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Modelos SIR
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SIR Models
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The SIR type models consider three types of populations, the susceptible S, the infected I and the recovered R.

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 three groups:

$N=S+I+R$

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Mechanism of contagion
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Probability that a person I find is infected is\\n\\n

$\displaystyle\frac{I}{N}$

\\n\\nthat is favorable cases I match for possible cases N. Normally the person has contact with C people so they give\\n\\n

$C\displaystyle\frac{I}{N}$

\\n\\nOpportunities to infect a healthy. The probability of contagion will depend on the time you have contact. Suppose we consider a time dt, in that case if the probability of contagion time is \beta the probability in time dt that a infected individual infects a healthy one is\\n\\n

$\beta dt$

\\n\\nWith this in time dt the probability that a healthy person becomes infected is for dt time equal to\\n\\n

$C\displaystyle\frac{I}{N}\beta$

\\n\\nTo obtain the total number of infected in time dt we have to consider that the newly calculated probability affects all those susceptible to infection. Therefore, the number of healthy S decreases in a time dt in an amount equal to\\n\\n

$C\displaystyle\frac{I}{N}S\beta$



that is

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

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Contagion mechanism - equation Susceptible SIR
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As the number of susceptible S is reduced as a function of those infected by time i\\n\\n

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



you have to with

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



the equation to calculate the development of the susceptible population is

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

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Number of those who remain infected
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If for the t' day i(t') people are infected, the total infected in one day t will be the sum of all infected from the beginning the appearance of the disease in a time 0, that is\\n\\n

$\displaystyle\int_0^tdt'i(t')$



If we want to estimate those that still remain infected, we must subtract all that have been recovered to date, those that are R(t). Therefore, the number that still remains infected in time t is

$I(t)=\displaystyle\int_0^tdtau i(tau)-R(t)$

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Variation in the number of infected
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To know the variation in time of the number of infected can be derived in time the equation

$I(t)=\displaystyle\int_0^tdtau i(tau)-R(t)$



with what you get the equation

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

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Mechanism of recovery/death
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The question now is how infected people (I) recover. In general, recovery/death occurs at an average time of infection. Modeling requires a probabilistic function of the person recovering / dying between t and t+dt after being infected. If the number that was between t' and t'+dt is i(t'), those that are recovered in the time t is equal to the sum of all the times in which the person may have been infected

$r(t)=\displaystyle\int_0^t dtau p(t-tau)i(tau)$

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Total recovered, SIR model
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If the number that is retrieved between the t' and t'+dt times is

$r(t)=\displaystyle\int_0^t dtau p(t-tau)i(tau)$



then the total recovered at the time t will be

$R(t)=\displaystyle\int_0^t dtau r(tau)$

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Recovered Differential Equation
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As the number of recovered is

$R(t)=\displaystyle\int_0^t dtau r(tau)$



and the daily number is

$r(t)=\displaystyle\int_0^t dtau p(t-tau)i(tau)$



the derivative of the number of recovered is equal to

$\displaystyle\frac{dR}{dt}=\int_0^t dtau p(t-tau)i(tau)$

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Simplified recovery model - equation recovered SIR
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The amount of daily recoveries was calculated by adding on all the historical i(t') weighted by the probability p that the recovery process will be completed on the day considered:

$r(t)=\displaystyle\int_0^t dtau p(t-tau)i(tau)$



which corresponded to the daily variation of the total recovered

$\displaystyle\frac{dR}{dt}=\int_0^t dtau p(t-tau)i(tau)$



As the number of infected daily is

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

\\n\\nit is necessary that the variation of recovered is of the form:\\n\\n

$\displaystyle\frac{dR}{dt}=\displaystyle\frac{\beta C}{N}\displaystyle\int_0^t dt' p(t-t')S(t')I(t')$

\\n\\nIf the infected are few compared to the susceptible and the probability function decays faster than the number of infected varies the integral can be replaced by a constant\\n\\n

$\displaystyle\frac{\beta C}{N}\displaystyle\int_0^t dt' p(t-t')S(t')I(t')\sim\left[\displaystyle\frac{\beta C}{N}\displaystyle\int_0^t dt' p(t-t')S(t')\right]I(t)\sim\gamma I(t)$



with what the recovery equation looks like

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

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Rewriting the second equation SIR
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If the third equation of the SIR model is replaced

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



in the second

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



and using

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



an equation is obtained in which one can understand the situations in which the disease spreads or is managed to control

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

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Approach of Survivors without contagion, SIR model
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As we saw, there is a limit in which the disease itself controls its expansion simply by the lack of susceptible to being infected. The question is what size is the group that is saved from infection. If we divide the equation

