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.

The origin of such models is the early 20th century, with important works being that of Ross^[1] in 1916, Ross and Hudson in 1917,^[2][3] Kermack and McKendrick in 1927,^[4] and Kendall in 1956.^[5] The Reed–Frost model was also a significant and widely overlooked ancestor of modern epidemiological modelling approaches.^[6]

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.

These models are used to analyze the disease dynamics and to estimate the total number of infected people, the total number of recovered people, and to estimate epidemiological parameters such as the basic reproduction number or effective reproduction number. Such models can show how different public health interventions may affect the outcome of the epidemic.

The SIR model

The SIR model^{[7][8][9][10]} 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.

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^[11] for infectious diseases that are transmitted from human to human, and where recovery confers lasting resistance, such as measles, mumps, and rubella.

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.^[11] Note that in the SIR model, ${\displaystyle R(0)}$ and ${\displaystyle R_{0}}$ are different quantities – the former describes the number of recovered at t = 0 whereas the latter describes the ratio between the frequency of contacts to the frequency of recovery.

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.^{[citation needed]}

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 (see diagram).

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 ${\displaystyle d(S/N)/dt=-\beta SI/N^{2}}$, where ${\displaystyle N}$ is the total population, ${\displaystyle \beta }$ 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 ${\displaystyle SI/N^{2}}$ 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.^[12])

Between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is ${\displaystyle \gamma I}$. If an individual is infectious for an average time period ${\displaystyle D}$, then ${\displaystyle \gamma =1/D}$. 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).^[13]

For the special case in which there is no removal from the infectious compartment (${\displaystyle \gamma =0}$), the SIR model reduces to a very simple SI model, which has a logistic solution, in which every individual eventually becomes infected.

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:^[8][14]

${\displaystyle \left\{{\begin{aligned}&{\frac {dS}{dt}}=-\beta IS,\\&{\frac {dI}{dt}}=\beta IS-\gamma I,\\&{\frac {dR}{dt}}=\gamma I,\end{aligned}}\right.}$

where ${\displaystyle S}$Zdroj:https://en.wikipedia.org?pojem=SIR_model
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