# Progression of the COVID outbreak in South Australia

Last updated: 2022-09-09 15:17 Adelaide timeUpdates daily

Contents

*R*

_{eff}estimate from all cases

*R*_{eff} estimate from all cases

South Australia is currently experiencing an outbreak of COVID, due to the omicron variant. The virus has been spreading in South Australia since it opened its borders on November 23rd to the states and territories with active COVID outbreaks.

The below plot shows how the effective reproduction number of the virus,
*R*_{eff} has changed over time in South Australia, as well as how
the daily cases have changed over time. An exponential trendline shows the
approximate trajectory daily case numbers would follow, were the reproduction
number of the virus to remain at its current level.

The same plot is shown twice, the first with cases on a linear scale, and the second on a log scale—the latter showing how consistently the caseload has followed exponential growth (which forms a straight line on a log scale).

### SIR model projection

The trendlines in the above plots simply project forward case numbers assuming that
*R*_{eff} remains at its current estimated value. What about the
effect of built-up immunity? Below is a plot with the projected trend taking into
account an estimated reduction of spread due to increasing immunity from
infections, which slows transmission once a non-negligible fraction of the
population has been infected. The projection assumes that only 35% of infections
are captured by testing, as such, for every projected case there are an additional
three to four infections implied that are not detected through testing.

This projection shows what might be possible if all other factors affecting
*R*_{eff}—such as restrictions—are held constant. If restrictions
are tightened, then *R*_{eff} will likely decrease and
growth will slow, resulting in a later and lower peak in cases than the current
projection.

Even if restrictions do not change, people may modify their behaviour once
caseloads become higher, which also may reduce *R*_{eff}.

This type of model is very simplistic, and should be taken with a grain of salt. The number of actual cases detected will depend strongly on what fraction of infections come forward for testing, which may change over time. Furthermore, this model treats the entire state as homogenous, whereas in reality some geographic areas or social networks will develop high immunity and thus slower growth before others. This can significantly affect the overall rate of spread, but is not captured in a simple model such as this.

The various effects not included in the model are also not included in the
projection's uncertainty range, which is solely due uncertainty in the current
*R*_{eff} estimate and average caseload.

The same plot is shown twice, the first with cases on a linear scale, and the second on a log scale.

### Animated projections over time

How have the above projections changed over time? Below are animated versions of the above projection, one with cases on a linear scale, and one with cases on a log scale, run on old data to show how the projections have changed over time. Note that these are not 100% identical to the projections actually made on previous days, as there have been some slight methodology changes - but they should be very close.### Disclaimer on trends

The plotted exponential trendlines in the above plots are simple extrapolations of
what will happen if *R*_{eff} remains at its current value. the SIR
model projection, although it accounts for the effect of increasing immunity,
still assumes that *R*_{eff} otherwise remains constant.

This does not take into account that things are in a state of flux. If
restrictions are tightened, the virus should have fewer opportunities for spread,
and *R*_{eff} will decrease. If restrictions are eased, it may
increase. Contact tracing may suppress spread to a greater or lesser degree over
time. The above plots specifically showing the effect of population immunity do
take into account a reduction in *R*_{eff} as immunity due to
infections increases, but ignores any other possible future changes in
*R*_{eff}.

Furthermore, when case numbers are small, the random chance of how many people
each infected person subsequently infects can cause estimates of
*R*_{eff} to vary randomly in time. As such the projections should be
taken with a grain of salt—it is merely an indication of the trend as it is right
now.

### Methodology

#### Padding, smoothing, and calculating *R*_{eff}

Before calculating *R*_{eff}, the daily case numbers are padded on the
right and smoothed. A symmetric smoothing method is used, which is why padding on the
right is necessary before smoothing.

The padding is an extrapolation based on a fit to the most recent 14 days of daily
case numbers *n*(*t*), with the following fit function:

*n*(*t*) = *A* exp[
*k*(
*t*-*t*_{today})
+ 1/2 *k̇* (*t*-*t*_{today}
)^{2}
]

where *A* and *k* are the fit parameters for the current daily caseload
and exponential growth rate, and *k̇* is the current expected rate of change of
the exponential growth rate due to vaccines:

*k̇* = 1 / 𝜏 (*s*_{vax}(*t*_{today}) /
*s*_{vax}(*t*_{yesterday}) - 1)

where 𝜏 = 5 days is the approximate mean generation time of the virus, and
*s*_{vax}(*t*_{today}) and
*s*_{vax}(*t*_{yesterday}) are the fraction of the
population not protected by vaccination from infection today and yesterday. See vaccination assumptions below for how
*s*_{vax}(*t*) is calculated from vaccination numbers.

