SIR model

Graphs of the three groups S, I and R with the starting values and infection rate and rate for the group R . The rates are in the unit day. ${\textstyle S(0)=997,I(0)=3,R(0)=0}$ ${\textstyle \beta =0{,}4}$ ${\textstyle \gamma =0{,}04}$

If neither medication nor a vaccination are available, the only way to reduce the number of infections is through appropriate measures. This animation shows the effect of reducing the infection rate by 76% (from to ) (the other parameters are as in the graphic above and ). The information on is adapted to the article text, the values ​​in the figure correspond with N = 1000. ${\textstyle \beta =0{,}5}$ ${\textstyle \beta =0{,}12}$ ${\textstyle S(0)=997,I(0)=3,R(0)=0}$ ${\textstyle \gamma =0{,}04}$ ${\displaystyle \beta }$ ${\displaystyle {\frac {\beta }{N}}}$

In mathematical epidemiology , a branch of theoretical biology , the SIR model (susceptible-infected-removed model) is a classic approach to describing the spread of infectious diseases with immunity development, which is an extension of the SI model . It is named after the grouping of the population into susceptible (S), infected (I) and people removed from the infection process (R), as explained below. The expansion of the SIR model to include those exposed, i.e. people who are infected but not yet contagious, is described with the SEIR model . Usually, a deterministic model, formulated by common differential equations linked to one another, is considered, in which the variables are continuous and correspond to large populations, but other, in particular stochastic models are also referred to as SIR, which have the grouping in common with the deterministic SIR model.

The model comes from William Ogilvy Kermack and Anderson Gray McKendrick (1927) and is sometimes named after both (Kermack-McKendrick model). Despite the simplicity of the model, the authors were able to model the data of a plague epidemic in Bombay in 1905/06 well.

Differential equations

When SIR model, three groups are distinguished individuals: At the time indicated the number of the disease non-immune healthy ( s usceptible worth individuals), the number of infectious infected ( i nfectious worth individuals) and the number of "remote" from the disease process people ( r emoved individuals). The latter occurs either through recovery with acquired immunity to the disease or through death. In another reading it is the resistant people. Furthermore, let the total number of individuals be, that is: ${\displaystyle t}$ ${\displaystyle S(t)}$ ${\displaystyle I(t)}$ ${\displaystyle R(t)}$ ${\displaystyle N}$

${\displaystyle N=I+S+R}$

To simplify the notation, the time dependency is omitted in the following. It applies to any time , , , . ${\displaystyle N=N(t),R=R(t),S=S(t),I=I(t)}$ ${\displaystyle N(t)\geq 0}$ ${\displaystyle S(t)\geq 0}$ ${\displaystyle R(t)\geq 0}$ ${\displaystyle I(t)\geq 0}$

A number of assumptions are made in the SIR model:

• Each individual can be infected by a pathogen only once and then either becomes immune or dies.
• Infected people are immediately contagious, an assumption that is not made in the SEIR model .
• The respective rates are constant.
• Those who died as a result of the infection are included in R, like those who were immunized.

Then the equations of the SIR model are:

${\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=\nu \,N-\beta {\frac {S\,I}{N}}-\mu \,S}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=\beta {\frac {S\,I}{N}}-\gamma \,I-\mu \,I}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}=\gamma \,I-\mu \,R}$

With the rates :

${\displaystyle \gamma }$ Rate at which infected people recover or die in the unit of time (since the dead are also included in R)

${\displaystyle \mu }$ general death rate per person in a population (ie "per capita")

${\displaystyle \nu }$ Birth rate per person in a population (ie "per capita")

${\displaystyle \beta }$, the rate that indicates the number of new infections caused by a first infectious case per unit of time. is also referred to as the transmission rate or transmission coefficient. ${\displaystyle {\frac {\beta }{N}}}$

${\displaystyle \beta }$can be further broken down :, with the contact rate and the likelihood of infection transmission on contact. ${\displaystyle \beta =q\cdot \kappa }$ ${\displaystyle \kappa }$ ${\displaystyle q}$

The infection rate (English: "force of infection") , that is, the probability per unit of time that a susceptible person will be infected is ${\displaystyle \lambda }$

${\displaystyle \lambda =\beta {\frac {I}{N}}}$

where represents the proportion of infected people in the total population and thus the probability of contact with an infected person. People are infected per unit of time. ${\displaystyle {\frac {I}{N}}}$ ${\displaystyle \lambda S}$

If one neglects the birth and death rates (N is then constant) the equations result: ${\displaystyle \mu =\nu =0}$

${\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=-\beta {\frac {S\,I}{N}}}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=\beta {\frac {S\,I}{N}}-\gamma \,I}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}=\gamma \,I}$

The equations are similar to the Lotka-Volterra equations in predator-prey systems and coupled balance equations in many other areas ( replicator equations ).

