Asthma Exacerbation Severity Model¶

Each asthma exacerbation is assigned one of four severity levels, classified retrospectively by the level of healthcare utilization required to treat it, following the framework described by the Global Initiative for Asthma (GINA) [{Global Initiative for Asthma}, 2023]:

Level	Name	Healthcare utilization
1	Mild	managed with reliever medication alone
2	Moderate	requires a physician visit and a prescription of oral corticosteroids (OCS)
3	Severe	requires an emergency department (ED) visit
4	Very severe	requires hospital admission

The very severe level is the one used to compute \(P(\text{hosp})\) and calibrate \(\alpha\) in the Calibration section of the Asthma Exacerbations Model, and is what the Hospitalization Data described in that section measures.

Dirichlet-Multinomial Model¶

Given the total number of exacerbations \(N_{\text{exacerbations}}^{(i)}\) that agent \(i\) experiences in a time interval (drawn from the Poisson model described in the Asthma Exacerbations Model), the number at each severity level is generated using a Dirichlet-Multinomial distribution. A preliminary probability vector across the four levels is first drawn from a Dirichlet distribution:

\[\mathbf{w}^{\text{pre},(i)} \sim \text{Dirichlet}(\boldsymbol{\delta})\]

The vector \(\mathbf{w}^{\text{pre},(i)}\) is defined as a length-4 vector of probabilities (summing to 1) giving this individual’s personal probability of each severity level:

\[\mathbf{w}^{\text{pre},(i)} := \begin{bmatrix} w^{\text{pre},(i)}_{\text{mild}} & w^{\text{pre},(i)}_{\text{moderate}} & w^{\text{pre},(i)}_{\text{severe}} & w^{\text{pre},(i)}_{\text{very severe}} \end{bmatrix}\]

For each agent with asthma, \(\mathbf{w}^{\text{pre},(i)}\) is sampled once, independently per agent, and held fixed for their simulated lifetime — representing individual heterogeneity in exacerbation severity, distinct from (and prior to) the adjustment for previous hospitalization described below. The actual exacerbation counts are determined later using \(N_{\text{exacerbations}}^{(i)}\) (the total count, from the Poisson model above) together with this probability vector.

The vector \(\boldsymbol{\delta}\) is the Dirichlet concentration vector:

\[\boldsymbol{\delta} = \kappa \cdot \mathbf{p}\]

where:

\[\begin{split}\mathbf{p} &= \begin{bmatrix} p_{\text{mild}} & p_{\text{moderate}} & p_{\text{severe}} & p_{\text{very severe}} \end{bmatrix} \\ &= \begin{bmatrix} 0.495 & 0.195 & 0.283 & 0.026 \end{bmatrix}\end{split}\]

are the same SYGMA II severity proportions used for \(P(\text{hosp})\) in the Calibration section of the Asthma Exacerbations Model [Lee, 2026], and \(\kappa = 100\) is an assumed concentration multiplier controlling how tightly an individual’s probabilities cluster around the population proportions \(\mathbf{p}\).

Adjustment for Previous Hospitalization¶

If the individual has previously been hospitalized for an asthma exacerbation, their probability of a very severe exacerbation is increased, and the remaining probability mass is redistributed proportionally across the other three levels:

\[\begin{split}w_j^{(i)} &:= \begin{cases} w^{\text{pre},(i)}_{j} \cdot \beta_{\text{prev hosp}}^{(i)} & \quad j = \text{very severe} \\ \dfrac{w^{\text{pre},(i)}_j}{\sum_{\ell \,\in\, \{\text{mild, moderate, severe}\}} w^{\text{pre},(i)}_{\ell}} \cdot (1 - w_{\text{very severe}}^{(i)}) & \quad j \in \{\text{mild, moderate, severe}\} \end{cases}\end{split}\]

where:

\(\beta_{\text{prev hosp}}^{(i)}\): \(\beta_{\text{prev hosp,pediatric}} = 1.79\) for individuals under 14 years of age, or \(\beta_{\text{prev hosp,adult}} = 2.88\) for individuals 14 years of age or older.

These rate multipliers are taken from a Canadian cohort study of the long-term natural history of severe asthma exacerbations [Lee, 2022], which found that a first follow-up severe exacerbation was associated with a 79% increase (rate multiplier 1.79, 95% CI 1.11–2.89) in the rate of subsequent exacerbations for pediatric patients, and a 188% increase (rate multiplier 2.88, 95% CI 1.35–5.15) for adult patients.

If the individual has no prior hospitalization, \(\mathbf{w}^{(i)} = \mathbf{w}^{\text{pre},(i)}\).

Since \(\mathbf{w}^{\text{pre},(i)}\) already sums to 1, inflating \(w_{\text{very severe}}^{(i)}\) by \(\beta_{\text{prev hosp}}^{(i)}\) alone would push the total above 1. The \(w_j^{(i)}\) formula corrects this: it proportionally shrinks the mild/moderate/severe probabilities so the full vector \(\mathbf{w}^{(i)}\) sums back to 1, while keeping their relative proportions to each other unchanged from \(\mathbf{w}^{\text{pre},(i)}\).

This raises the question of how “previously hospitalized” is determined for an agent whose asthma history was not directly simulated cycle-by-cycle — for example, an agent assigned an asthma label and a diagnosis age all at once when they enter the simulation. See Step 4: Check hospitalizations for how this is initialized in that case.

Finally, the number of exacerbations at each severity level is drawn from a Multinomial distribution, using \(N_{\text{exacerbations}}^{(i)}\) as the number of trials:

\[(n_{\text{mild}}^{(i)}, n_{\text{moderate}}^{(i)}, n_{\text{severe}}^{(i)}, n_{\text{very severe}}^{(i)}) \sim \text{Multinomial}(N_{\text{exacerbations}}^{(i)}, \mathbf{w}^{(i)})\]