If you want to generalize the first equation of the SIR model that describes the evolution of the susceptible S

equation=4068\\n\\nwhere \beta is the probability of infection, number of contacts C, number of infected I and number of total people N.\\n\\nIf those born are considered these will be proportional to the number of people in the society N, that is\\n\\n

$\left(\displaystyle\frac{dS}{dt}\right)_{nacer}=\mu_bN$

\\n\\nwhere \mu_b is the constant of proportionality. Similarly, the number of people who die from another cause will be\\n\\n

$\left(\displaystyle\frac{dS}{dt}\right)_{morir}=-\mu_dS$

where \mu_d is the proportionality constant and in this case it represents a fraction of the number of susceptible persons that dies. The negative sign reminds us that death leads to a reduction of susceptible.

In this way the first equation of the modified SIR model is

equation

In the case of the second equation of the SIR model, the equation must be modified

equation=4079\\n\\nin which I are infected, \beta the probability of contagion, C the number of contacts, N the population size and \gamma the factor that models recovery.\\n\\nIf it is assumed that people are only born healthy, the second equation of the model may only include infected people who die for another reason than the disease being studied. Therefore the variation of those infected by death should be\\n\\n

$\left(\displaystyle\frac{dI}{dt}\right)_{morir}=-\mu_dI$

where \mu_d is the proportionality constant and in this case it represents a fraction of the number of susceptible persons that dies. The negative sign reminds us that death leads to a reduction of susceptible.

Therefore the second equation is written as

equation

In the case of the third equation of the SIR model, the equation must be modified

equation=4072\\n\\nin which R is the population of the recovered, I the infected and \gamma the factor that models the recovery.\\n\\nIf it is assumed that people are only born healthy, the third equation of the model may only include recovered people who die for another reason than the disease being studied. Therefore the variation of those recovered by death must be\\n\\n

$\left(\displaystyle\frac{dR}{dt}\right)_{morir}=-\mu_dR$

where \mu_d is the proportionality constant and in this case it represents a fraction of the number of susceptible persons that dies. The negative sign reminds us that death leads to a reduction of susceptible.

Therefore the third equation is written as

equation

As in the case of the SIR model, there are a number of susceptible under which the disease does not find enough victims to grow. This occurs at the moment that the slope of the infected is null:\\n\\n

$\displaystyle\frac{dI}{dt}=\displaystyle\frac{\beta C}{N}I(t)S(t)-(\gamma+\mu_d)I(t)=0$

in which it can be cleared in S giving the value of susceptible that is critical to control the disease is:

equation

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.

If we look at the second equation of the modified SIR model that describes the evolution

equation=4079\\n\\nWe see that the sign of the factor in parentheses determines whether the number of infected continues to grow or decrease. The disease is considered in the process of being controlled if the factor is negative or\\n\\n

$\displaystyle\frac{\beta C}{N}S(t)-(\gamma+\mu_d)<0$

\\n\\nor\\n\\n

$\displaystyle\frac{S(t)}{N}\displaystyle\frac{\beta C}{(\gamma+\mu_d)}< 1.0$

At the beginning of the spread, the susceptible population is largely the entire population (S(0)\sim N), so the disease is contained to the extent that \beta C/(\gamma+\mu_d) is less than one. Therefore, the reproduction factor is defined as

equation

This is the condition that the reproduction factor must exist.

equation=4083

is greater than zero but also that R_0 has to be less than one which implies that more than those who die from other causes must be born:

equation

Another application of the equation is to allow estimating the necessary measures to avoid the epidemic. In general, any change from \beta, C, \gamma, \mu_d and S is required, leads to Z\leq 1. The reduction of S is associated with what vaccination is. If it is assumed that the general public does not modify their customs to reduce \beta and C, and we have no medicines to increase the \gamma factor We can pass people from the S state to the R via vaccination. If q is the fraction to be vaccinated, that is, qS will be vaccinated, we will achieve control if\\n\\n

$Z=1=\displaystyle\frac{S-qS}{N}\displaystyle\frac{\beta C}{(\gamma+\mu_d)}$

or clearing q

equation

If the asymptotic number of infected is observed\\n\\n

$\displaystyle\frac{I_{\infty}}{N}=\displaystyle\frac{\mu_b}{\gamma+\mu_d}-\displaystyle\frac{\mu_d}{\beta C}$

We note that, depending on the parameters, the value could become negative, which makes no sense. In case there is no situation that the asymptotic number is zero or positive there is no static asymptotic solution. The condition that the static solution exists is

equation

In the case of reaching the situation in which the infected begin to descend, the first equation of the modified SIR model is obtained.\\n\\n

$\displaystyle\frac{dS}{dt}=-\left(\displaystyle\frac{\beta C}{N}I(t)+\mu_d\right)S(t)+\mu_bN=0$

\\n\\nin which it can be cleared in I giving the value is\\n\\n

$I_{crit}=\left(\displaystyle\frac{\mu_b N}{S_{\infty}}-\mu_d\right)\displaystyle\frac{N}{\beta C}$

\\n\\nAs the limit of susceptible is\\n\\n

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

it has

equation

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

equation=4078

equation=4079

equation=4080

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