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Aplicaciones del Modelo SIRD
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ID:(891, 0)


SARS Case 2003
Description

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In 2003 a SARS pandemic occurred that started in Chine and spread via Hong Kong to the rest of the world.

The WHO data, which covers the whole world in particular, has a relatively simple structure for the case of Hong Kong (a single focus). The data that can be downloaded from the general report of [WHO SARS 2003] (http://www.who.int/csr/sars/country/en/) in which is the cumulative number of:

• infected

• dead

• recovered

By date and country.

The number of deaths and accumulated recoveries correspond to the D and R of the SIRD model respectively.

The accumulated number of infected J does not correspond to the I of the SIRD model since the latter represents the infected existing at a given time and not the historical accumulated.

To fully describe the model we must, based on the experimental data, determine the factors:

• $\bar{\beta}\equiv\beta C$ which is the infection rate

• $\gamma$ recovery rate

• $\delta$ death rate

• $N$ the number of the social group or cell in which it is propagated

if it is assumed that initially there was only one infected.

ID:(8226, 0)


Definition of contagion rate
Equation

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As in the infection spread equation in the SIRD model

$\displaystyle\frac{ dI }{ dt }=\left(\displaystyle\frac{ \beta C }{ N } S - \gamma - \delta \right) I $



The number of contacts C and the probability that the contact, if infected, infects /beta in the form of a product, it is impossible to determine both parameters separately. Therefore, the probability of total infection that considers both parameters is introduced:

$\bar{\beta}=\beta C$

ID:(8228, 0)


Number of Infected
Equation

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To calculate the number of infected I the number of accumulated infected J can be taken by subtracting the number of recovered R and dead D< /tex>:

$ I = J - R - D $

ID:(8227, 0)


Recovery rate determination
Equation

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As the data of both the infected I_i and the recovered R_i is available and it must be fulfilled that

$\displaystyle\frac{ dR }{ dt }= \gamma I $

\\n\\nyou can make an adjustment for least squares in which you look for a \gamma that minimizes\\n\\n

$min \sum_i\left(\displaystyle\frac{dR_i}{dt}-\gamma I_i\right)^2$



what happens if the recovery rate is

$\gamma=\displaystyle\frac{\displaystyle\sum_iI_i\displaystyle\frac{dR_i}{dt}}{\displaystyle\sum_iI_i^2}$

ID:(8229, 0)


Determination of the death rate
Equation

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As the data of both the infected I_i and the dead D_i is available and it must be fulfilled that

$\displaystyle\frac{ dD }{ dt }= \delta I $

\\n\\nyou can make an adjustment for least squares in which you look for a \delta that minimizes\\n\\n

$min \sum_i\left(\displaystyle\frac{dD_i}{dt}-\delta I_i\right)^2$



what happens if the death rate is

$\delta=\displaystyle\frac{\displaystyle\sum_iI_i\displaystyle\frac{dD_i}{dt}}{\displaystyle\sum_iI_i^2}$

ID:(8230, 0)


Infection propagation equation
Equation

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The infection spread equation

$\displaystyle\frac{ dI }{ dt }=\left(\displaystyle\frac{ \beta C }{ N } S - \gamma - \delta \right) I $



can be rewritten with

$\bar{\beta}=\beta C$



how

$\displaystyle\frac{dI}{dt}=\left(\bar{\beta}\displaystyle\frac{S}{N}-(\gamma+\delta)\right)I$

ID:(8231, 0)


Infected rate equation
Equation

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If the point at which the number of infected reaches a maximum I_{crit} is known, the derivative of the number of infected is nil

$\displaystyle\frac{dI}{dt}=\left(\bar{\beta}\displaystyle\frac{S}{N}-(\gamma+\delta)\right)I$

\\n\\nand with that\\n\\n

$\bar{\beta}\displaystyle\frac{S_{crit}}{N}-(\gamma+\delta)=0$



so with

$ N = S + I + R + D $



you have that the infection rate would be equal to

$\bar{\beta}=\displaystyle\frac{(\gamma+\delta)N}{N-(I_{crit}+R_{crit}+D_{crit})}$

Therefore \bar{\beta} will always be less than the sum of \gamma and \delta.

