CDS - SIR Model for Epidemic Deseases

Remember:

SIR: Susceptible, Infectious, or Recovered

Reduced SI Model:$\{\begin{array}{l}\frac{ds}{dt}=-\beta sx\\ \frac{dx}{dt}=\beta sx\end{array}$

SIR Model:$\{\begin{array}{l}\frac{ds}{dt}=-\beta sx\\ \frac{dx}{dt}=\beta sx-\gamma x\\ \frac{dr}{dt}=\gamma x\end{array}$

$R_0$ index: the simplest definition is:$R_0=\frac{\beta }{\gamma }$And it defines:

• If $R_0<1$, the infection is not spreding.
  The recovery rate $(\gamma )$ is grater then the infection rate $(\beta )$.
• If $R_0>1$, the infection is spreading.
• $R_0=1$ is called “epidemic transition”.

Extended SIR Model (1):$\{\begin{array}{l}\frac{ds}{dt}=-\frac{\beta }{c+k_1}sx\\ \frac{dx}{dt}=\frac{\beta }{c+k_1}sx-\gamma x\\ \frac{dr}{dt}=\gamma x\\ \frac{dc}{dt}=kc(1-c)(x-r)\end{array}$

Extended SIR Model (2):$\{\begin{array}{l}\frac{ds}{dt}=-\beta sx\\ \frac{dx}{dt}=\beta sx-{\gamma }_{1}x-{\gamma }_{2}x-{\gamma }_{3}x\\ \frac{dq}{dt}={\gamma }_{1}x+{\delta }_{1}h-{\delta }_{2}q-{\delta }_{3}q\\ \frac{dh}{dt}=-{\delta }_{1}h+{\gamma }_{2}x+{\delta }_{2}q-{\eta }_{1}h-{\eta }_{2}h\\ \frac{dr}{dt}={\gamma }_{3}x+{\delta }_{3}q+{\eta }_{1}h\\ \frac{dd}{dt}={\eta }_{2}h\end{array}$

In the simplest case we can solve this problem analytically and find the solution $x(t)$, which wil be in the form: $x(t)=\frac{x_0{e}^{\beta t}}{1-x_0+x_0{e}^{\beta t}}$, and as you can see for $\beta >0$, then $x(t)$ will tend to $1$ for $t\to \infty$. We also know that this also from the fact that $x^{\ast }$ is an asymptotic steady state of the logistic equation. In the SI equation, $\beta$ represents a rate of contacts, then it assumed positive.

The SRI Model:

We have used this model to predict the infection rate for the coronavirus in Italy, however, this is still a simplied model, and does not represent correctly what is really happanening, the estimation is not correct:

This represents the data quite well:

The extended SRI Model:

• This model accounts also for quarantined individuals $(q)$ and for hospitalized individuals $(h)$.
• $\beta sx$ : number of contacts between susceptible and infected.
• ${\gamma }_{1}x$ : individuals from infected to quarantine.
• ${\gamma }_{2}x$ : infected ⇒ hospitalized.
• …
• ${\eta }_{2}h$: hospitalized ⇒ deceased.
• NOTE: That for each equation, the negative part all depend on the respective variable itself, otherwise the variable would become negative in the long run, and in this case each parameter/variable has to be $\ge 0$. (we cannot have a negative number of people)
  ~Ex.: $\frac{dx}{dt}=\beta sx-{\lambda }_{1}x-{\lambda }_{2}x-{\lambda }_{3}x$, the negative terms are ${\lambda }_{1}x{\lambda }_{2}x,{\lambda }_{3}x$ and they all depend on $x$.not-sure-about-this But $\beta sx$ depends on $x$ and is positive?? The positive sings means some people that enters in this class from a different one, and the negative sign means that they go from this class to another. Take for example the $\pm {\eta }_{2}h$ in hospitalied and dead, they have opposite signs, meaning that some hospitalized people die.

---

• Analytic solution.

• The susceptible converge to $0$, meaning all people have become infected (and then recovered)

• Different parameters $\beta$ and $\gamma$.

• $R_0$: strength of infection.

• $f$ is the least quadratic model function.
• This is can also be considered a machine learning program.

• This are the parameters $\beta$ and $\gamma$ found by the previous formula.

• As we can see the function we found, does not fit correctly the following data.

• A more complex, but more accurate model.
• The last formula consider the social distancing and mask used to protect the susceptible individuals.

• Now the function fits the data much better.

• When the lockdown ends we need to expect a second peak.
• The black line represents the parameter $c$.

• Since we have more data (all the regions combined), the resulting found function, fits the data a lot better.

• This model accounts also for quarantined individuals $(q)$ and for hospitalized individuals $(h)$.
• $\beta sx$ : number of contacts between susceptible and infected.
• ${\gamma }_{1}x$ : individuals from infected to quarantine
• ${\gamma }_{2}x$ : infected ⇒ hospitalized
• …
• ${\eta }_{2}h$: hospitalized ⇒ deceased.
• NOTE: That for each equation, the negative part all depend on the respective variable itself, otherwise the variable would become negative in the long run, and in this case each parameter/variable has to be $\ge 0$. (we cannot have a negative number of people)
  ~Ex.: $\frac{dx}{dt}=\beta sx-{\lambda }_{1}x-{\lambda }_{2}x-{\lambda }_{3}x$, the negative terms are ${\lambda }_{1}x{\lambda }_{2}x,{\lambda }_{3}x$ and they all depend on $x$.