity (MSIR). Compartmental models can also be used to model multiple risk groups, and even the interaction of multiple pathogens.
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 individuals become susceptible again. We have the model: dSdt=−βSIN+γIdIdt=βSIN−γI Note that denoting with N the total population it holds that: dSdt+dIdt=0⇒S(t)+I(t)=N .It follows that: dIdt=(β−γ)I−βNI2 ,i.e. the dynamics of infectious is ruled by a logistic function, so that ∀I(0)>0 lim lim t→+∞I(t)=(1−γβ)N.
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 I(t)=I∞1+Ve−χt .where I∞=(1−γ/β)N is the endemic infectious population, χ=β−γ , and V=I∞/I0−1 . As the system is assumed to be closed, the susceptible population is then S(t)=N−I(t) Whenever the integer nature of the number of agents is evident (populations with fewer than tens of thousands of individuals), inherent fluctuations in the disease spreading process caused by discrete agents result in uncertainties. In this scenario, the evolution of the disease predicted by compartmental equations deviates significantly from the observed results. These uncertainties may even cause the epidemic to end earlier than predicted by the compartmental equations.
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 to: dIdt∝I⋅(N−I). In the long run, in the SI model, all individuals will become infected. The SIRD model The Susceptible-Infectious-Recovered-Deceased model differentiates between Recovered (meaning specifically individuals having survived the disease and now immune) and Deceased. This model uses the following system of differential equations: dSdt=−βISN,dIdt=βISN−γI−μI,dRdt=γI,dDdt=μI, where β,γ,μ are the rates of infection, recovery, and mortality, respectively.
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)ISN−γ(t)I,dRdt=γ(t)I,dVdt=v(t)S, where β,γ,v are the rates of infection, recovery, and vaccination, respectively. For the semi-time initial conditions S(0)=(1−η)N , I(0)=ηN , R(0)=V(0)=0 and constant ratios k=γ(t)/β(t) and b=v(t)/β(t) the model had been solved approximately. The occurrence of a pandemic outburst requires k+b<1−2η and there is a critical reduced vaccination rate bc beyond which the steady-state size S∞ of the susceptible compartment remains relatively close to S(0) . Arbitrary initial conditions satisfying S(0)+I(0)+R(0)+V(0)=N can be mapped to the solved special case with R(0)=V(0)=0 The numerical solution of this model to calculate the real-time reproduction number Rt of COVID-19 can be practiced based on information from the different populations in a community. Numerical solution is a commonly used method to analyze complicated kinetic networks when the analytical solution is difficult to obtain or limited by requirements such as boundary conditions or special parameters. It uses recursive equations to calculate the next step by converting the numerical integration into Riemann sum of discrete time steps e.g., use yesterday's principal and interest rate to calculate today's interest which assumes the interest rate is fixed during the day. The calculation contains projected errors if the analytical corrections on the numerical step size are not included, e.g. when the interest rate of annual collection is simplified to 12 times the monthly rate, a projected error is introduced. Thus the calculated results will carry accumulative errors when the time step is far away from the reference point and a convergence test is needed to estimate the error. However, this error is usually acceptable for data fitting. When fitting a set of data with a close time step, the error is relatively small because the reference point is nearby compared to when predicting a long period of time after a reference point. Once the real-time Rt is pulled out, one can compare it to the basic reproduction number R0 . Before the vaccination, Rt gives the policy maker and general public a measure of the efficiency of social mitigation activities such as social distancing and face masking simply by dividing RtR0 . Under massive vaccination, the goal of disease control is to reduce the effective reproduction number Re=RtSN<1 , where S is the number of susceptible population at the time and N is the total population. When Re<1 , the spreading decays and daily infected cases go down.
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 additionally through colostrum). This is called passive immunity. This added detail can be shown by including an M class (for maternally derived immunity) at the beginning of the model.
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 as tuberculosis never completely recover and continue to carry the infection, whilst not suffering the disease themselves. They may then move back into the infectious compartment and suffer symptoms (as in tuberculosis) or they may continue to infect others in their carrier state, while not suffering symptoms. The most famous example of this is probably Mary Mallon, who infected 22 people with typhoid fever. The carrier compartment is labelled C.
