What’s new?
Today’s announcement was 2287 new local cases, the highest on record. It is, however, quite consistent with the models, especially the new Gompertz model. My interpretation is that this seemingly spike in cases needs to be interpreted in the light that the lower than expected case numbers from the past couple of days were likely artifactually low.
Both models indicate that we are approaching the peak in this outbreak in Melbourne. With the more limited data, I’m not very confident with the projection estimates, other than the general shape of their trajectories. The rapid fall in cases in the Richards’ growth curve model isn’t plausible, however, my intuition is that the Gompertz curve might be overly conservative and is overestimating.
It should be noted I made a substantial change in the model on 13 October 2021. A few days prior to this, I changed the model to fit data from later in the outbreak, from mid-August 2021, via non-linear regression to the Gompertz and Richards’ growth curves. The logic was the the transmission dynamics and context of the outbreak at changed with a number of large public protests which might have resulted in new infections. Reviewing the history of this further, it actually made more sense to limit the series to start from mid- to late-September 2021, which was when a series of large protests occurred in a run. The model now only includes data from 20 September 2021 onward.
I expect that the models will improve as more data is included. The Gompertz model estimates around 12,000-14,000 new cases in the coming week.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 14 October 2021
What is this?
The blue charts the the Gompertz model. The top image 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 20 September 2021 to a 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.
As I’ve included a minimum of 14 days of data for each of the models (with data starting 20 September 2021), we only have a few day’s worth of 7-day projections. The Gompertz model is performing quite well. Insufficient number of days have passed to evaluate the 14-day projection errors.
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 14 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 both seem to indicate that we are approaching the peak. The rapid and sharp fall off in the Richard’s growth curve projection is not plausible.
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) -1.9048e-01 4.3612e-02 -4.3676 0.0002979 ***
c:(Intercept) 2.3854e+03 1.0436e+03 2.2857 0.0333198 *
d:(Intercept) 5.2763e+04 4.2674e+03 12.3642 8.002e-11 ***
e:(Intercept) 7.1916e+01 9.1752e-01 78.3804 < 2.2e-16 ***
f:(Intercept) 4.6906e-01 1.6176e-01 2.8997 0.0088600 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
177.6716 (20 degrees of freedom)
Gompertz model
summary(model.g)
Gompertz model - model.g
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -5.3098e-02 3.1173e-03 -17.033 9.058e-14 ***
c:(Intercept) 6.2931e+03 3.6192e+02 17.388 6.038e-14 ***
d:(Intercept) 1.0103e+05 7.3489e+03 13.748 5.708e-12 ***
e:(Intercept) 7.3090e+01 1.3096e+00 55.812 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
207.5825 (21 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