Asthma Exacerbations Model¶

Data¶

Population Data¶

We use the Statistics Canada population data that was generated and saved as: processed_data/birth/initial_pop_distribution_prop.csv.

Column	Type	Description
`timepoint`	`int`	the starting date / time of the time interval that the data applies to
`age`	`int`	the age of the person in years
`province`	`str`	the province of the person (e.g., `AB` = Alberta, `BC` = British Columbia, etc.)
`n_age`	`int`	the number of people in a given age group, time interval, province, and projection scenario
`n_birth`	`int`	the number of births in that time interval, province, and projection scenario
`prop`	`float`	the proportion of the population in that age group, time interval, province, and projection scenario relative to the number of births in that time interval, province, and projection scenario
`prop_male`	`float`	the proportion of the population in a given age group, time interval, province, and projection scenario who are male
`projection_scenario`	`str`	the projection scenario used to generate the data

Occurrence Data¶

We use the occurrence data that was generated and saved as: processed_data/asthma_occurrence_predictions.csv

See Occurrence Model 1: Crude Occurrence for more details about this dataset.

Column	Type	Description
`timepoint`	`int`	the starting date / time of the time interval that the data applies to
`sex`	`str`	`F` = female, `M` = male
`age`	`int`	the age of the patient in years
`incidence`	`float`	the predicted asthma incidence for the given time interval, age, and sex
`prevalence`	`float`	the predicted asthma prevalence for the given time interval, age, and sex

Hospitalization Data¶

The data is from the Hospital Morbidity Database (HMDB) from the Canadian Institute for Health Information (CIHI).

The hospitalization data was collected from patients presenting to a hospital in Canada due to an asthma exacerbation. We will use this data to calibrate the exacerbation model.

The hospitalization rate in this table is the hospitalization rate per 100 000 people. For example, in the category F_90+, the value would be the number of people hospitalized who are female and over 90 during the given year. This can be calculated:

\[\text{rate} = \dfrac{\text{count}}{N} \times 100000\]

Column	Type	Description
`fiscalYear`	`int`	the year the data was collected
`N`	`float`	the hospitalization rate for all ages and sexes in that year.
`M`	`float`	the rate for all ages who are male in that year.
`F`	`float`	the rate for all ages who are female in that year.
`0`	`float`	the rate for all sexes aged 0 in that year.
`1`	`float`	the rate for all sexes aged 1 in that year.
`...`	`...`	…
`90`	`float`	the rate for all sexes aged 90 in that year.
`90+`	`float`	the rate for all sexes aged over 90 in that year.
`F_0`	`float`	the rate for all females aged 0 in that year.
`...`	`...`	…
`F_90+`	`float`	the rate for all females aged over 90 in that year.
`M_0`	`float`	the rate for all males aged 0 in that year.
`...`	`...`	…
`M_90+`	`float`	the rate for all males aged over 90 in that year.

Model¶

The number of exacerbations in a given time interval is modelled using a Poisson distribution. The formula is:

\[N_{\text{exacerbations}} \sim \text{Poisson}(\lambda) = \dfrac{\lambda^k e^{-\lambda}}{k!}\]

Here \(\lambda\) is the expected number of exacerbations per time interval. To obtain \(\lambda\), we must perform a Poisson regression. The Poisson regression assumes that the value we are interested in can be approximated using the following formula:

\[\ln(\lambda) = \ln(\alpha) + \beta_0 + \sum_{i=1}^3 \beta_i c_i\]

where:

Variable	Description
\(\alpha\)	the calibration multiplier that adjusts the model to match the hospitalization data
\(\beta_0\)	a constant randomly chosen from the normal distribution \(\mathcal{N}(0, 1)\)
\(c_i\)	relative time spent in control level \(i\)
\(\beta_i\)	control level constant calculated from the Control Model

To calculate the \(\beta_i\) values, we consider the discrete random variable \(S\), which is the severity of an asthma exacerbation, and the continuous random variable \(R\), which is the rate of exacerbations per time interval.

Calibration¶

We are interested in calculating \(\alpha\). If we rewrite the equation, the meaning of \(\alpha\) becomes more apparent:

\[\lambda = \alpha \cdot e^{\beta_0} \prod_{i=1}^3 e^{\beta_i c_i}\]

How do we obtain \(\alpha\)? We again assume that the mean value has the same form as in a Poisson regression, with the following formula:

\[\ln(\lambda_{C}) = \sum_{i=1}^3 \beta_i c_i\]

\(\lambda_C\): the average number of exacerbations in a given time interval
\(c_i\): relative time spent in control level \(i\)
\(\beta_i\): control level constant

Here, the \(\beta_i\) values were calculated from the Economic Burden of Asthma (EBA) study and are given by:

\[\begin{split}\beta_1 &:= \ln(0.1880058) \\ \beta_2 &:= \ln(0.3760116) \\ \beta_3 &:= \ln(0.5640174)\end{split}\]

The number of exacerbations predicted by the model is then:

\[\begin{split}N_{\text{exac}}^{\text{(pred)}} &= \lambda_C \cdot N_{\text{asthma}} \\ N_{\text{asthma}} &= N \cdot \eta_{\text{prev}}\end{split}\]

\(N_{\text{asthma}}\): the number of people in a given time interval, age, sex with asthma
\(N\): the number of people in a given time interval, age, and sex
\(\eta_{\text{prev}}\): the prevalence of asthma in a given time interval, age, and sex, from Occurrence Model 1: Crude Occurrence

and number of hospitalizations is:

\[N_{\text{hosp}}^{\text{(pred)}} = N_{\text{exac}}^{\text{(pred)}} \cdot P(\text{hosp})\]

\(N_{\text{exac}}^{\text{(pred)}}\): the predicted number of exacerbations (of any severity) for a given time interval, age, and sex
\(P(\text{hosp})\): the probability of hospitalization due to asthma given the patient has an asthma exacerbation

Finally, \(\alpha\) can be computed:

\[\alpha(a, s, y) = \dfrac{N_{\text{hosp}}(a, s, y)}{N_{\text{hosp}}^{\text{(pred)}}(a, s, y)}\]