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 the latency period is a random variable with exponential distribution with parameter a (i.e. the average latency period is a−1 ), and also assuming the presence of vital dynamics with birth rate Λ equal to death rate Nμ (so that the total number N is constant), we have the model: dSdt=μN−μS−βISNdEdt=βISN−(μ+a)EdIdt=aE−(γ+μ)IdRdt=γI−μR. We have S+E+I+R=N, but this is only constant because of the simplifying assumption that birth and death rates are equal; in general N is a variable. For this model, the basic reproduction number is: R0=aμ+aβμ+γ. Similarly to the SIR model, also, in this case, we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently from biologically meaningful initial conditions (S(0),E(0),I(0),R(0))∈{(S,E,I,R)∈[0,N]4:S≥0,E≥0,I≥0,R≥0,S+E+I+R=N} it holds that: lim t→+∞(S(t),E(t),I(t),R(t))=DFE=(N,0,0,0), lim t→+∞(S(t),E(t),I(t),R(t))=EE. In case of periodically varying contact rate β(t) the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients: dE1dt=β(t)I1−(γ+a)E1dI1dt=aE1−(γ+μ)I1 is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane). The SEIS model The SEIS model is like the SEIR model (above) except that no immunity is acquired at the end. S→E→I→S In this model an infection does not leave any immunity thus individuals that have recovered return to being susceptible, moving back into the S(t) compartment. The following differential equations describe this model: dSdt=Λ−βSIN−μS+γIdEdt=βSIN−(ϵ+μ)EdIdt=εE−(γ+μ)I The MSEIR model For the case of a disease, with the factors of passive immunity, and a latency period there is the MSEIR model. M→S→E→I→R dMdt=Λ−δM−μMdSdt=δM−βSIN−μSdEdt=βSIN−(ε+μ)EdIdt=εE−(γ+μ)IdRdt=γI−μR The MSEIRS model An MSEIRS model is similar to the MSEIR, but the immunity in the R class would be temporary, so that individuals would regain their susceptibility when the temporary immunity ended.
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. Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic.
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 the school holidays the probability of getting such a disease dramatically decreases. As a consequence, for many classes of diseases, one should consider a force of infection with periodically ('seasonal') varying contact rate F=β(t)IN,β(t+T)=β(t) with period T equal to one year.
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 dynamical systems may undergo very interesting and complex phenomena of nonlinear parametric resonance. It is easy to see that if: lim t→+∞(S(t),I(t))=DFE=(N,0), whereas if the integral is greater than one the disease will not die out and there may be such resonances. For example, considering the periodically varying contact rate as the 'input' of the system one has that the output is a periodic function whose period is a multiple of the period of the input.
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 the endemic equilibrium. Remarkably, in some cases, the behavior may also be quasi-periodic or even chaotic.
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 cases, this is done by combining the SIR model with a diffusion equation ∂tS=DS∇2S−βISN,∂tI=DI∇2I+βISN−γI,∂tR=DR∇2R+γI, where DS , DI and DR are diffusion constants. Thereby, one obtains a reaction-diffusion equation. (Note that, for dimensional reasons, the parameter β has to be changed compared to the simple SIR model.) Early models of this type have been used to model the spread of the black death in Europe. Extensions of this model have been used to incorporate, e.g., effects of nonpharmaceutical interventions such as social distancing.
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 models for each subgroup may be used that are mutually connected through interaction links. Such Interacting Subpopulation SEIR models have been used for modeling the COVID-19 pandemic at continent scale to develop personalized, accelerated, subpopulation-targeted vaccination strategies that promise a shortening of the pandemic and a reduction of case and death counts in the setting of limited access to vaccines during a wave of virus Variants of Concern.
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 which an individual has a probability p of being infected by each of his infected neighbors in a given time step leads to results similar to giant component formation on Erdos Renyi random graphs.
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 epidemics and alarm, they begin to comply with the various restrictions and the spreading of epidemics may decline. Over time, some people get tired/frustrated by the restrictions and stop following them (exhaustion), especially if the number of new cases drops down. After resting for some time, they can follow the restrictions again. But during this pause the second wave can come and become even stronger than the first one. Social dynamics should be considered. The social physics models of social stress complement the classical epidemics models.
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 (Sexh) that do not react on the external stimuli (this is a sort of refractory period). In other words: S(t) = Sign(t) + Sres(t) + Sexh(t). Symbolically, the social stress model can be presented by the "reaction scheme" (where I denotes the infected individuals): Sign+2I→Sres+2I – mobilization reaction (the autocatalytic form here means that the transition rate is proportional to the square of the infected fraction I); Sres→Sexh – exhaustion process due to fatigue from anti-epidemic restrictions; Sexh→Sign – slow relaxation to the initial state (end of the refractory period).The main SIR epidemic reaction S...+I→2I has different reaction rate constants β for Sign, Sres, and Sexh. Presumably, for Sres, β is lower than for Sign and Sign.
