[sir_model] Some suggestions on code and wording #1049

Hi @jstac

I was reading through the sir_model lecture and noted some possible updates.

Overview and model description

• Change both occurrences of “Modeling COVID 19” to “Modeling COVID-19,” including the index entry to make them consistent with the main text and also consistent with Wikipidea.

• update the “COVID-19 Working papers and code” link to https://sites.google.com/site/andyatkeson/home/covid-work, currently, this link goes to the personal website of Andrew Atkeson instead of this paper and code.

• update the four-state model as an SEIR (Susceptible-Exposed-Infected-Removed) model rather than an SIR model. In this lecture, we uses 4 state instead of three states as the name SIR suggested.

• [qe-writing-006] Use sentence case for the lower-level headings at lectures/sir_model.md:68,82,236,284,347:

  • The SEIR model
  • Time path
  • Experiment 1: constant R0 case
  • Experiment 2: changing mitigation
  • Ending lockdown

State definitions and assumptions

• At line 74, we could update the compartment symbols in math delimiters: susceptible ($S$), exposed ($E$), infected ($I$), and removed ($R$). Currently, we have (R) for the removed ones and markdown rendered this as a trademark ®.

• At line 76-80, replace the “Comments:” label with a prf:assumption environment since they are all assumptions used later in this lecture:

```{prf:assumption}
- Those in state $R$ were previously infected and have either recovered or died.
- Those who have recovered are assumed to have acquired immunity.
- Those in the exposed group are not yet infectious.
```

Time path and parameters

• At line 86-87, replace the claim that positive transmission and $i(0)>0$ imply everyone is eventually infected as this is not true in this model And we didn't introduce the meaning of $i$ yet. Suggested text:

  When the effective reproduction number exceeds one, infections initially increase but eventually decline as the susceptible population falls.

• Apply the following small prose corrections at line 91,94,112,114,131:

  • Change “lower case” to “lowercase.”
  • Add the missing period to the $\sigma$ definition.
  • Capitalize “The dot symbol.” as it is a beginning of a new sentence.
  • Write “for a suitable definition of $F$.” by adding the missing article "a"

• [qe-math-001] At line 103, 105 , replace the raw Unicode symbols inside the math directive with \sigma and \gamma.

• At line 113, call $\gamma$ the removal rate, since the transition includes both recovery and death. Currently it is calle d recovery rate which doesn't reflect that this also include dealth.

Implementation

• At line 192-199, explain the initial-condition calibration. With a population of 330 million, $i_0=10^{-7}$ represents 33 initially infected people, $e_0=4i_0$ represents 132 exposed people, and $s_0=1-i_0-e_0$ sets the initial removed fraction to zero. This makes the calibrated number economically meaningful.

• At line 207, format odeint as inline code.

Experiments and figures

• At line 244, render the zero in the legend variable as a subscript:

  labels = [f'$R_0 = {r:.2f}$' for r in R0_vals]

• [qe-fig-008] At line 261,317, add lw=2 to the line plots.

• At line 256-265,313-320, add axis labels and units so that reader can know the unit of the plot. The shared plot_paths helper could accept a y-axis label, while all relevant plots use days on the x-axis. Quantity-specific y-axis labels should distinguish:

  • current cases as a fraction of the population;
  • cumulative cases as a fraction of the population;
  • active infections as a fraction of the population;
  • cumulative deaths;
  • deaths per day.

  The standalone mitigation plot should use $R_0$ on the y-axis.

• At line 301-309, explain that $\eta$ is measured per day because $t$ is measured in days. Its reciprocal is the characteristic adjustment period, so the specified values correspond to adjustment periods of 5, 10, 20, 50, and 100 days.

• After line 345, add some economic interpretation, currently it finishes with the plot.

  Faster implementation of mitigation mainly delays the infection peak, with a smaller effect on its height.

Ending lockdown and mortality

• At line 383, describe the plotted quantity as a fraction rather than a number:

  Here is the fraction of the population with an active infection:

• After the active-infections plot at line 386, add some economic interpretation.

  Both scenarios produce approximately the same infection peak, but the longer lockdown delays it.

• At line 391, add the missing period after “Suppose that 1% of cases result in death.”

• At line 397-401, correct the cumulative-deaths calculation. Since $c(t)=i(t)+r(t)$, the current calculation counts a fraction of active infections as deaths and begins with 250 cumulative deaths even though the initial removed fraction is zero. Cumulative deaths should be $\nu r(t)N=\nu[c(t)-i(t)]N$:

  paths = [(c_path - i_path) * ν * pop_size
           for c_path, i_path in zip(c_paths, i_paths)]

  This is consistent with the daily-deaths formula $\nu\gamma i(t)N$. Label the y-axis cumulative deaths.

• At line 404-408, change the introduction to:

  This is the number of deaths per day:

instead of using daily death rate as the following plot has y-axis as the number of death per day as unit and keep the existing calculation and label the y-axis deaths per day.

• At line 411, write “the peak of the curve.”

What do you think? I'd be grateful if you could take a look when you have a chance.
Happy to put up a PR.

Best,
Longye