In the literature, a variant of the equations is sometimes used in which the transmission coefficient is included and is often also referred to as although it has a different value : If, for example, and our coefficient is above , then the value must be inserted into the variant of the differential equation . If we use a different identifier for the sake of clarity , the first differential equation is written in the variant: ${\textstyle {\frac {\beta }{N}}}$ ${\textstyle \beta }$ ${\textstyle N=1000}$ ${\textstyle \beta =0{,}4}$ ${\textstyle 0{,}0004}$ ${\textstyle {\tilde {\beta }}}$

${\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=-{\tilde {\beta }}S\,I}$

Base reproduction number and course of an epidemic

The base reproduction number is

${\displaystyle R_{0}={\frac {\beta }{\gamma }}}$

The usual abbreviation for the basic reproduction number is used here (it should not be confused with the initial value of the number of resistant people at the time , which is sometimes also referred to as). ${\displaystyle R(0)}$ ${\displaystyle t=0}$ ${\displaystyle R_{0}}$

The base reproduction number indicates the number of further infections caused by an infected person (during the entire duration of their infectious period) in the beginning of the epidemic. In addition , there is also the factor that indicates the duration of the infectious period. In the beginning of an epidemic, birth and death rates can be roughly neglected, i.e. set; then one obtains the form of the SIR equations given at the end of the last section. For the beginning of an epidemic it must be and consequently (since applies at the beginning ) according to the SIR equations and thus (see also the following section about the discretized form of the equations) ${\displaystyle R_{0}}$ ${\displaystyle \beta }$ ${\displaystyle {\frac {1}{\gamma }}}$ ${\displaystyle \mu =\nu =0}$ ${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}>0}$ ${\displaystyle N\approx S}$ ${\displaystyle \beta >\gamma }$ ${\displaystyle R_{0}={\frac {\beta }{\gamma }}>1}$

As the process progresses, the number of infected people increases according to the SIR equations , if , so and with it ${\displaystyle I}$ ${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=I\,(\beta {\frac {S}{N}}-\gamma )>0}$ ${\displaystyle \beta {\frac {S}{N}}>\gamma }$

${\displaystyle R_{0}{\frac {S}{N}}>1}$

On the left is the product of the basic reproduction number and the proportion of the susceptible in the population. The latter is at the same time the probability of meeting an infectable person during contact. The inequality is synonymous with ${\displaystyle {\frac {S}{N}}}$

${\displaystyle S>{\frac {N}{R_{0}}}}$

The growth of the infected decreases (subsidence or end of the epidemic) if it falls below the value . With a value of the basic reproduction number this would be half of the population and with a third, so that in the former case half and in the latter case 2/3 of the population are infected or resistant, i.e. H. are no longer susceptible to infection; one then speaks of “ herd immunity ”. ${\displaystyle S}$ ${\displaystyle \rho ={\frac {N}{R_{0}}}}$ ${\displaystyle R_{0}=2}$ ${\displaystyle R_{0}=3}$

In influenza , for example, the base reproduction numbers are usually between 2 and 3.