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 the fitting of the SIRSS model to publicly available data The KdV-SIR equation Based on the classical SIR model, a Korteweg-de Vries (KdV)–SIR equation and its analytical solution have been proposed to illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions. The KdV-SIR equation is written as follows: d2Idt−σo2I+32σo2ImaxI2=0 Here, σo=γ(Ro−1) ,Ro=βγSoN and Imax=So2(Ro−1)2Ro2 .So indicates the initial value of the state variable S . Parameters σo (σ-naught) and Ro (R-naught) are the time-independent relative growth rate and basic reproduction number, respectively. Imax presents the maximum of the state variables I (for the number of infected persons). An analytical solution to the KdV-SIR equation is written as follows: I=Imaxsech2(σo2t) which represents a solitary wave solution.
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.
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 (with a vaccine giving lifelong immunity) at a rate P∈(0,1) :dSdt=νN(1−P)−μS−βINSdIdt=βINS−(μ+γ)IdVdt=νNP−μV where V is the class of vaccinated subjects. It is immediate to show that: lim t→+∞V(t)=NP, thus we shall deal with the long term behavior of S and I , for which it holds that: lim t→+∞(S(t),I(t))=DFE=(N(1−P),0) lim t→+∞(S(t),I(t))=EE=(NR0(1−P),N(R0(1−P)−1)).
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 for a disease whose basic reproduction number may be as high as 18 one should vaccinate at least 94.4% of newborns in order to eradicate the disease.
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 damages from the vaccine. In order to assess whether this behavior is really rational, i.e. if it can equally lead to the eradication of the disease, one may simply assume that the vaccination rate is an increasing function of the number of infectious subjects: 0.
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 only on the current low incidence due to high vaccine coverage, instead taking into account future resurgence of infection due to coverage decline.
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 condition: P≥1−(1+ρμ)1R0 Pulse vaccination strategy This strategy repeatedly vaccinates a defined age-cohort (such as young children or the elderly) in a susceptible population over time. Using this strategy, the block of susceptible individuals is then immediately removed, making it possible to eliminate an infectious disease, (such as measles), from the entire population. Every T time units a constant fraction p of susceptible subjects is vaccinated in a relatively short (with respect to the dynamics of the disease) time. This leads to the following impulsive differential equations for the susceptible and vaccinated subjects: dSdt=μN−μS−βINS,S(nT+)=(1−p)S(nT−),n=0,1,2,…dVdt=−μV,V(nT+)=V(nT−)+pS(nT−),n=0,1,2,… It is easy to see that by setting I = 0 one obtains that the dynamics of the susceptible subjects is given by: S∗(t)=1−p1−(1−p)E−μTE−μMOD(t,T) and that the eradication condition is: R0∫0TS∗(t)dt<1
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 classes s(t,a),i(t,a),r(t,a) (to limit ourselves to the susceptible-infectious-removed scheme) such that: S(t)=∫0aMs(t,a)da I(t)=∫0aMi(t,a)da R(t)=∫0aMr(t,a)da (where aM≤+∞ is the maximum admissible age) and their dynamics is not described, as one might think, by "simple" partial differential equations, but by integro-differential equations: ∂ts(t,a)+∂as(t,a)=−μ(a)s(a,t)−s(a,t)∫0aMk(a,a1;t)i(a1,t)da1 ∂ti(t,a)+∂ai(t,a)=s(a,t)∫0aMk(a,a1;t)i(a1,t)da1−μ(a)i(a,t)−γ(a)i(a,t) ∂tr(t,a)+∂ar(t,a)=−μ(a)r(a,t)+γ(a)i(a,t) where: F(a,t,i(⋅,⋅))=∫0aMk(a,a1;t)i(a1,t)da1 is the force of infection, which, of course, will depend, though the contact kernel k(a,a1;t) on the interactions between the ages.
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,t)+φr(a)r(a,t))da where φj(a),j=s,i,r are the fertilities of the adults.
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 demographic equilibrium the following necessary and sufficient condition linking the fertility φ(.) with the mortality μ(a) must hold: exp ⁡(−∫0aμ(q)dq)da and the demographic equilibrium is exp ⁡(−∫0aμ(q)dq), automatically ensuring the existence of the disease-free solution: DFS(a)=(n∗(a),0,0).
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.
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 as vertical transmission. The influx of additional members into the infected category can be considered within the model by including a fraction of the newborn members in the infected compartment.
