What’s new?
Today’s announcement was 867 new local cases in Victoria fits very well with the Gompertz model and is less compatible with the Richards’ growth curve model. The Richards’ model suggests the approaching of a peak, but this has proven incorrect over the past week. Although a peak must occur at some time in the future, when this is likely to occur is uncertain and the models are likely unreliable.
The Gompertz model projects an ongoing trajectory of case growth, although the rate of growth may have slowed. The model estimates around 6500 new cases in the coming week.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 28 September 2021
What is this?
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 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 Richards’ growth curve model has been quite unstable in the last few days, which is an indication that its projections (from 7 days and 14 days ago) have been a poor description of reality. The Gompertz model projections have been describing the case series well with high accuracy for over a week.
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 28 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. It looks like that the Richards’ growth curve peak is premature. 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) -7.1867e-02 1.1037e-02 -6.5115 3.216e-08 ***
c:(Intercept) 2.8948e+00 3.0452e+01 0.0951 0.9246393
d:(Intercept) 3.9958e+04 5.9982e+03 6.6616 1.864e-08 ***
e:(Intercept) 5.2140e+01 2.7909e+00 18.6819 < 2.2e-16 ***
f:(Intercept) 1.8515e+00 4.6987e-01 3.9404 0.0002482 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
65.17146 (51 degrees of freedom)
Gompertz model
summary(model.g)
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -2.8284e-02 2.0147e-04 -140.3908 < 2.2e-16 ***
c:(Intercept) 1.1055e+02 1.7306e+01 6.3882 4.67e-08 ***
d:(Intercept) 1.2023e+05 2.6272e+03 45.7637 < 2.2e-16 ***
e:(Intercept) 8.3049e+01 5.0936e-01 163.0459 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
76.25605 (52 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