Discretized form of differential equations

The discretized form of the differential equations with time step is: ${\displaystyle \Delta t}$

${\displaystyle S(t)-S(t-\Delta t)=-\beta \,\Delta t\,{\frac {S(t-\Delta t)}{N}}\,I(t-\Delta t):=-I_{\text{new}}(t)}$

${\displaystyle R(t)-R(t-\Delta t)=\gamma \,\Delta t\,I(t-\Delta t):=R_{\text{new}}(t)}$

${\displaystyle I(t)-I(t-\Delta t)=\left(\beta \,{\frac {S(t-\Delta t)}{N}}-\gamma \right)\,\Delta t\,I(t-\Delta t)=I_{\text{new}}(t)-R_{\text{new}}(t)}$

${\displaystyle I_{\text{new}}(t)}$corresponds to the number of newly infected persons in the observation period , i.e. what appears in the official statistics as the number of newly infected people , whereby in practice corrections are made for delayed reporting and other things. A day is often selected and set as the time unit and observation period for the report . ${\displaystyle \Delta t}$ ${\displaystyle \Delta t=1}$

For the derivation of the basic reproduction number, consider the discretized form (step ) of the differential equation for : ${\displaystyle \Delta t}$ ${\displaystyle I(t)}$

${\displaystyle I(t)-I(t-\Delta t)=\left(\beta \,{\frac {S(t-\Delta t)}{N}}-\gamma \right)\,\Delta t\,I(t-\Delta t)}$

With the duration of the infectious period used for and according to the definition of the base reproduction number or net reproduction number : ${\displaystyle D={\frac {1}{\gamma }}}$ ${\displaystyle \Delta t}$ ${\displaystyle R_{0}}$ ${\displaystyle R}$

${\displaystyle I(t)=R\,I(t-D)}$

where at the beginning of the epidemic, as is known, results ${\displaystyle R}$ ${\displaystyle R_{0}}$

${\displaystyle I(t)-I(t-D)=\left(R-1\right)\,I(t-D)=\left(\beta \,{\frac {S(t-D)}{N}}-\gamma \right)\,D\,I(t-D)}$

So is

${\displaystyle R={\frac {\beta }{\gamma }}\,{\frac {S}{N}}}$

and for , at the beginning of the epidemic , it results: ${\displaystyle R_{0}}$ ${\displaystyle S\approx N}$

${\displaystyle R_{0}={\frac {\beta }{\gamma }}}$

Mathematical treatment

With the help of the SIR model, we can determine for given initial values whether the course of the disease will lead to an epidemic . This question is equivalent to the question of whether the number of infected people is increasing at the time . Consider the derivation: ${\displaystyle I_{0},S_{0},\beta ,\gamma }$ ${\displaystyle 0}$

${\displaystyle \left.{\frac {\mathrm {d} I}{\mathrm {d} t}}\right|_{t=0}=I_{0}({\frac {\beta S_{0}}{N}}-\gamma )>0\ \ {\text{für }}S_{0}>\rho :={\frac {\gamma }{\beta }}N}$.

Here we call the threshold of an epidemic because of for all time the inequality for all this and for abating the epidemic: ${\displaystyle \rho }$ ${\displaystyle S(t)\leq S_{0}}$ ${\displaystyle {\dot {S}}\leq 0}$ ${\displaystyle t\geq 0}$ ${\displaystyle S_{0}<\rho }$

${\displaystyle {\dot {I}}(t)=I({\frac {\beta S}{N}}-\gamma )\leq 0}$.

for everyone . ${\displaystyle t\geq 0}$

In the SIR model, an epidemic occurs exactly when is. This is an essential statement of the model, also known as the threshold theorem. In order to start an epidemic, there must be a minimum density of infectables. Conversely, if the number of those who can be infected is pushed below this threshold during the course of the epidemic, the epidemic will expire. ${\displaystyle S_{0}>\rho }$

Maximum number of infected people

From the above differential equations for and it follows: ${\displaystyle I}$ ${\displaystyle S}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} S}}=-{\frac {(\beta S-\gamma N)I}{\beta SI}}=-1+{\frac {\rho }{S}}}$.

Integration by separating the variables provides:

${\displaystyle I(t)+S(t)-\rho \ln S(t)=I_{0}+S_{0}-\rho \ln S_{0}={\text{const.}}}$

with the logarithm function . The function is a first integral of the system and is constant on the trajectories of the system in the phase space given by I and S. The maximum number of infected people is obviously for and with . With the above equation assuming : ${\displaystyle \ln }$ ${\displaystyle I+S-\rho \ln S}$ ${\displaystyle S_{0}\geq \rho }$ ${\displaystyle S(t)=\rho }$ ${\displaystyle R(0)=0}$

${\displaystyle I_{\text{max}}=-\rho +\rho \ln(\rho )+I_{0}+S_{0}-\rho \ln S_{0}=N-\rho +\rho \ln \left({\frac {\rho }{S_{0}}}\right)}$

