What’s new?
Today’s announcement was 1965 new local cases is the highest yet, and unfortunately remains consistent with the trend of ongoing growth in cases.
Today, I revised the models used in the Victorian COVID-19 outbreak (August 2021) resulting in the sixth Victorian lockdown. Those who have followed this modelling series would have observed that the original models, which fitted the cumulative case numbers starting from 4 August 2021 (the day immediately prior to the first case) to the Gompertz equation, and the Richards’ growth curve using non-linear regression, wasn’t working well in the past week. This was particularly the case with the Richards’ growth curve model. I believe this is due to a substantial change in the underlying growth dynamics from the beginning of the series in early August, to more recent times.
In the new models, the model only includes the cumulative case numbers starting from the 19 August 2021. The rationale for this choice is that this is the day after the first of a number of large public protests in Victoria: https://en.wikipedia.org/wiki/COVID-19_protests_in_Australia#2021_2
One of the assumptions when fitting a the cumulative case numbers to a simple epidemic growth curve is that the underlying growth dynamics driving the observed phenomenon will largely remain constant throughout the series. There has been a clear escalation in cases and a change in the trajectory recently. The very early case numbers and trajectory have lesser direct mechanistic impact on the current and future numbers, and if anything, provide a bias given the assumptions inherent in modelling using regression to the epidemiologic growth curves.
Both the new Gompertz and Richards’ growth curve models show ongoing growth in cases over the next seven days without an indication that the peak will arrive soon. The magnitude of the growth in cases is different between the models, and it isn’t clear which is more likely to be more accurate, though it does seem that the Richards’ growth curve model may have a better fit to the recent data.
The Gompertz model estimates around 12,000-13,500 new cases in the coming week, while the Richards’ growth curve model estimates 15,000-16,500.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 9 October 2021
What is this?
Green charts are the Richards’ growth curve model, and the blue charts the the Gompertz model. The top image of each colour is a chart of the cumulative (total) COVID-19 cases in Victoria, starting from 4 August 2021, and the lower image is a chart of the daily new cases. Only local cases are included (i.e., excluding cases identified in quarantine). Projections are given for the next 7 days. It should be noted that estimates have high levels of uncertainty beyond a few days and must be interpreted cautiously.
The projections are made using a model by fitting the cumulative case data since 19 August 2021 to a Richards’ growth curve and Gompertz equation using non-linear regression. The dark central dashed lines are the model estimates, with 95% confidence intervals of the estimate. On the lower chart, the colour gradations can be understood as the degree of uncertainty in the model projections.
Gompertz and Richards’ growth curve
The Gompertz function is a type of sigmoid, or “S”-shaped curve. It’s been around since the early 19th century and was initially used to describe and model demographic mortality curves, and hence, well known to actuaries. The Gompertz function can also be used to accurately model biological growth (e.g., epidemics, tumour size, enzymatic reactions). I have chosen to use this model to help with creating insights as earlier in the pandemic, it was found to be useful in modelling cumulative cases of COVID-19 from the Chinese outbreaks (Jia et al. arXiv:2003.05447v2 [q-bio.PE]).
The Richards’ growth curve (or the generalised logistic function), which is a broad family of sigmoid (S-shaped) curves that can describe well many types of growth, including epidemics. It has also been demonstrated to have utility in modelling COVID-19 outbreaks in 2020 (Lee et al. PLoS One 2020 doi: 10.1371/journal.pone.0236860).
Gompertz equation:
Richard’s growth curve / generalised logistic function:
- B, C, D, E, F are the parameters of the model
- e is Euler’s number
- x in the model is the date (day number)
- These are the parameterisation of these functions used in the drc package: https://cran.r-project.org/web/packages/drc/drc.pdf
Why the changes?
I’ve undertaken some assessment of the degree of predictive error in both the Gompertz and Richards’ growth curve models. These charts compare the 7- and 14-day total case projections of the models, to what actually occurred in reality 7 and 14 days later. For interpretation, above the 0% error line means that the model provided an over-estimate compared to reality, and below the 0% error line an underestimate.
Like seen in the Sydney 2021 models in early July, the very early data from about the first month results in models that lack stability, with a tendency to underestimate, before “catching up” with the data. The Richards’ growth curve model has been quite unstable and is currently broken. The Gompertz model projections had been describing the case series well but has started underestimating.
Daily case trends
Comparison between the Gompertz and Richards’ growth curve model projections, along with smoothed data trends (7-day simple moving average, and GAM) with data up to 9 October 2021
The generalised additive model gives a descriptive “reality check” to the models. The GAM can be considered as an advanced smoothed trend of the daily counts. The models seem to be doing well but there is some indication that the Gompertz model is relatively underestimating.
Model summaries
Richards’ growth curve model
summary(model.r)
Model fitted: Generalised logistic (ED50 as parameter) (5 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -8.0568e-02 3.7118e-03 -21.706 < 2.2e-16 ***
c:(Intercept) -8.4165e+02 9.9477e+00 -84.608 < 2.2e-16 ***
d:(Intercept) 1.7963e+05 1.0000e+01 17963.369 < 2.2e-16 ***
e:(Intercept) 8.9728e+01 5.5112e-01 162.811 < 2.2e-16 ***
f:(Intercept) 8.8838e-01 4.9794e-02 17.841 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
185.1624 (47 degrees of freedom)
Gompertz model
summary(model.g)
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -2.1761e-02 1.4485e-04 -150.2242 < 2.2e-16 ***
c:(Intercept) 3.8847e+02 8.8686e+01 4.3803 6.405e-05 ***
d:(Intercept) 4.1041e+05 1.3652e+04 30.0624 < 2.2e-16 ***
e:(Intercept) 1.1161e+02 6.5646e-01 170.0241 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
282.7926 (48 degrees of freedom)
Want to know more?
Primary data source is from Victoria Government Department of Health for daily new cases. The analysis is performed using RStudio Cloud using R version 4.1.0.
Today’s charts
Data: au_covid_vic
R code: models_vic