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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)