# COVID-19: A Changing Curve

As was the case a few weeks ago, we’ve recently made some changes to the Monitoring COVID-19 In Canada Dashboard. At the time, the changes provided more functionality. Today, we’ve updated the dashboard to better represent the current state of the pandemic.

What exactly does that mean?

During the early stages of a pandemic, especially before mitigation strategies are in place, a virus typically spreads through a population in an exponential fashion. This type of exponential growth is seen throughout nature. From a modelling point of view, this period of growth can be described by the aptly named exponential model: $y=\alpha e^{\beta t},$

where $t$ is time (in days), $y$ is the number of new or cumulative cases or deaths, and $\alpha$ and $\beta$ are parameters that describe the curve. We are most interested in estimating $\beta$ as it provides insight into how fast the disease is spreading through the population. Different values of $\beta$ indicate that the population of infected (or dead) is growing ( $\beta>0$), staying the same ( $\beta=0$), or shrinking ( $\beta<0$), as seen in Figure 1. Figure 1: Several exponential curves given different values of . When , the curve continues to grow as time passes. When , the curve remains flat. And when , the curve decays towards 0.

As we observe new case or mortality counts, we can estimate $\beta$ using standard regression tools. To estimate the daily or cumulative growth rate, we would simply calculate $(e^{\beta}-1)\times 100\%$. For example, if $\beta=0.5$, then we would expect the daily or cumulative case counts to grow by ~65% each day (a huge increase). If $\beta=-0.5$, we would expect them to decrease by ~40% per day. To estimate the time it takes for the number of cases (or deaths) to double, we simply calculate $\frac{\ln{2}}{\beta}$. If $\beta=0.5$, then we’d expect cases (or deaths) to double approximately every 1.4 days.

Using the exponential model to understand a pandemic works extremely well until our pattern of growth changes. That is, when successful mitigation measures or limited resources begin to slow or prevent growth, we have to look to better models because the exponential model can’t adapt1. The basic exponential model (as described above) can’t increase, level off, and then decrease to match the pattern of cases as a pandemic waxes and wanes.

This also means that we shouldn’t use the exponential model to estimate the growth rate or the doubling time as was described previously.

And so, given that physical distancing, hand washing, and other mitigation strategies have slowed the spread of COVID-19, it’s necessary to update the models we use to understand the growth rate and the doubling time. As such, our previous plots under the Statistics menu on the dashboard have been updated as follows:

• Growth Rate now displays the raw day over day change in cases (see Figure 2). What we want to see here is a growth rate that decreases over time to 0. In the early exponential growth phases of the pandemic, raw growth rates were averaging between 20% and 30%. However, thanks to mitigation measures, that growth rate has dropped to around 2% to 3% in recent weeks.
• Doubling Time has been updated to display the time it takes for cases to double based on the raw growth rate. As the pandemic wanes and mitigation measures take effect, we want to and expect to see the doubling time increase. The higher this value, the better. For clarity, we’ve also included a trend line on this plot that is generated using a local polynomial smoothing function (LOESS). These types of functions respond well to changes in the pattern of the data we are plotting (see Figure 3). Interestingly, the LOESS trend line begins to bend upward approximately 2 weeks after many of the provinces called for Canadians to begin physical distancing.