# Modeling notes: Airports versus treatment plants

## Background

You may recall from some months back that we’ve distinguished between a few mechanisms that could favor monitoring a city’s airplane trit vs its treatment plant.

Epi dynamics mechanisms include

1. the pandemic is likely to start elsewhere, in which case global prevalence is likely to lead local prevalence for a time. the effective prevalence seen in WWTP samples is the local prevalence, whereas that in airport trit samples will (ideally) be close to the global prevalence.
2. frequent flier class has a higher disease prevalence early on and contributes disproportionately to airport samples, elevating the effective prev in airport samples
3. lots of fliers return home (round trip). if deliberate release at airport, then might catch these people shedding when returning

others?

#1 is what I generally had in mind thinking about this problem, and #2 and #3 were suggested by Kevin.

Other mechanisms include

1. greater P2RA factor for airplane samples because of less microbial growth or non-human inputs into the waste
2. stochastic dynamics, e.g. if a single infected person defecates on a flight — importance depends on detection model
3. others?

See also

* [Charlie's july 2023 update](https://docs.google.com/document/d/1IRxyRXiaYpZi_gNtALK6FtUzVqWhd_7I9Ktf7R8b3ZE/edit)

## Simple deterministic model of a pandemic and local monitoring

Aim is to explore the intuition that, if a pandemic starts at a random location in the world, and we are monitoring air traffic and the local WWTP at a single city (e.g. Boston), then there can be an advantage to monitoring air traffic early on due to a higher disease prevalence among travelers than the local community.

Definitions

* Population sizes: Let $N_g$ be the global population and $N_l$ be the local population
* Rate of flying: Let $F$ be the rate per person per day of flying anywhere, and $f$ be the rate per person per day of flying into the focal city.
* Let $r$ be the exponential growth rate of the pathogen
* Let $I_g$ be total number of global infections and $i_g = I_g / N_g$ be the fraction of the population that is infected (i.e., the pathogen prevalence); similarly for $i$.

I suppose exponential growth at rate $r$ in the global number of infections and in the local number of infections.
I suppose new infections fly into the focal city at rate $I_g f$.
I ignore any reduction in $I_l$ due to flights out of the focal city, which I _think_ is reasonable for the scenarios of interest.
I suppose that $I_g(0) = 1$ and $I_l(0) = 0$.

The infection dynamics follow a system of ODEs,

\begin{align}
\frac{dI_g}{dt} &= r I_g \\
\frac{dI_l}{dt} &= f I_g + r I_l,
\end{align}

with solution

\begin{align}
I_g &= e^{rt} \\
I_l &= f t e^{rt}. \\
\end{align}

The corresponding prevalence dynamics are

\begin{align}
i_g &= \frac{e^{rt}}{N_g} \\
i_l &= \frac{f t e^{rt}}{N_l}.
\end{align}

The ratio of local to global prevalence is

\begin{align}
\frac{i_l}{i_g} = \frac{N_g}{N_l} f t.
\end{align}

What is a reasonable guess at $f$ for Boston?
Suppose that a traveler travels into the focal city with a probability $N_l / N_g$, to reflect the crude assumption idea that choice of destination city is in proportion to the population size.
Then $f = F N_l / N_g$, in which case

\begin{align}
\frac{i_l}{i_g} = F t.
\end{align}

This argument suggests that the local prevalence remains less than the global prevalence for $\sim 1/F$ days.
The above result suggests the local prevalence is larger than the global prevalence after $t > 1/F$; however, I think this behavior is an artefact of the model only allow infections to migrate into the focal city.
If we had symmetric travel then we'd instead have $i_l$ converge to $i_g$.

Charlie estimates that approximately 1 in 300 people in the world fly each day, or $F \approx \frac{1}{300}$, so that global prevalence leads local prevalence for $\sim 300$ days.

Monitoring wastewater of air travelers coming into a city will, in the idealized case, see an effective prevalence equal to $i_g$.
Whereas we expect monitoring at the WWTP will see an effective prevalence of $i_l$.
If a pandemic starts outside of the focal city (as is most likely), then $i_g$ leads $i_l$ by $\sim 1/F$ days, giving a window of $\sim 1/F$ in which monitoring travelors outperforms monitoring the local city.

### Detection (very in progress)

Following the detection model I describe [here](https://naobservatory.github.io/mikes-notebook/posts/2023-02-06-basic-detection-theory/#cumulative-reads-over-a-given-period).

Suppose that

* we need to see a cumulative number of reads $M^*$ to detect the pathogen.
* we sequence $\mathcal M$ total reads per day
* the p2ra factor for municipal is $s_l$ and for airport samples is $s_g$
* we can think of the p2ra factor as shedding times protocol bias, of the pathogen relative to background microbiome.
* it might be different in airport samples due to different ratios of defecation to urine, and less opportunity for non-pathogen growth and non-human input

Below I used subscript $x$ to stand for either $g$ or $l$.

The cumulative per-capital infection hours in the respective catchments is $\int_0^T i_x(t) \; dt$.

Call $Q_l$ and $Q_g$ the (expected) cumulative reads in municipal or airport samples and is approximately

\begin{align}
Q_x \approx s_x \mathcal M \int_0^T i_x(t) \; dt
\end{align}

Call $t^*_x$ the time when the cumulative reads hits the threshold for each strategy.

Cumulative incidence at detection

## Other thoughts

A lesson from the above, also observed by Charlie and Dan's work together, is that the advantage to monitoring airports goes away once there's been enough time for travel to bring the pathogen levels in the focal city in sync with the global population.
Interestingly, this suggests a reason why airport monitoring may have a weaker advange in an airport-release scenario than in the above 'random-origin' scenario: If the pathogen is release in a major international airport, then there is a higher chance that infected passengers arrive in the focal city right away and start an outbreak there immediately.