What’s new?
Today’s announcement was 1466 new local cases is rather lower than projected, which is very welcome. This might indicate that we are approaching the peak in this outbreak in Melbourne and I am quietly hopeful. The Gompertz and Richards’ growth curve models give different and incompatible trajectories. The Gompertz model suggests ongoing growth. The inclusion of the last two day’s data points has resulted in a major trajectory change in the Richards’ growth curve model, suggesting that we are approaching, or have approached, the peak.
We do need to be cautious, as we have seen this pattern before, in mid-September. The case numbers appeared to be peaking, but then took off again. This could potentially be explained by the large public protests which occurred in mid- to late- September: https://en.wikipedia.org/wiki/COVID-19_protests_in_Australia#2021_2
Which model is a better description of reality will become increasingly obvious in the coming days.
The Gompertz model estimates around 15,000-16,000 new cases in the coming week, while the Richards’ growth curve model estimates 13,000-17,000.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 12 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 models have been underestimating but seen to be “catching up”.
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 12 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.
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) -1.3100e-01 1.6213e-02 -8.0803 1.250e-10 ***
c:(Intercept) -8.0474e+02 1.1717e+02 -6.8680 9.654e-09 ***
d:(Intercept) 7.8363e+04 8.9407e+03 8.7648 1.118e-11 ***
e:(Intercept) 7.8742e+01 1.5164e+00 51.9285 < 2.2e-16 ***
f:(Intercept) 5.4371e-01 7.5682e-02 7.1842 3.092e-09 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
205.9913 (50 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) -2.0416e-02 1.0032e-04 -203.5201 < 2.2e-16 ***
c:(Intercept) 3.7567e+02 8.4881e+01 4.4259 5.076e-05 ***
d:(Intercept) 5.2862e+05 1.4266e+04 37.0545 < 2.2e-16 ***
e:(Intercept) 1.1887e+02 5.1739e-01 229.7419 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
277.4954 (51 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