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_05_seir_model.qmd
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_05_seir_model.qmd
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------------------------------------------------------------------------
::::: columns
::: {.column width="30%"}
:::
::: {.column width="70%"}
- Some diseases have an [latent/exposed period]{style="color:tomato;"} during which individuals are [infected but not yet infectious]{style="color:tomato;"}. Examples include pertussis, COVID-19, and Ebola.
- Disease transmission does not occur during the latent period because of low levels of the virus in the host.
:::
:::::
------------------------------------------------------------------------
------------------------------------------------------------------------
::::: columns
::: {.column width="50%"}
:::
::: {.column width="50%"}
- The SEIR model extends the SIR model to include an exposed compartment, [$E$]{style="color:blue;"}.
- [$E$]{style="color:blue;"}: infected but are not yet infectious.
- Individuals stay in [$E$]{style="color:blue;"} for [$1/\sigma$]{style="color:blue;"} days before moving to $I$.
:::
:::::
------------------------------------------------------------------------
::::: columns
::: {.column width="60%"}
{width="80%"}
:::
::: {.column width="40%"}
Model equations: \begin{align*}
\frac{dS}{dt} & = -\beta S I \\
\frac{dE}{dt} & = \beta S I - \color{orange}{\sigma E} \\
\frac{dI}{dt} & = \color{orange}{\sigma E} - \gamma I \\
\frac{dR}{dt} & = \gamma I
\end{align*}
:::
:::::
------------------------------------------------------------------------
### SEIR model with births and deaths
- Let's relax the assumption about births and deaths in the population.
- We will assume that the susceptible population is replenished with new individuals at a constant rate, $\mu$.
- We will also assume that everyone dies at a constant rate, $\mu$.
------------------------------------------------------------------------
Our model schematic now looks like this:
::: notes
Explain in terms of inflows and outflows
:::
------------------------------------------------------------------------
::::: columns
::: {.column width="50%"}
:::
::: {.column width="50%"}
The model equations now become:
\begin{align}
\frac{dS}{dt} & = \color{green}{\mu N} - \beta S I - \color{blue}{\mu} S \\
\frac{dE}{dt} & = \beta S I - \sigma E - \color{blue}{\mu} E \\
\frac{dI}{dt} & = \sigma E - \gamma I - \color{blue}{\mu} I \\
\frac{dR}{dt} & = \gamma I - \color{blue}{\mu} R
\end{align}
:::
:::::
<!-- ------------------------------------------------------------------------ -->
<!-- #### Practicals -->
<!-- - Let's implement the SEIR model in R using the script `02_seir.Rmd`. -->
------------------------------------------------------------------------
::: {.callout-caution collapse="true" icon="false"}
#### Discussion
What is the $R0$ of the SEIR model?
:::
------------------------------------------------------------------------
### The R0 of the SEIR Model
- Beyond the SIR model, calculating $R0$ for more complex models can be challenging due to the presence of multiple compartments.
- For complex models, we use the [next generation matrix]{style="color:tomato;"} approach [@diekmann1990definition; @diekmann2010construction].
::: {.callout-note collapse="true" icon="false"}
Using the next generation matrix approach, we can show that the SEIR model with constant births and deaths has $$R0 = \dfrac{\beta \sigma}{(\gamma + \mu)(\sigma + \mu)}$$.
:::
------------------------------------------------------------------------
<!-- {{< include _05_01_next_generation_matrix.qmd >}} -->
<!-- ------------------------------------------------------------------------ -->
### Numerical simulations
#### R Practicals
- We can use the same approach as the SIR model to simulate the SEIR model.
- Modify the `sir.Rmd` script to simulate the SEIR model.