If you’ve been following The Dossier you’ll know that I’ve been collecting data on the Covid-19 omicron outbreak that we are in the midst of. I’ve made observations that confirm that omicron causes milder symptoms than other forms of Covid-19. But the less severe nature of omicron is offset, at least in part, by omicron being highly transmissible. Even though on average omicron causes less severe symptoms, there is always a percentage of individuals that show more severe symptoms than the average.
This percentage, though small, can easily represent a large number of people presenting to hospitals and requiring treatment in ICU’s across the country.
Indeed, I’ve shown in my article: Further Steep Increases in Covid-19 Cases, observations that there are increased hospitalisations already occurring in NSW as well as a worrying uptick in hospitalisations and deaths in other States and Territories. Given these factors, quantitatively determining the transmissibility of the omicron variant is critical to the management of the current outbreak.
In this article, we’ll attempt to do just that using the Covid-19 data presented in the chart below which shows case numbers for Australian States and Territories over the month ending Friday 7th January. As you can see from this chart, case numbers are showing rapid growth but is that growth exponential?
Source: https://www.covid19data.com.au/cases-last-28-days
The early phase of an outbreak of an infectious agent is often characterised by exponential growth. This has been investigated and found to be true for the original Covid-19 outbreak in countries such as Italy and Ukraine [1]. It’s of interest to know whether the same exponential growth would apply for the spread of a new Covid-19 variant, omicron, in a population that has been widely vaccinated against earlier forms of the virus (the 12+ population in Australia 90% or more have been fully vaccinated).
Mathematically, exponential growth follows the equation:
where V is the number of cases, t is time expressed in days, c and b are regression parameters to be determined from the observed data (chart above). The time frame chosen for the analysis is arbitrarily selected from 16th Dec 2021 to 7th Jan 2022.
The significance of the mathematical modelling is to obtain td which is the time-to-double the number of infections, which is given by:
where ln 2 is the natural logarithm (base e) of 2 and c is the first of the parameters found from regression analysis.
The exponential fitting results are shown in the chart below, for NSW (solid blue line) and for VIC (solid red line). The observed case data are shown with blue circles (NSW) or red circles (VIC).
Exponentially fitted data for NSW (blue) and VIC (red) from 16th Dec to 7th Jan.
I’ll present the statistics for goodness-of-fit a bit later in this article but worth just spending a moment examining the chart. The NSW data appears to fit the exponential curve quite well from visual inspection. On the other hand, the VIC data fits obviously less well. The VIC data might fit better if we had selected a time period that captured the time of outbreak better. For instance, the 21st Dec to 10th Jan.
The other thing to notice from the above chart is that the exponential increases to infinity which, of course, is impossible. This is because when the virus outbreak first occurs there are large numbers of hosts for it to infect; its growth is unrestrained. After a while the virus runs out of new hosts in a given locality because hosts are sickened, hosts recover and gain immunity from further infection, or in a worst-case scenario, hosts die.
We’ll consider how to model later stages of the current Covid-19 wave of infections in a later article. For the moment we’re only considering the initial growth phase of the virus for which an exponential law applies.
Linear regression of log10 no. of cases versus time in days for States and Territories.
The standard method to carry out an exponential fit is to “linearise” the data by first taking log10 of the V data (the number of cases) vs x (time in days) as is shown in the chart above for States and Territories, except WA. The time period is the 16th Dec to 7th Jan, as before.
The regression equation used, in the analysis of case data for each State and Territory, in the chart above, is as follows:
where the constant term = 0434285 arises from converting between loge and log10. The important parameter is c from which we can determine td the time-to-double the number of infections for the current (mostly) omicron outbreak.
Statistical Analysis
Linear regression analysis of the Covid-19 data was carried out using the LINEST array function in Microsoft Excel®. The important results are summarised in the table below:
Table: Regression Summary
| State | c | td | r² | F-test |
| NSW | 0.063 | 11.0 | 0.97 | 0.0000 |
| VIC | 0.059 | 11.8 | 0.88 | 0.0000 |
| QLD | 0.128 | 5.4 | 0.96 | 0.0000 |
| SA | 0.094 | 7.4 | 0.93 | 0.0000 |
| TAS | 0.145 | 4.8 | 0.97 | 0.0000 |
| NT | 0.093 | 7.5 | 0.88 | 0.0000 |
| ACT | 0.096 | 7.2 | 0.94 | 0.0000 |
Table of results from the statistical analysis.
From the slope parameter, c, the time-to-double, td, has been calculated using the expression above. The values of td, of 11 and 12 days for NSW and VIC, respectively, have been found For QLD and TAS the td, values are 5.4 and 4.8 days respectively. The r² values returned from LINEST are also shown in the table above. All r² values are close to 1.0 indicating that the model accounts for most of the variability in the y = log10 case data. The F-values returned from LINEST have been converted to probabilities using the Excel FDIST function and reported in the table above [2]. The linear models reported here for all States and Territories are all highly significant at the α = 0.05 level.
By way of comparison, the original Covid-19 outbreak in Feb to Apr 2020 the value of td, was 2.31 days for the USA which was the fastest spreading region worldwide. For Covid-19 originally, values of td, were 2.91 and 3.65 days in Europe and globally, respectively [1]. Despite our best efforts in Australia to vaccinate most of the population (currently at 90% for 12+). Despite control measures such as social distancing and wearing masks. Covid-19 had found a way to spread itself about half to a third as rapidly today, with the omicron variant, as it did when it originally surfaced in Feb 2020 when there were no community immunity and scarcely any control measures to stop it.
Conclusions
The work reported here has demonstrated that the omicron variant of Covid-19 is spreading exponentially. The results found confirm that an exponential growth model for Covid-19 fits the case data with a high-level of significance.
The time-to-double for the exponential growth for omicron in States and Territories varies from 5 days to 12 days. These findings were found for case data reported during the period 16th Dec to Jan 7.
The most surprising finding of this work is that despite national efforts in vaccination of 90% of the population, most of the population exercising social distancing and wearing masks, that the spread of Covid-19 through the omicron variant has only been slowed by a factor of around a half to a third of what it was during the original outbreak of Covid-19 in Feb-Apr 2020.
The virus is continues to surprise us with it’s resilience to our efforts to control and eliminate it.
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[1] Nesteruk I. COVID-19 Pandemic Dynamics: Mathematical Simulations. Springer Nature; 2021.
[2] For F(v1, v2), v1 = m – 1, where m = the number of parameters to fit, in this linear model: c and b, so m = 2; v2 = n – m – 1 where n = the number of case data, so n = 22.