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



by

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

\\n\\nan independent equation of time (or variable to temporal u) of the form can be obtained\\n\\n

$\displaystyle\frac{dI}{dS}=-1+\displaystyle\frac{\gamma}{\beta C}\displaystyle\frac{N}{S}$

\\n\\nIf this equation is integrated assuming initial conditions S_0 and I_0 it is obtained that\\n\\n

$I - I_0 = (S_0 - S) + \displaystyle\frac{\gamma N}{\beta C}\ln\displaystyle\frac{S}{S_0}$

\\n\\nTo determine the number of survivors we are interested in the case in which the situation has already stabilized and we no longer have infected ( I = 0 ). It can also be assumed that the initial infected are a few, that is, I_0\ll S_0\\n\\nWith this the equation for the calculation of survivors is reduced\\n\\n

$(S_0-S)+\displaystyle\frac{\gamma N}{\beta C}\ln\displaystyle\frac{S}{S_0}=0$



This equation cannot be solved in an exact way, however if the logarithm is expanded by Taylor until the second order is achieved, the size of the population is not infected.

$S_{\infty}=S_0\left(3-2\displaystyle\displaystyle\frac{\beta C}{\gamma}\displaystyle\displaystyle\frac{S_0}{N}\right)$

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Stationary condition
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Initially it can be considered that S is similar to N so that the system starts to spread out of control if in

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

\\n\\nthe factor\\n\\n

$\displaystyle\frac{C\beta}{\gamma}\displaystyle\frac{S}{N}-1 > 0$

\\n\\nis positive. This occurs if the factor\\n\\n

$\displaystyle\frac{C\beta}{\gamma}\displaystyle\frac{S}{N}$

\\n\\nIt is greater than one. As initially the susceptible population S is similar to the entire population considered N, the system has to be spread out of control if:\\n \\n

$\displaystyle\frac{C\beta}{\gamma} > 0$



On the other hand, over time, the disease itself will reduce the number of susceptible people causing it to self-control due to the reduction of new victims. These situations will occur if the number of susceptible reaches

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

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.

The number of critically susceptible subjects shows that preventive vaccination helps control the disease by creating a situation in which the initial susceptible are such that the disease is self-controlled.

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Reproduction Factor SIR
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Being critical is the factor that compares the factors of contagion \beta C with the recovery factor \gamma which is called the reproduction factor and designates by letter

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

As a reference, reproduction factors of the most typical diseases are listed:

DiseaseTransmission$R_0$
Measlesdroplets in suspension12-18
Whooping coughdroplets in suspension12-17
Diphtheriasaliva6-7
Smallpoxdroplets in suspension5-7
Poliofecal-oral route5-7
Rubelladroplets in suspension5-7
Mumpsdroplets in suspension4-7
HIV / AIDSsexual contact2-5
SARSdroplets in suspension2-5
Influenza (1918 pandemic strain)droplets in suspension2-3
Ebolabody fluids1.5-2.5

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Limit Containment
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Another application of the equation is to allow estimating the necessary measures to avoid the epidemic. In general, the susceptible population needs to be less than the critical value.

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

\\n\\nThe reduction of S is achieved with a preventive vaccination. 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 it should be such that\\n\\n

$S_{crit}=(1-q)S$



With the recovery factor

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

\\n\\nit has to\\n\\n

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



If it is assumed that initially the susceptible are equal to the population (S\sim N), the universe to be vaccinated is equal to

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

which gives us an idea of the level of mass that the vaccination campaign should have.

As an example in the diseases considered, we must:

| Disease | Transmission | Vaccination |

|: --------- |: ---------- |: ---------: |

| Measles | droplets in suspension | 92-94% |

| Whooping cough | droplets in suspension | 92-94% |

| Diphtheria | saliva | 83-86% |

| Smallpox | droplets in suspension | 80-86% |

| Polio | fecal-oral route | 80-86% |

| Rubella | droplets in suspension | 80-86% |

| Mumps | droplets in suspension | 75-86% |

| HIV / AIDS | sexual contact | 50-80% |

| SARS | droplets in suspension | 50-80% |

| Influenza (1918 pandemic strain) | droplets in suspension | 50-67% |

| Ebola | body fluids | 34-60% |

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

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



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



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

where t is the time \beta the contagion cup, \gamma the recovery cup, C the number of contacts and N the population.

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SIR model to describe SARS 2003 in Hong Kong
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If the susceptible, infected and 'recovered' (who heal or die) are observed, the typical curves of the SIR model are observed:

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