ID:(8234, 0)


Regression for the calculation of the affected population
Equation

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To search for the number of people in the circle N you can search for the infection spread equation

$\displaystyle\frac{dI}{dt}=\left(\bar{\beta}\displaystyle\frac{S}{N}-(\gamma+\delta)\right)I$



with the condition

$ N = S + I + R + D $



and the relationship for $\bar{\beta}$

$\bar{\beta}=\displaystyle\frac{(\gamma+\delta)N}{N-(I_{crit}+R_{crit}+D_{crit})}$



minimization of quadratic deviation

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$

ID:(8232, 0)


$N^2$ factor
Equation

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If the expression develops

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$



the coefficient is obtained

$S_1=\sum_i\left(\displaystyle\frac{dI_i}{dt}+(\gamma+\delta)I_i\right)^2$

for the term in $N^2$.

ID:(8236, 0)


$\bar{\beta}^2N^2$ factor
Equation

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If the expression develops

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$



the coefficient is obtained

$S_2=\sum_iI_i^2$

for the term in $\bar{\beta}^2N^2$.

ID:(8237, 0)


$\bar{\beta}^2$ factor
Equation

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If the expression develops

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$



the coefficient is obtained

$S_3=\sum_iI_i^2(I_i+R_i+D_i)^2$

for the term in $\bar{\beta}^2$.

ID:(8238, 0)


$\bar{\beta}N^2$ factor
Equation

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If the expression develops

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$



the coefficient is obtained

$S_4=-2\sum_i\left(\displaystyle\frac{dI_i}{dt}+(\gamma+\delta)I_i\right)I_i$

for the term in $\bar{\beta}N^2$.

ID:(8239, 0)


$\bar{\beta}^2N$ factor
Equation

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If the expression develops

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$



the coefficient is obtained

$S_5=-2\sum_iI_i^2(I_i+R_i+D_i)$

for the term in $\bar{\beta}^2N$.

ID:(8240, 0)


$\bar{\beta}N$ factor
Equation

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If the expression develops

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$



the coefficient is obtained

$S_6=2\sum_i\left(\displaystyle\frac{dI_i}{dt}+(\gamma+\delta)I_i\right)I_i(I_i+R_i+D_i)$

for the term in $\bar{\beta}N$.

ID:(8241, 0)


Regression equation
Equation

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The equation

$min \sum_i\left(N\displaystyle\frac{dI_i}{dt}-\bar{\beta}(N)(N-I_i-R_i-D_i)I_i+(\gamma+\delta)I_iN\right)^2$



can be rewritten with

$S_1=\sum_i\left(\displaystyle\frac{dI_i}{dt}+(\gamma+\delta)I_i\right)^2$



$S_2=\sum_iI_i^2$



$S_3=\sum_iI_i^2(I_i+R_i+D_i)^2$



$S_4=-2\sum_i\left(\displaystyle\frac{dI_i}{dt}+(\gamma+\delta)I_i\right)I_i$



$S_5=-2\sum_iI_i^2(I_i+R_i+D_i)$



$S_6=2\sum_i\left(\displaystyle\frac{dI_i}{dt}+(\gamma+\delta)I_i\right)I_i(I_i+R_i+D_i)$



giving

$min (S_1N^2+(S_6+S_4N)N\bar{\beta}+(S_3+S_5N+S_2N^2)\bar{\beta}^2)$

where \bar{\beta} depends on N.

ID:(8235, 0)


Number of people in cell
Equation

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The condition

$min (S_1N^2+(S_6+S_4N)N\bar{\beta}+(S_3+S_5N+S_2N^2)\bar{\beta}^2)$



It can be applied differentiating from N, considering that

$\bar{\beta}=\displaystyle\frac{(\gamma+\delta)N}{N-(I_{crit}+R_{crit}+D_{crit})}$



and matching zero with what you get

$N=\displaystyle\frac{(S_6+S_0S_4)S_0-(\gamma+\delta)(2S_3+S_0S_5)}{S_6+S_0S_4+(\gamma+\delta)(S_5+2S_0S_2)}$

ID:(8242, 0)


SARS simulator - adjustment of a SEIR Model
Php

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This simulator contains the SARS epidemic data for the case of Hong Kong and allows searching the parameters of a SEIR model by adjusting the curves to the actual values:

ID:(9659, 0)