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# Progression of the COVID outbreak in the Australian Capital Territory

**New: Progression of the COVID outbreak in New Zealand**

**New: Progression of the COVID outbreak in Victoria**

**See also: Progression of the COVID outbreak in New South Wales**

**See also: Australian vaccination rollout**

**See also: The road to a COVID-free Victoria (old 2020 second wave plots)**

Contents

*R*

_{eff}estimate from all cases

*R*_{eff} estimate from all cases

The Australian Capital Territory is currently experiencing an outbreak of COVID, due to the delta variant. Since August 12th, the Australian Capital Territory has been under lockdown restrictions.

How have restrictions in the ACT affected the spread of the virus? The below
plot shows how the effective reproduction number of the virus,
*R*_{eff} has changed over time in the ACT, as well as how
the daily cases have changed over time. A 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).

### Projected effect of vaccine rollout

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 vaccines? Below is a plot with the projected trend taking into account an
estimated reduction of spread due to vaccination levels increasing in the ACT
through the remainder of the year. This projection is a stochastic SIR model,
and thus also takes into account immunity due to infections, which would decrease
transmission once a non-neglibible fraction of the population has been infected.

The rate of vaccination assumed in the model is 1.3 doses
per 100 population per day, up to a maximum of 85% of the
population. See below for plots of how these assumed rates compare to the
vaccination rollout in the ACT so far. The vaccines are assumed to reduce
*R*_{eff} by 40% per dose per capita—that is, overall spread is
reduced by 40% for a partly-vaccinated individual and 80% for a fully vaccinated
individual.

This projection shows what might be possible if all other factors affecting
*R*_{eff}—such as restrictions—are held constant. If the ACT
eases restrictions, however, then *R*_{eff} will likely increase and
the effect of vaccines in reducing spread will be delayed compared to this
projection. On the other hand, if contact tracing is successful in decreasing
*R*_{eff}, case numbers may decline faster than this projection.

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

Below are plots of the vaccination rollout in the ACT to date, in terms of daily and cumulative doses per 100 population, with the assumed future rate used for the above projections also shown.

### 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 trendlines are simple extrapolations of what will happen if
*R*_{eff} remains at its current value. This does not take into account
that things are in a state of flux. As restrictions take effect, 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 vaccines do take into account a reduction in *R*_{eff} as
vaccination coverage increases, but ignore 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 projection should be taken with a grain of
salt—it is merely an indication of the trend as it is right now.

### Methodology

#### Smoothing, calculating *R*_{eff} and projections

Daily case numbers have been smoothed with 4-day Gaussian smoothing:

*N*_{smoothed}(*t*) = *N*(*t*) ∗
(2*πT*_{s}^{2})^{-1/2} exp(-*t*^{2} /
2*T*_{s}^{2})

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

Before smoothing, the daily case numbers are padded on the right with an extrapolation based on a exponential fit to the most recent 14 days of data.

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

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

*N*

_{smoothed}(

*t*) /

_{i}*N*

_{smoothed}(

*t*

_{i-1}))

^{Tg}

where

*T*

_{g}= 5 days is the approximate generation time of the virus.

The uncertainty in *R*_{eff} has contributions from the uncertainty
in the above-mentioned exponential 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.

The extrapolation of daily case numbers is based on 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)/Tg}

#### Vaccination model

**Note 2021-07-30**: I have
changed the model used for the projections of the vaccine rollout to a stochastic
SIR model, and the description below is out of date. I haven't yet documented the
new model on this page, but the code can be seen here
in the meantime. The projected outcomes with the new model are very similar to the
old model, which was valid in the limit of infections only reaching a small
fraction of the total population.

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

*V*_{effective}(*t*) = *V*(*t*) ∗
(2*πσ*^{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 susceptible to the virus is then:

*s*(*t _{i}*) = 1 - 0.4 ×

*D*(

*t*

_{i})

where

*D*(

*t*

_{i}) is the cumulative number of doses per capita on each day.

*R*_{eff} is then estimated for any future date as:

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

*s*(t

_{today})

And case numbers extrapolated from one day to the next, beginning with
*N*_{smoothed}(*t*_{today}), according to:

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

*N*

_{extrap}(

*t*)

_{i-1}*R*

_{eff}(

*t*

_{i})

^{1/Tg}

### 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.