What’s new?
Today’s announcement was 1838 new local cases is consistent with the new Gompertz model. 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 Richards’ growth curve model has significant uncertainty.
I expect that the models will improve as more data is included. The Gompertz model estimates around 13,000-14,500 new cases in the coming week.
It should be noted that the model and projections assume that there isn’t a major change in the underlying transmission dynamics. For instance, in the setting of major policy changes (e.g., reducing measures that control transmission), it would be expected that the model will give underestimates.
Projection of new daily cases, and cumulative counts of COVID-19 with data up to 17 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 17 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) -9.2218e-02 2.7965e-02 -3.2977 0.003148 **
c:(Intercept) 3.3419e+03 1.5442e+03 2.1642 0.041079 *
d:(Intercept) 8.0214e+04 1.2207e+04 6.5711 1.051e-06 ***
e:(Intercept) 6.9374e+01 7.1955e+00 9.6414 1.516e-09 ***
f:(Intercept) 1.3072e+00 8.7497e-01 1.4940 0.148762
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
213.9493 (23 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) -4.7518e-02 2.0862e-03 -22.777 < 2.2e-16 ***
c:(Intercept) 5.8198e+03 3.3069e+02 17.599 3.151e-15 ***
d:(Intercept) 1.1811e+05 6.4863e+03 18.210 1.410e-15 ***
e:(Intercept) 7.5995e+01 1.0906e+00 69.682 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
207.5897 (24 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