Describing our approach to modeling an arbitrary infectiousness distribution

docs/0 contents.md

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+# Table of contents
+
+1. Synthetic population
+2. [Transmission](transmission.md)
+    - [Time-varying infectiousness](time-varying-infectiousness.md)
+    - Estimating infectiousness from viral load
+3. Contact
+4. Natural immunity
+5. Interventions
+    - Mask wearing
+    - Isolation
+    - Vaccines

docs/time-varying-infectiousness.md

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+# Time-varying infectiousness
+
+## Motivation
+
+Many epidemiological studies have shown that infectiousness varies widely over the course
+of the individual's infection. We detail the process for sampling an arbitrary number of
+infection attempts appropriately from an arbitrary infectiousness curve in an agent-based model (ABM).
+We describe an approach that provides great generalizability, enabling sampling when both the
+underlying contact rate and infectiousness distribution change over the course of an infection.
+We motivate our discussion with a simple example of an infectious person who takes an
+antiviral partially through their infection and is hospitalized.
+
+## Assumptions
+
+1. We describe the approach for sampling from an arbitrary infectiousness dsitribution for the uniform
+distribution from zero to one -- $\mathcal{U}(0, 1)$.
+    - Using [inverse transform sampling](https://en.wikipedia.org/wiki/inverse_transform_sampling), a
+uniformly-distributed random variable can be transformed to any distribution by passing samples through
+the inverse cumulative distribution function (CDF).
+    - This requires that the infectiousness over time distribution has an inverse CDF or we are
+    able to approximate it (i.e., from empirical data).
+2. Infectiousness over time is a relative quantity, so it is completely separate from the absolute degree
+to which someone is infectious, represented by the total number of secondary infection attempts they have.
+That quantity, denoted $C_i$ in the generality, is directly related to $R_0$.
+3. We require $C_i$ ordered draws from the infectiousness over time/generation interval distribution (time
+from an individual becoming infected to their secondary contacts becoming infected). We schedule infection
+attempts at these drawn times.
+
+## Overall Methodology
+
+We could just take $C_i$ draws from the generation interval (GI) and order them, scheduling an infection
+attempt at each time drawn. However, we argue (in detail [below](#why-do-we-need-order-statistics)) that
+it is more robust to draw the minimum of our $C_i$ draws, evaluate an infection attempt at that time,
+and _then_ schedule the next infection attempt -- drawing the second smallest of our $C_i$ ordered draws
+from the GI.
+
+Drawing the smallest, the second smallest, the third smallest, etc. of a set number of random draws
+from a distribution is the problem of having the distribution's
+[order statistics](https://en.wikipedia.org/wiki/Order_statistic)", and it is readily solvable for the
+case of the uniform distribution.
+
+## Drawing ordered samples from a uniform distribution
+
+Let us begin by considering the minimum of $n$ draws from a uniform distribution. We denote the sorted
+first of these draws as $X_{(1)}$, the sorted second as $X_{(2)}$, etc. Note that this is different from $X_1$
+which is the unsorted first random draw from the distribution, unsorted. We are interested in the distribution of
+$X_{(1)}$. Let us consider the cumulative distribution function of $X_{(1)}$ since it is a continuous random variable.
+
+$$\mathbb{P} \\{X_{(1)} \leq x \\}$$
+
+We know that if the minimum is below some value, $x$, that could mean that just the minimum is below $x$ or that
+all $n$ of our random samples are below the minimum. There are too many values to enumerate, so let us consider
+the opposite instead -- that $X_{(1)}$ is greater than some value $x$.
+
+$$\mathbb{P} \\{X_{(1)} \leq x \\} = 1 - \mathbb{P} \\{X_{(1)} > x \\}$$
+
+In this case, if $X_{(1)} > x$, we know that all of sampled values are at least above $x$. Recall that
+each of the samples is independent and identically-distributed.
+
+$$= 1 - \mathbb{P} \\{X_i > x \\}^n$$
+
+$$= 1 - (1 - F_X(x))^n$$
+
+For a uniform distribution:
+
+$$= 1 - (1 - x)^n$$
+
+This is the CDF for a Beta distribution with $alpha = 1$ and $beta = n$. More generally, the distribution
+of the $k$th infection attempt from $n$ total infection attempts is $\beta(k, n - 1 + k)$.
+
+However, we cannot just independently draw the time of a next infection attempt from the corresponding Beta
+distribution. Consider that we draw the first infection attempt time from $\beta(1, 5)$ and happen to get a large value.
+We need to take that into account when drawing the second infection attempt time. The second infection
+attempt time must be greater than the first, so we cannot just draw from $\beta(2, 6)$ but rather $\beta(2, 6) | x_{(1)}$.
+
+We can set $x_{(1)}$ as the minimum of a new uniform
+distribution, $\mathcal{U}(x_{(1)}, 1)$, from which we need to draw an infection attempt. Because this is a new distribution,
+we want the first of $n - 1$ infection attempt times on this distribution. We can do that by drawing the
+minimum of $n - 1$ infection attempts from $\mathcal{U}(0, 1)$, and scaling that value to be on $(x_{(1)}, 1)$.
+In other words, we shrink the available uniform distribution with each infection attempt and ask what
+would be the _next_ infection attempt on that distribution. Concretely, if $x_{(1)}$ is
+on $(0, 1)$, and $x_{(2)}$ is also on $(0, 1)$, we use the following equation:
+
+$$x_{(2)} := x_{(1)} + x_{(2)} * (1 - x_{(1)})$$
+
+For further detail, the Wikipedia [page on order statistics](https://en.wikipedia.org/wiki/Order_statistic) is a
+great reference as are various notes on the subject from
+[advanced statistics courses](https://colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf).
+
+## Workflow
+
+1. Draw a number of secondary infection attempts for the agent, $C_i$. This can be equal to $R_i$ if the sampling times
+are from the generation interval exactly, or $C_i$ can be greater than $R_i$ if sampling from a