If you place and and you get: ${\displaystyle S_{o}=N}$ ${\displaystyle I_{0}=0}$ ${\displaystyle \rho ={\frac {N}{R_{0}}}}$

${\displaystyle I_{\text{max}}={\frac {N}{R_{0}}}\left(R_{0}-1-\ln R_{0}\right)}$

The equation for ("final size equation") also results from the first integrals : ${\displaystyle S(\infty )}$

${\displaystyle \ln {\frac {S(\infty )}{N}}=R_{0}\cdot ({\frac {S(\infty )}{N}}-1)}$

from the values ​​for (with ) and (with ). The equation can be used to determine . In particular, it results for the solution , that is, there is no breakout. ${\displaystyle t=0}$ ${\displaystyle S_{0}=N,I_{0})=0}$ ${\displaystyle t=\infty }$ ${\displaystyle I(\infty )=0}$ ${\displaystyle S(\infty )}$ ${\displaystyle R_{0}<1}$ ${\displaystyle {\frac {S(\infty )}{N}}=1}$

Number of "survivors"

The question also arises as to whether the epidemic will be “survived” at all, that is, whether there will be susceptible substances left in the end. For this we calculate , i.e. with time towards infinity ( ). Analogously results from the above differential equations ${\displaystyle S(\infty )}$ ${\displaystyle S(t)}$ ${\displaystyle t}$ ${\displaystyle \infty }$

${\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} R}}=-{\frac {S}{\rho }}}$, whose solution is, with the exponential function . ${\displaystyle S=S_{0}\exp \left(-{\frac {R}{\rho }}\right)\geq S_{0}\exp \left(-{\frac {N}{\rho }}\right)>0}$ ${\displaystyle \exp }$

This obviously means that the entire population will not be infected. From it also follows . It turns out that at the end of an epidemic there is less of a lack of susceptibles than of infected people! ${\displaystyle 0<S(\infty )<N}$ ${\displaystyle \lim _{t\to \infty }{\frac {dS}{dt}}=-{\frac {\beta }{N}}I(\infty )S(\infty )=0}$ ${\displaystyle I(\infty )=0}$

Approximations: Reduce the number of parameters

The square of the hyperbolic secant (red) occurring in the solutions of the SIR model has an even steeper slope than the hyperbolic secant itself

If we know the initial values , we can quickly determine the dynamics of a disease using the differential equations above. Often, however, it is precisely these constants that are difficult to determine, which is why we want to approximate the above equations in the following. ${\displaystyle I_{0},S_{0},\beta ,\gamma }$

It follows immediately from the differential equations discussed

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} R}{\mathrm {d} t}}=\gamma (N-R-S)=\gamma \left(N-R-S_{0}\exp \left(-{\frac {R}{\rho }}\right)\right).\end{aligned}}}$

The equation simplifies to a riccatischen differential equation if through the first 3 summands of the Taylor series to is approximated: ${\displaystyle \exp \left(-{\frac {R}{\rho }}\right)}$ ${\displaystyle {\frac {R}{\rho }}=0}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}=\gamma \left(N-S_{0}\right)+\gamma \left({\frac {S_{0}}{\rho }}-1\right)R-\gamma {\frac {S_{0}}{2\rho ^{2}}}R^{2}=-\gamma {\frac {S_{0}}{2\rho ^{2}}}R^{2}+\gamma \left({\frac {S_{0}}{\rho }}-1\right)R+\gamma \left(N-S_{0}\right)}$

so

${\displaystyle R(t)={\frac {\rho ^{2}}{S_{0}}}\left(\left({\frac {S_{0}}{\rho }}-1\right)+\alpha \operatorname {tanh} \left({\frac {\alpha \gamma t}{2}}-\phi \right)\right)}$

where were introduced:

${\displaystyle \alpha ={\sqrt {\left({\frac {S_{0}}{\rho }}-1\right)^{2}+2{\frac {S_{0}(N-S_{0})}{\rho ^{2}}}}}}$

${\displaystyle \phi =\tanh ^{-1}\left({\frac {1}{\alpha }}\left({\frac {S_{0}}{\rho }}-1\right)\right)}$

