What’s new?
Today’s announcement was 507 new local cases in Victoria is very compatible with the Richards’ growth curve models, and less compatible with the Gompertz model. The two models still have relatively good fit with the data, but provide different and incompatible projections for the next week.
The Gompertz model projects an ongoing trajectory of case growth. The Richards’ growth curve model projections that the peak in new cases is approaching, and maybe even has been reached. Both models are still plausible, but my interpretation of the analysis patterns is that the trend indicated by Richards’ growth curve (i.e., that the cases have slowed and possibly peaked) is more likely to be true.
Which is correct is likely to become obvious in the coming days. The Gompertz model estimates around 5000-5500 new cases in the coming week. The Richards’ growth curve model projects substantially few new cases, which would be what is expected if the peak is indeed approaching. Like the Sydney outbreak projections, if the Richards’ growth curve model is correct, it is likely to be overly optimistic in the near future. That is, the projected rapid decline in cases will likely be substantially slower than currently indicated in the projection. This seems to be a bias in the model.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 19 September 2021
What is this?
Green charts are the Richards’ growth curve model. Blue charts the the Gompertz model. For each colour, 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 4 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 interesting pattern of the Richards’ growth curve 7-day projection error (bottom left panel) moving from an underestimate towards 0% error, and then overestimate in a smooth (almost “linear”) pattern was seen as the Sydney outbreak reached its peak number of cases. This gives me some hope that we may truly be approaching the peak in new cases in Victoria.
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 19 September 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. There is an indication of “peaking” in both the simple moving average and GAM charts, which may indicate that the Richards’ growth curve model is a better description of the trend. At present, there is notable uncertainty in the models. The Gompertz and Richards’ growth curve models are incompatible with each other at the moment.
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) -0.374276 0.053033 -7.0574 1.203e-08 ***
c:(Intercept) -45.984790 14.589149 -3.1520 0.002989 **
d:(Intercept) 9718.794921 547.504762 17.7511 < 2.2e-16 ***
e:(Intercept) 47.913140 0.452465 105.8935 < 2.2e-16 ***
f:(Intercept) 0.278154 0.042258 6.5822 5.772e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
37.68248 (42 degrees of freedom)
Gompertz model
summary(model.g)
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -2.7458e-02 2.2017e-04 -124.7136 < 2.2e-16 ***
c:(Intercept) 1.1870e+02 1.9780e+01 6.0009 3.645e-07 ***
d:(Intercept) 1.3744e+05 5.4723e+03 25.1155 < 2.2e-16 ***
e:(Intercept) 8.5776e+01 6.3587e-01 134.8941 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
76.95162 (43 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