After padding, the daily case numbers—after adding an offset and taking their log— are smoothed with 4-day Gaussian smoothing. The result is then exponentiated and the offset subtracted off again to obtain smoothed daily case numbers. This is a compromise between ordinary Gaussian smoothing for small case numbers, and Gaussian smoothing in log space for larger case numbers.

log(*n*_{smoothed}(*t*)+10) = log(*n*(*t*)+10) ∗
(2𝜋*T*_{s}^{2})^{-1/2} exp(-*t*^{2} /
2*T*_{s}^{2})

where *T*_{s} = 4 days and ∗ is the convolution operation.

*R*_{eff} is then calculated for each day as:

*R*_{eff}(*t _{i}*) =
(

*n*

_{smoothed}(

*t*) /

_{i}*n*

_{smoothed}(

*t*

_{i-1}))

^{𝜏}.

The uncertainty in *R*_{eff} has contributions from the uncertainty in
the above-mentioned fit, as well as uncertainty in daily case numbers. The latter is
considered to be Poisson noise scaled by a constant, chosen so as to make the
reduced chi squared between raw and smoothed daily case numbers equal to 1.0.

#### Trend based on *R*_{eff} only

The plot(s) with a simple exponential extrapolation of daily case numbers, without
taking into account the expected effects of vaccines or population immunity, is
exponential growth/decay using the most recent value of *R*_{eff} and
its uncertainty range:

*n*_{extrap} (*t _{i}*) =

*n*

_{smoothed}(

*t*

_{today})

*R*

_{eff}(

*t*

_{today})

^{(ti - ttoday) / 𝜏}

#### SIR model with projected effect of vaccines

To project the effect of vaccines and community immunity on the progression of daily cases in the future, I use a stochastic SIR model. The model is run 1000 times to project daily case numbers, and the median result and 68% range of outcomes shown in the projection.

In each run of the model, a random value for each of
*R*_{eff}(*t*_{today}) and
*n*_{smoothed}(*t*_{today}) is drawn from a multivariate
normal distribution, using the mean values and covariance of the estimates of the two
parameters.

The initial infectious population is taken to be:

*i*(*t*_{today}) =
𝜏 *n*_{smoothed}(*t*_{today})

The initial recovered population *r*(*t*_{today}) is taken to be the
cumulative number of cases so far in the outbreak, minus the currently infectious
population.

Each day, the fraction of the population susceptible to infection *s*(*t*) is
taken to be

*s*(*t*) = *s*_{vax}(*t*) [1 - (*r*(*t*) +
*i*(*t*)) / *N*]

where *s*_{vax}(*t*) is the fraction of the population not
protected from infection by vaccines (see See below
for how this is calculated), and *N* is the total population multiplied by the
fraction of infections assumed to come forward for testing (35%).

Assuming that *R*_{eff} is proportional to the susceptible fraction of
the population, we can estimate what the reproduction number of the virus would be
in the absence of any vaccines or immunity from infection:

*R*_{naive} = *R*_{eff}(*t*_{today}) /
*s*(*t*_{today})

And compute an *R*_{eff} for each future day based on the susceptible
fraction of the population that day:

*R*_{eff}(*t*) = *s*(*t*) *R*_{naive}

Each day of the model, the infectious population is increased by the number of new infections, and decreased by the number of new recoveries. The recovered population is increased by the number of new recoveries.

The number of new infections each day is drawn from a Poission distribution with mean
equal to *i*(*t*) *R*_{eff}(*t*) / 𝜏.

The number of new recoveries each days is drawn from a binomial distribution with
number of trials equal to the infectious population *i*(*t*), and with
probability 1 / 𝜏.

#### Vaccination assumptions

To model vaccines taking effect after a delay, daily vaccine dose numbers
*v*(*t*) are convolved with a Gaussian function offset 1.5 weeks in the
future, with standard deviation 0.5 weeks:

*v*_{effective}(*t*) = *v*(*t*) ∗
(2𝜋*σ*^{2})^{-1/2} exp(-(*t* - *μ*)^{2} /
2*σ*^{2})

where *μ* = 10.5 days, *σ* = 3.5 days, and ∗ is the convolution
operation. This causes the effect of a vaccine dose to over the course of the
second week after it is adminsitered, reaching nearly full effectiveness
approximately two weeks after administration.

I assume that one dose of any vaccine reduces spread by 40%, and two doses reduce spread by 80%. The proportion of the population not protected by the vaccine against infection is then:

*s*_{vax}(*t _{i}*) = 1 - 0.4 ×

*d*(

*t*

_{i})

where *d*(*t*_{i}) is the cumulative number of doses per
capita on each day.

As of Dec 2021, these vaccine assumptions are not particularly relevant to the projections, as the 1st and 2nd dose rollouts are mostly complete, and omicron likely escapes most vaccine-induced immunity against infection in any case. I am not yet including a modelled effect of 3rd doses, but may do so if the omicron waves last long enough for it to plausibly make a difference to spread, or if 3rd doses are brought forward in time.

### Data sources and contact

Source for case numbers: covidlive.com.au

Plots/analysis by Chris Billington. Contact: chrisjbillington [at] gmail [dot] com

Python script for producing the plots can be found at GitHub.