The function is the hyperbolic secant and the hyperbolic tangent , its inverse function. ${\displaystyle \operatorname {sech} }$ ${\displaystyle \operatorname {tanh} }$ ${\displaystyle \tanh ^{-1}}$

This means that the differential equation for can be expressed with only three parameters: ${\displaystyle R}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}={\frac {\gamma \alpha ^{2}\rho ^{2}}{2S_{0}}}\operatorname {sech} ^{2}\left({\frac {\alpha \gamma t}{2}}-\phi \right)=B\cdot \operatorname {sech} ^{2}\left({\frac {A\cdot t}{2}}-\phi \right)}$

So these three parameters are (with the initially exponential growth corresponding to the doubling time), the phase and . Depending on the data, the differential equation or the implicit equation for can be used. ${\displaystyle A=\alpha \gamma }$ ${\displaystyle A={\frac {\ln 2}{t_{d}}}}$ ${\displaystyle t_{d}}$ ${\displaystyle \phi }$ ${\displaystyle B={\frac {\gamma \alpha ^{2}\rho ^{2}}{2S_{0}}}=\gamma \cdot I_{\text{max}}}$ ${\displaystyle R(t)}$

If you place and you get and thus: ${\displaystyle I_{0}=0}$ ${\displaystyle S_{0}>\rho }$ ${\displaystyle \alpha ={\frac {S_{0}}{\rho }}-1}$

${\displaystyle R(t)={\frac {\rho ^{2}}{S_{0}}}\left({\frac {S_{0}}{\rho }}-1\right)\left(1+\operatorname {tanh} \left({\frac {\gamma }{2}}\left({\frac {S_{0}}{\rho }}-1\right)t-\phi \right)\right)}$

With you get an approximate value for the "extent of the epidemic" : ${\displaystyle \lim _{t\to \infty }\operatorname {tanh} \left(at+b\right)=1}$ ${\displaystyle R(\infty )=S_{0}-S(\infty )}$

${\displaystyle R(\infty )=S_{0}-S(\infty )=2\rho \left(1-{\frac {\rho }{S_{0}}}\right)}$

This gives the second part of the threshold theorem. Be with me at the beginning , then the "extent of the epidemic" is: ${\displaystyle S_{0}=\rho +v}$ ${\displaystyle v>0}$

${\displaystyle R(\infty )=2\rho {\frac {v}{\rho +v}}\approx 2v}$

and . In the end, the number of susceptible people is reduced by compared to the level before the epidemic. ${\displaystyle S(\infty )=\rho -v}$ ${\displaystyle 2v}$

According to the last differential equation in the SIR model, the number of infected people is:

${\displaystyle I(t)={\frac {1}{\gamma }}{\frac {\mathrm {d} R}{\mathrm {d} t}}={\frac {\alpha ^{2}\rho ^{2}}{2S_{0}}}\operatorname {sech} ^{2}\left({\frac {\alpha \gamma t}{2}}-\phi \right)}$

The curve of has the shape of a bell curve with an initially exponential increase. For example, Kermack and McKendrick found for the plague epidemic in Bombay 1905/06 (with almost always fatal outcome, so that a week was taken as the unit of time for the rates) with: ${\displaystyle I(t)}$ ${\displaystyle \gamma \approx 1}$

${\displaystyle I(t)\approx {\frac {\mathrm {d} R}{\mathrm {d} t}}=890\cdot \operatorname {sech} ^{2}\left(0{,}2\cdot t-3{,}4\right)}$

The good agreement made this a frequently cited example in mathematical epidemiology, but it has also been criticized.

David George Kendall found exact solutions for and the SIR model in 1956 , but the differential equations are mostly solved numerically. ${\displaystyle R(t)}$

Extension of the model

This SIRD model shows the time course of the four groups S, I, R and D for the starting values and the infection rate (identical to the values ​​in the graphic for the SIR model above), a rate for group R and the mortality rate ${\textstyle S(0)=997,I(0)=3,R(0)=0}$ ${\displaystyle \beta =0{,}4}$ ${\displaystyle \gamma =0{,}035}$ ${\displaystyle \mu =0{,}005}$

If you want to consider the dead separately (instead of adding them to group R ), you can expand it to the SIRD model (Susceptible-Infected-Recovered-Deceased-Model) . Here, the group includes R only the individuals who have survived the disease and have become immune, and the dead form their own group D .

