MODEL_DESCRIPTION.md

Infant-trained Bayesian hierarchical model summary

These are the assumptions we are currently making about the biology:

1. We are striving to estimate time since infection.
2. Individuals are infected in-utero, at birth, or through breastfeeding.
3. For individuals infected in-utero, time points closer to the third trimester (of pregnancy) are most likely to be the infection time.
4. Individuals infected through breastfeeding are infected at some point after one month of age.
5. APD is defined to be the measure of average pairwise diversity at the third codon position using only sites at which the sum of all minor variants is greater than 0.01.
6. APD is zero when time since infection is zero (infection time).
7. APD of the sequence increases with time for most individuals.
8. The rate of APD change over time may be different for each individual.
9. The rate of APD change over time may be different for each sequence fragment.
10. For individuals infected in-utero, time since infection will be greater than the observed time (age at sampling time) *
11. For individuals infected at birth, time since infection will be equal to observed time *
12. For individuals infected through breastfeeding (after birth), time since infection will be less than the observed time (age at sampling time) *

* -- see time clarification example in the model explanation section.

Data processing/filtering for training and testing the model:

• We are excluding data when, for a given subject and sequence fragment, we only have one sample time point.
• For samples which have more than 2 replicates, we are including only the first 2 replicates for which we have data.
• We are averaging these two sample replicates for each subject and sequence fragment to get the final APD value used in model fitting

Here are some definitions we will use in the model explanation:

Key definitions:

• observation_count: the number of observations in the data.
• subject_count: the number of individuals (or subjects) in the data.
• fragment_count: the number of sequencing regions (or fragments) in the data.
• subject: the unique identifier of an individual or subject.
• fragment: the unique identifier of a sequencing region or fragment.
• observed_time: the age of the individual at sampling time.
• time_since_infection: the time since infection
• apd: APD measurements (cutoff 0.01) from all runs for each individual (not an average)
• is_utero: TRUE/FALSE indicating whether the individual was known to be infected in utero
• is_post: TRUE/FALSE indicating whether the individual was known to be infected after birth

Other definitions (to be defined in the following model explanation)

• predicted_time_since_infection
• time_since_infection_variance_estimate
• total_slope
• baseline_slope
• subject_slope_delta
• subject_slope_delta_mean_estimate
• subject_slope_delta_variance_estimate
• fragment_slope_delta
• fragment_slope_delta_mean_estimate
• fragment_slope_delta_variance_estimate
• observed_time_to_time_since_infection_correction
• predicted_observed_time

Model Explanation:

1. We model time_since_infection to be normally distributed around a predicted_time_since_infection with a time_since_infection_variance_estimate.
   • We choose to model time_since_infection_variance_estimate as a cauchy distribution with mean = 0 and standard deviation = 0.5.
2. We model the function apd -> predicted_time_since_infection as a linear function with predicted_time_since_infection = total_slope * apd.
3. We model total_slope to be the sum of a baseline_slope, a subject_slope_delta, and a fragment_slope_delta.
4. We model the baseline_slope of this function as a uniform distribution, restricted to be positive.
5. We define subject_slope_delta to be the difference in slope (or APD rate of change) from the baseline_slope defined uniquely for each subject. We model the subject_slope_delta of this function as a normal distribution, with mean, subject_slope_delta_mean_estimate, and variance, subject_slope_delta_variance_estimate.
   • We choose to model the subject_slope_delta_mean_estimate as a normal distribution with mean = 0 and standard deviation = 1.
   • We choose to model the subject_slope_delta_variance_estimate as a cauchy distribution with mean = 0 and standard deviation = 20.
6. We define fragment_slope_delta to be the difference in slope (or APD rate of change) from the baseline_slope defined uniquely for each fragment. We model the fragment_slope_delta of this function as a normal distribution, with mean, fragment_slope_delta_mean_estimate, and variance, fragment_slope_delta_variance_estimate.
   • We choose to model the fragment_slope_delta_mean_estimate as a normal distribution with mean = 0 and standard deviation =1.
   • We choose to model the fragment_slope_delta_variance_estimate as a cauchy distribution with mean = 0 and standard deviation = 5.
7. Since the data does not directly measure time_since_infection, for training the model, we define an observed_time_to_time_since_infection_correction for each subject which will act to convert between time_since_infection and predicted_observed_time.
   • We choose to model the observed_time_to_time_since_infection_correction for individuals infected in utero as a beta(1, 5) distribution bounded between 0 and 0.75 (years).
   • We choose to model the observed_time_to_time_since_infection_correction for individuals infected after birth as a uniform distribution between -1 and -0.0833 (years).
8. We model predicted_observed_time to be the difference between time_since_infection and a subject specific observed_time_to_time_since_infection_correction. See time clarification example below...
9. Lastly, we model observed_time to be normally distributed around a predicted_observed_time with standard deviation = 0.1.

TIME CLARIFICATION EXAMPLE: Infants can either be infected in utero, at birth, or post birth.

Say we have an infant that was infected two months before birth. Say we have a sample taken for sequencing at six months post birth. At six months, this individual will have an observed_time of 6 months. However, since the individual was infected eight months prior to sampling, their time_since_infection will be 8 months at sampling time. The difference between the time_since_infection and the observed_time will be two months. In the model, we define these two months as the observed_time_to_time_since_infection_correction--shown as "observed time correction" in the figure below.

In equation form, this relationship for infants infected in utero is defined as follows:

Say we have an infant that was infected two months post birth. Say we have a sample taken for sequencing at six months post birth. At six months, this individual will have an observed_time of 6 months. However, since the individual was infected four months prior to sampling, their time_since_infection will be 4 months at sampling time. The difference between the time_since_infection and the observed_time will be two months. In the model, we define these two months as the observed_time_to_time_since_infection_correction--shown as "observed time correction" in the figure below.

In equation form, this relationship for infants infected post birth is defined as follows:

Say we have an infant that was infected at birth. Say we have a sample taken for sequencing at six months post birth. At six months, this individual will have an observed_time of 6 months. Since the individual was infected at birth, their time_since_infection will also be 6 months at sampling time. This is shown in the figure below:

In equation form, this relationship for infants infected at birth is defined as follows:

Note that these relationship are approximate due to noise in fitting the model.

The main stan model can be found here; we refer to this model as the "infant-trained Bayesian hierarchical regression model" in our manuscript associated with this work.