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Modelos SEIR

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Modified SEIR models

Description

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Equation of susceptible in the SEIR model

Equation

In the case of those susceptible, the process of generating latent people in the SEIR model is equivalent to that of creating infected in the SIR model. Therefore in this case the equation that describes the susceptible is the same in both models:

$\displaystyle\frac{dS}{dt}=-\displaystyle\frac{\beta C}{N}I(t)S(t)+\mu_bN-\mu_dS(t)$

ID:(4086, 0)


Latent equation model SEIR

Equation

In the case of the equation of latent cases, one must first consider those that have been infected and that the SIR model led to those infected\\n\\n

$-\displaystyle\frac{\beta C}{N} I(t) S(t $

\\n\\nwhere \beta is the probability of infected, C the number of contacts, S the number of those susceptible, I the number of those infected and N the number of the population.\\n\\nThe number of latents will decrease depending on the fraction \sigma that will show the symptoms and be part of those infected with symptoms I\\n\\n

$-\sigma E(t)$

\\n\\nSimilarly those who die from another cause should be considered\\n\\n

$-\mu_d E(t)$



so the equation to describe the latent will be

$\displaystyle\frac{dE}{dt}=\displaystyle\frac{\beta C}{N}I(t)S(t)-(\sigma+\mu_d)E(t)$

ID:(4087, 0)


Equation of Infected in the SEIR Model

Equation

In the case of the equation of infected cases, first consider those that are latent E and that in the proportion \sigma become infected

\sigma E(t)

The number of infected will decrease depending on the \gamma fraction of the infected I that is recovered

-\gamma I(t)

Similarly those who die from another cause should be considered

-\mu_d I(t)

so the equation to describe the infected will be

$\displaystyle\frac{dI}{dt}=\sigma E(t)-(\gamma+\mu_d)I(t)$

ID:(4088, 0)


Equation of recovered in the SEIR model

Equation

In the case of the equation of recovered cases, it is necessary to first consider those that are infected I and that in the proportion \gamma become recovered\\n\\n

$\gamma I(t)$

\\n\\nSimilarly those who die from another cause should be considered\\n\\n

$-\mu_d R(t)$



so the equation to describe the infected will be

$\displaystyle\frac{dR}{dt}=\gamma I(t)-\mu_dR(t)$

ID:(4089, 0)


Reproduction factor in SEIR model

Equation

If the probability of becoming infected is \beta, the number of contacts C, the reproduction factor \gamma, the step factor from latent to infected and \mu_d the death factor from other causes, the reproduction factor is

$R_0=\displaystyle\frac{\beta C\sigma}{(\sigma+\mu_d)(\gamma+\mu_d)}$

ID:(4093, 0)


Number of critical susceptible, SEIR model

Equation

In the critical case where the system becomes stable the number of latex E will not vary

$\displaystyle\frac{dE}{dt}=\displaystyle\frac{\beta C}{N}I(t)S(t)-(\sigma+\mu_d)E(t)$



and the infected I

$\displaystyle\frac{dI}{dt}=\sigma E(t)-(\gamma+\mu_d)I(t)$



where S is the number of susceptible, \beta is the probability of infecting, C number of contacts, N population number, \gamma is the recovery factor, \mu_b the birth factor and \mu_d the death factor.

The number of critical latents can be cleared from the second equation

$\displaystyle\frac{E_{crit}}{N}=\displaystyle\frac{(\gamma+\mu_d)}{\sigma}\displaystyle\frac{I_{crit}}{N}$



If this value is replaced in the first equation, the critical value for the susceptible ones is obtained

$\displaystyle\frac{S_{crit}}{N}=\displaystyle\frac{(\sigma+\mu_d)(\gamma+\mu_d)}{\beta C\sigma}$

which corresponds to the situation in which the infected curve reaches its maximum. In other words, the number of critically susceptible is the number of susceptible that remain at the moment that the number of infected reaches its maximum.

ID:(4090, 0)


Number of critical latents, SEIR model

Equation

From the equation of the infected I

$\displaystyle\frac{dI}{dt}=\sigma E(t)-(\gamma+\mu_d)I(t)$



where \gamma is the recovery factor, \sigma the latent to infected step factor, E the latent ones and \mu_d the factor of the dead.

As the number of susceptible in the critical case is

$\displaystyle\frac{I_{crit}}{N}=\displaystyle\frac{\mu_bN-\mu_dS_{crit}}{{\beta C S_{crit}}}$



you can calculate the critical number of those infected

$\displaystyle\frac{E_{crit}}{N}=\displaystyle\frac{(\gamma+\mu_d)}{\sigma}\displaystyle\frac{I_{crit}}{N}$

which corresponds to the situation in which the infected curve reaches its maximum. In other words, the number of critical latents is the number of latents that remain at the moment that the number of infected reaches its maximum.

ID:(4092, 0)


Number of critically infected, SEIR model

Equation

From the equation of the susceptible S

$\displaystyle\frac{dS}{dt}=-\displaystyle\frac{\beta C}{N}I(t)S(t)+\mu_bN-\mu_dS(t)$



where \beta is the probability of infection, C the number of contacts, N the number of the population, I those infected, \mu_b the factor of births and \mu_d the factor of the dead.

As the number of susceptible in the asymptotic case is

$\displaystyle\frac{S_{crit}}{N}=\displaystyle\frac{(\sigma+\mu_d)(\gamma+\mu_d)}{\beta C\sigma}$



the asymptotic number of those infected can be calculated

$\displaystyle\frac{I_{crit}}{N}=\displaystyle\frac{\mu_bN-\mu_dS_{crit}}{{\beta C S_{crit}}}$

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SEIR model curve

Image

In the case of SEIR models there are four curves, that of susceptible, latent, infected and recovered:

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Curvas del modelo SEIR

Html

The model can solve numerically the equations for susceptible S, infected I, latent E and recovered R:

$\displaystyle\frac{dS}{dt}=-\displaystyle\frac{\beta C}{N}I(t)S(t)+\mu_bN-\mu_dS(t)$



$\displaystyle\frac{dE}{dt}=\displaystyle\frac{\beta C}{N}I(t)S(t)-(\sigma+\mu_d)E(t)$



$\displaystyle\frac{dI}{dt}=\sigma E(t)-(\gamma+\mu_d)I(t)$



$\displaystyle\frac{dR}{dt}=\gamma I(t)-\mu_dR(t)$

where t is the time \beta the cup of contagion, \sigma the cup of emergence of the symptoms in the infected, \gamma the recovery rate, C the number of contacts, N the population, \mu_b per capita birth rate and \mu_d per capita mortality.

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