The following system of differential equations has to be solved:

${\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=-\beta \cdot {\frac {S\cdot I}{N}}}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=\beta \cdot {\frac {S\cdot I}{N}}-\gamma \cdot I-\mu \cdot I}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}=\gamma \cdot I}$

${\displaystyle {\frac {\mathrm {d} D}{\mathrm {d} t}}=\mu \cdot I}$

At the time indicates the number of people who have recovered and the number of people who have died from the disease. Also means the rate at which infected people recover and the mortality rate at which infected people die. , and transmission rate have the same meaning as in the SIR model . ${\textstyle t}$ ${\textstyle R(t)}$ ${\textstyle D(t)}$ ${\textstyle \gamma }$ ${\textstyle \mu }$ ${\textstyle S(t)}$ ${\textstyle I(t)}$ ${\textstyle {\frac {\beta }{N}}}$

Another modification takes into account vaccination of newborns with a proportion : ${\displaystyle p}$

${\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=\nu (1-p)\,N-\beta {\frac {S\,I}{N}}-\mu \,S}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=\beta {\frac {S\,I}{N}}-\gamma \,I-\mu \,I}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}=\gamma \,I+p\nu N-\mu \,R}$

There are also variants in which two (or more) population groups are considered, for example the interaction of a core group, which is particularly active in promoting an infection, with the rest of the population.

Stochastic SIR models are used to study smaller populations that cannot be treated well with deterministic models. Only integer values ​​of the population proportions and statistical distributions for the transition rates such as . The course of epidemics is not determined deterministically here, i.e. H. an epidemic can also be applied to stop, if by chance in the infection period (infectious period, d. e. the period in which a Infizierter can transmit the infection) there are no contacts. Usually simulations (usually with the Monte Carlo method ) are carried out several times with the same parameters and the results are then statistically evaluated. ${\displaystyle \beta }$ ${\displaystyle R_{0}>1}$

A variant that takes quarantine measures and isolation measures such as social distancing into account was used to explain subexponential growth, i.e. growth of the infected in accordance with a power law in time, for Covid-19 in China from the end of January 2020 (SIR-X model). The differential equations are (with adaptation to the form of the SIR equations used here):

${\displaystyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=-\beta {\frac {S\,I}{N}}-\kappa _{0}S}$

${\displaystyle {\frac {\mathrm {d} I}{\mathrm {d} t}}=\beta {\frac {S\,I}{N}}-\gamma \,I-\kappa _{0}I-\kappa I}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}=\gamma \,I+\kappa _{0}S}$

${\displaystyle {\frac {\mathrm {d} X}{\mathrm {d} t}}=(\kappa +\kappa _{0})I}$

A newly introduced group of symptomatic infected people is in quarantine. The general measures to reduce contact are also described (social distancing, etc.) and affect infected and non-infected people equally, the special quarantine measures for infected people with the coefficient . The respective measures are no longer applicable . A new effective one results with an effective infection period . This new effective is smaller than . Another method to simulate the influence of isolating measures is to use approaches that vary over time. ${\displaystyle X}$ ${\displaystyle \kappa _{0}}$ ${\displaystyle \kappa }$ ${\displaystyle \kappa =\kappa _{0}=0}$ ${\displaystyle R_{0,e}={\frac {\beta }{{\gamma }_{e}}}=\beta \cdot T_{i,e}}$ ${\displaystyle T_{i,e}={\frac {1}{{\gamma }_{e}}}={\frac {1}{\gamma +\kappa +\kappa _{0}}}}$ ${\displaystyle R_{0,e}}$ ${\displaystyle R_{0}}$ ${\displaystyle \beta }$

See also

• Mathematical Modeling of Epidemiology (introductory article)
• SI model (contagion without recovery)
• SIS model (spread of infectious diseases without building up immunity)
• SEIR model (spread of contagious diseases with immunity development, in which infected people are not immediately infectious)
• Dynamic system (mathematical generic term)

literature

• NF Britton: Essential Mathematical Biology . 1st edition. Springer, Berlin 2003, ISBN 1-85233-536-X .
• Michael Li: An introduction to mathematical modeling of infectious diseases, Springer, 2018

Web links

• Kermack-McKendrick Model, mathworld
• Numberphile: The Coronavirus Curve on YouTube , March 25, 2020, accessed May 16, 2020.

