What’s new?
Today’s announcement was 445 new local cases in Victoria is mostly compatible with both the Gompertz and Richards’ growth curve models, though it is still early days. It is likely that both models are underestimating, but improving, especially the Richards’ growth curve model. The projections of the cumulative number of cases for today from 7 days ago underestimated by 21% and 6% for the Gompertz and Richards’ growth curve models respectively.
It is not clear which is the better model, though the Richards’ growth curve model is more likely to be correct in the very short term (the model has substantially lower residuals), but the Gompertz model might be more likely to be correct in the slightly longer term trend. Although the Richards’ growth curve model seems to suggest a peak in the near future, this is highly unreliable with early data.
Daily case numbers are likely to continue to grow. In the next week, the Gompertz model estimates around 3500-4000 new cases, while the Richards’ growth curve model estimates 4000-6000. If we factor in the historical underestimating, then both models suggests around 5500 new cases. This might be the more likely magnitude of new cases.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 14 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.
Q: Why are the confidence intervals so wide on the Richards’ growth curve?
A: The confidence intervals are computed by varying the parameter estimates (Richards’ curve has five parameters, and the Gompertz curve has four) in a normal distribution according to their standard error from the non-linear regression, and then simulating new curves by bootstrapping. The parameter in the Richards’ growth curve that defines the upper asymptote (“d” from below) has a very large standard error with the limited data in regression, which results in curves (and hence, a confidence interval) that includes regions that are not especially compatible with the data.
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.
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 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. At present, there is notable uncertainty in the models.
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) -3.4084e-01 6.3809e-02 -5.3416 4.890e-06 ***
c:(Intercept) -5.0557e+01 1.3091e+01 -3.8619 0.0004369 ***
d:(Intercept) 1.8039e+04 8.9250e+03 2.0211 0.0505454 .
e:(Intercept) 5.4058e+01 4.8758e+00 11.0869 2.560e-13 ***
f:(Intercept) 3.0237e-01 5.6533e-02 5.3486 4.785e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
35.70463 (37 degrees of freedom)
Gompertz model
summary(model.g)
Model fitted: Gompertz (4 parms)
Parameter estimates:
Estimate Std. Error t-value p-value
b:(Intercept) -2.7549e-02 3.1667e-04 -86.9943 < 2.2e-16 ***
c:(Intercept) 1.0834e+02 2.3598e+01 4.5911 4.719e-05 ***
d:(Intercept) 1.2773e+05 9.5511e+03 13.3735 5.216e-16 ***
e:(Intercept) 8.4902e+01 8.9459e-01 94.9061 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
79.82352 (38 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