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 model must include both species, generally requiring many more compartments than a model for direct transmission. Others Other occurrences which may need to be considered when modeling an epidemic include things such as the following: Non-homogeneous mixing Variable infectivity Distributions that are spatially non-uniform Diseases caused by macroparasites
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 limit, where the population is effectively infinite. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system. In a true system then, the pathogen may not propagate, as no host will be infected. But, in deterministic mean-field models, the number of infected can take on real, namely, non-integer values of infected hosts, and the number of hosts in the model can be less than one, but more than zero, thereby allowing the pathogen in the model to propagate. The reliability of compartmental models is limited to compartmental applications.
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 the COVID-19 pandemic.
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 temperature of the system. The distribution is expressed in the form: exp ⁡(−εikT) where pi is the probability of the system being in state i, exp is the exponential function, εi is the energy of that state, and a constant kT of the distribution is the product of the Boltzmann constant k and thermodynamic temperature T. The symbol {\textstyle \propto } denotes proportionality (see § The distribution for the proportionality constant).
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 distribution shows that states with lower energy will always have a higher probability of being occupied.
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 the statistical mechanics of gases in thermal equilibrium. Boltzmann's statistical work is borne out in his paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium" The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902.The Boltzmann distribution should not be confused with the Maxwell–Boltzmann distribution or Maxwell-Boltzmann statistics. The Boltzmann distribution gives the probability that a system will be in a certain state as a function of that state's energy, while the Maxwell-Boltzmann distributions give the probabilities of particle speeds or energies in ideal gases. The distribution of energies in a one-dimensional gas however, does follow the Boltzmann distribution.
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 probability of state i, εi is the energy of state i, k is the Boltzmann constant, T is the absolute temperature of the system, M is the number of all states accessible to the system of interest, Q (denoted by some authors by Z) is the normalization denominator, which is the canonical partition function It results from the constraint that the probabilities of all accessible states must add up to 1.The Boltzmann distribution is the distribution that maximizes the entropy subject to the normalization constraint and the constraint that {\textstyle \sum {p_{i}{\varepsilon }_{i}}} equals a particular mean energy value (which can be proven using Lagrange multipliers).
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 higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states i and j is given as where: pi is the probability of state i, pj the probability of state j, εi is the energy of state i, εj is the energy of state j.The corresponding ratio of populations of energy levels must also take their degeneracies into account.
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 that, if we pick a random particle from that system and check what state it is in, we will find it is in state i. This probability is equal to the number of particles in state i divided by the total number of particles in the system, that is the fraction of particles that occupy state i.
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 the fraction of particles in state i as a function of the energy of that state is This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line of atoms or molecules undergoing transitions from one state to another. In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state. This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition.
Boltzmann distribution	The distribution	The softmax function commonly used in machine learning is related to the Boltzmann distribution: softmax ⁡[−ε1kT,…,−εMkT]
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 statistical mechanics to describe canonical ensemble, grand canonical ensemble and isothermal–isobaric ensemble. The generalized Boltzmann distribution is usually derived from principle of maximum entropy, but there are other derivations.The generalized Boltzmann distribution has the following properties: It is the only distribution for which the entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics.
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.
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. Some special cases (derivable from the canonical ensemble) show the Boltzmann distribution in different aspects: Canonical ensemble (general case) The canonical ensemble gives the probabilities of the various possible states of a closed system of fixed volume, in thermal equilibrium with a heat bath. The canonical ensemble has a state probability distribution with the Boltzmann form.
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 the collection. The canonical ensemble has the property of separability when applied to such a collection: as long as the non-interacting subsystems have fixed composition, then each subsystem's state is independent of the others and is also characterized by a canonical ensemble. As a result, the expected statistical frequency distribution of subsystem states has the Boltzmann form.
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 statistics give the expected number of particles found in a given single-particle state, in a classical gas of non-interacting particles at equilibrium. This expected number distribution has the Boltzmann form.Although these cases have strong similarities, it is helpful to distinguish them as they generalize in different ways when the crucial assumptions are changed: When a system is in thermodynamic equilibrium with respect to both energy exchange and particle exchange, the requirement of fixed composition is relaxed and a grand canonical ensemble is obtained rather than canonical ensemble. On the other hand, if both composition and energy are fixed, then a microcanonical ensemble applies instead.
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 collective states of the entire system considered as a whole, provided the entire system is in thermal equilibrium.
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 grand canonical ensemble the state-filling statistics of quantum gases are described by Fermi–Dirac statistics or Bose–Einstein statistics, depending on whether the particles are fermions or bosons, respectively.

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