Individual evidence

1. ↑ Kermack, McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A, Vol. 115, 1927, pp. 700-721
2. ^ Sebastian Möhler: Spread of infectious diseases . ( tu-freiberg.de [PDF; accessed on March 12, 2020]).
3. ↑ Regina Dolgoarshinnykh: Introduction to Epidemic Modeling . ( columbia.edu [PDF; accessed March 12, 2020]).
4. ↑ ^{a b c d e} Eichner, Kretzschmar, Mathematische Modelle in der Infektionsepidemiologie, in A. Krämer, R. Reintjes (Eds.), Infektionsepidemiologie, Springer 2003, pp. 81–94
5. ↑ Michael Li, An introduction to mathematical modeling of infectious diseases, Springer, 2018, Section 2.1, Kermick-McKendrick model.
6. ↑ ^{a b} Viola Priesemann et al., Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions , Science, May 15, 2020
7. ^ Norman Bailey, The mathematical theory of infectious diseases, Griffin and Company 1975, p. 11, threshold theorem
8. ↑ Odo Diekmann, Hans Heesterbeek, Tom Britton, Mathematical tools for understanding infectious disease dynamics, Princeton UP 2013, p. 15
9. ↑ The following derivation with the associated formulas can be found e.g. B. in Michael Li, An introduction to mathematical modeling of infectious diseases, Springer, 2018, p. 45
10. ↑ Kermack, McKendrick, A contribution to the mathematical theory of epidemics, Proc. Royal Soc. A, Volume 115, 1927, p. 714
11. ↑ Nicholas Bacaer: The model of Kermack and McKendrick for the plague epidemic in Bombay and the type reproduction number with seasonality, Journal of Mathematical Biology, Volume 64, 2012, pp. 403-422. According to this, the constant parameters obtained are unrealistic and the plague epidemic occurred seasonally in Bombay from 1897 to at least 1911, coupled to the rat population, so that a more complex model is necessary
12. ^ DG Kendall, Deterministic and stochastic epidemics in closed populations, Proc. Third Berkeley Symposium Math. Stat. & Prob., Vol. 4, 1956, University of California Press, pp. 149-165, Project Euclid
13. ↑ Amenaghawon Osemwinyen, Aboubakary Diakhaby: Mathematical Modeling of the Transmission Dynamics of Ebola Virus . July 2015 ( researchgate.net [accessed March 12, 2020]).
14. ↑ Benjamin Maier, Dirk Brockmann, Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China, Science, April 8, 2020, online
15. ↑ For example Q. Lin et al., A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action, Int. J. Infectious Diseases, Volume 93, 2020, pp. 211-216

Remarks

1. ↑ This form (which does not occur) is used by Kermack and McKendrick for their SIR model. On p. 713 you use the functions for and for infection rate and the rate for group R and write ${\textstyle N}$ ${\textstyle x(t),y(t),z(t)}$ ${\textstyle S(t),I(t),R(t)}$ ${\textstyle \kappa ,l}$

   ${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=-\kappa xy}$

   ${\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}=\kappa xy-ly}$

   ${\displaystyle {\frac {\mathrm {d} z}{\mathrm {d} t}}=ly}$

2. ↑ For the sake of comparability, the differential equations given in the source have been brought into the same form as in the SIR model above. There is no uniform use of the parameters in the literature; next to the above DGL is z. B. also (infection rate named here for the sake of clarity ) common. The identity then applies . In the figures used above , and the descriptions on Wikimedia Commons refer to the latter variant of the DGL, so that the infection rates mentioned there are only the values ​​according to the nomenclature in the article (the captions have been adapted for the article ). ${\textstyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=-\beta {\frac {S(t)\cdot I(t)}{N}}}$ ${\textstyle {\frac {\mathrm {d} S}{\mathrm {d} t}}=-{\tilde {\beta }}\cdot S(t)\cdot I(t)}$ ${\textstyle {\tilde {\beta }}}$ ${\textstyle {\tilde {\beta }}={\frac {\beta }{N}}}$ ${\textstyle N=1000}$ ${\textstyle {\frac {1}{1000}}}$