+[proposal distribution](#rejection-sampling-from-an-arbitrary-generation-interval) and rejecting some attempts to leave
+a total of $R_i$ infection attempts.
+2. Draw the time for the first of $n$ remaining infection attempts of $\mathcal{U}(0, 1)$ by taking a random value from
+$\beta(1, m)$. $m = C_i - $ (the number of infection attempts that have occured).
+3. Scale the value on $\mathcal{U}(0, 1)$ to be on $\mathcal{U}(x_{(i)}, 1)$ where $x_{(i)}$ is the previous draw.
+4. Convert the uniform value to generation interval space by passing it through the inverse CDF of the generation interval,
+and schedule the next infection attempt at the specified time. Wait until that time has occured in the simulation before
+proceeding.
+
+    The result of passing the uniform time through the GI's inverse CDF is the time _since_ the agent first become
+infectious at which the given $n$th infection attempt occurs.
+
+5. Repeat from step two until $m = 0$.
+
+## Why do we need order statistics?
+
+We provide a detailed justification of why scheduling the next infection attempt after the last (i.e., sequentially
+drawing infection attempt times from the GI) is the better approach compared to taking all the draws
+from the GI at the beginning of an individual's infectiousness period and at once scheduling the infection
+attempts based on those times.
+
+### Changes in the number of infection attempts in the middle of an agent's infectious course
+
+Imagine an agent dies while they are still infectious. They can no longer infect others. (Or, if the
+disease in question is Ebola, there are well-defined processes by which they may still be infectious and
+those processes should not be lumped with the infection generation interval.)
+
+To ensure a dead person does not infect others, we would like to remove the plans where they infect others.
+If we had pre-scheduled all infections, we would have had to store the plan IDs for all the infections in
+`HashMap<PersonId, Vec<PlanId>>`. Then, each time we had executed one of these plans, we would have to
+remove it from the vector, so that the entry for each `PersonId` tells us the plans we have _left_ for a given
+person. Then, when the agent dies, we could iterate through the remaining plans and cancel them.
+
+Clearly, this is cumbersome, requiring us to store a `HashMap` and iterate
+through a vector every time an infection attempt occurs. Alternatively, if we scheduled infection events
+sequentially, we would only have to store the next infection attempt time. We could instead use a
+`HashMap<PersonId, PlanId>`, removing the need to iterate through a vector and instead enabling us to
+directly use the `.insert` method on the `HashMap`.
+
+More generally, this idea applies to any case where we need to change the number of infection attempts
+partway through an infection course. Sequentially scheduling the attempts makes it possible to accomodate
+changes to the number of infection attempts that may happen in the middle of an infection course.
+
+Why not just check whether the agent is alive or not at the beginning of the infection attempt? If they are
+not alive, skip the infection attempt. There are two reasons this is not the desired solution. First,
+it is cleaner to handle removing an agent at the point at which they become no longer relevant rather than
+continuously checking on whether they are alive. Secondly, there are other cases when the number
+of infection attempts may change partially through an infection, such as being hospitalized and having
+a lower contact rate. Sequentially scheduling infection attempts enables arbitrary changes to the
+effective contact rate while an agent is infectious in the simplest way.
+
+### Rejection sampling from an arbitrary generation interval
+
+Let us consider a case where the infectiousness distribution changes over time. Concretely, imagine that
+infectiousness is a function of the viral load, and an agent gets an antiviral partially through their
+infection course. Depending on when the agent gets the antiviral, they have a different reduction in viral
+load. Therefore, even if we know that an agent will get an antiviral, we don't know their infectiousness
+distribution in advance.
+
+It is still straightforward to sample the appropriate infectiousness distribution without needing to change
+the generation interval. We can use [rejection sampling](https://en.wikipedia.org/wiki/Rejection_sampling).
+In this case, we still sample infection attempt times from the pre-antiviral infectiousness distribution, and
+we accept those sampled times as actual infection attempts with probability $a(t) / g(t)$ where $a(t)$ is the
+post-antiviral infectiousness distribution and $g(t)$ is the underlying pre-antiviral infectiousness
+distribution.
+
+However, rejection sampling is inherently inefficient. It requires that we draw plans at a faster rate than events
+actually happen, and then we need to evaluate at the time of the plan whether we mean for something to actually
+happen. Nevertheless, there are ways to improve efficiency. The trivial case of drawing plans at a faster rate
+than events actually happen is to use the maximum event rate as the sampling rate for all $t$. But, if the
+actual event rate is much smaller than the maximum for nearly all $t$, (i.e., a disease where infectiousness is highly
+concentrated around a specific time or $a(t) << g(t)$), this is particularly inefficient because we are rejecting the majority of samples. Instead, we may try making our proposal distribution better fit our underlying distribution by making it, say, a linear approximation of $a(t)$.
+
+This approach is only possible if we sequentially sample infection attempts. If we sample all
+infection attempts at the beginning of an agent's infection, we are locked into using a singular sampling rate. Thus,
+sequential sampling enables us to most generally model a generation interval or infectiousness distribution
+that may change over the course of an agent's infection in a non pre-defined way.
+
+These examples underscore that sequentially sampling infection attempts rather than pre-scheduling enables
+the modeler to consider changes in both the contact rate and the generation interval without needing to change
+the transmission model.