The SIRD type models consider four types of populations, the susceptible S, the infected I and the recovered R and the dead D.

As in

• the infection is not fatal,
• the model does not include birth
• the model does not include death from another cause

The total number of the population will be equal to the sum of the four groups:

$ N = S + I + R + D $

Basically, the SIRD model is a simple generalization of the original SIR model. His interest lies in studying the imbalances of the propagation of the populations of recovered R and dead D.

(ID 8218)

In the SIRD model the only difference with respect to the SIR model is in the generation of two populations (recovered and dead) from the same infected population. Therefore, the dynamics of the evolution of the susceptible S is identical to that of the SIR model. Therefore, the equation is governed by

$\displaystyle\frac{ dS }{ dt }=- C \displaystyle\frac{ I }{ N } S \beta $

(ID 8219)

In the case of the recovered R you can model your cup as proportional to the universe of infected that exists at a time I. If the proportionality constant is also referred to as \gamma in this case, the population of recoveries will be described by

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

(ID 8220)

In analogy to the case of the recovered R, the death rate can be modeled as proportional to the universe of infected that exists at a time I. If the proportionality constant of denominated \delta has to be that the population of recovered will be described by

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

(ID 8221)

In the case of the SIR model, the dynamics of the infected are described by the equations

$\displaystyle\frac{dI}{dt}=i(t)-\displaystyle\frac{dR}{dt}$

where

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

In the case of the SIRD model to the recovered R, the dead D must be added so the equation becomes

\displaystyle\frac{dI}{dt}=C\beta\displaystyle\frac{I}{N}S-\displaystyle\frac{dR}{dt}-\displaystyle\frac{dD}{dt}

but with

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

Y

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

this equation can be written as

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

(ID 8222)

Infection rate

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

its sign when the factor in parentheses is zero. This occurs when the population of susceptible reaches a critical number such that

$\displaystyle\frac{ S_{crit} }{ N }=\displaystyle\frac{ \gamma + \delta }{ \beta C }$

In this circumstance the epidemic begins to be controlled. The number S_{crit} can be reached either by infection or by preventive vaccination.

(ID 8224)

The reproduction factor is defined as the inverse factor of the proportion of critical susceptible and the size of the social group N

$\displaystyle\frac{ S_{crit} }{ N }=\displaystyle\frac{ \gamma + \delta }{ \beta C }$

So you have to:

$ R_0 =\displaystyle\frac{ \beta C }{ \gamma + \delta }$

(ID 8223)

To contain the spread, the number of susceptible S must be reduced to the critical number

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

Therefore the fraction to be vaccinated is equal to

q = \displaystyle\frac{S-S_{crit}}{N}

that in the case of the entire population N susceptible equal to

q = 1-\displaystyle\frac{S_{crit}}{N}

or with the recovery factor

$ R_0 =\displaystyle\frac{ \beta C }{ \gamma + \delta }$

It can be written as:

$ q =1-\displaystyle\frac{1}{ R_0 }$

(ID 8225)