CDS - Lecture 15

• 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 $x$.