Challenges in modelling interventions for COVID-19
Introduction
To begin with a disclaimer, all simulations within this post are for educational purposes only. Although I have tried to use parameters consistent with the current covid-19 outbreak where possible, there are other factors such as incubation period, heterogeneity of the population, or importation of cases that I haven't explicitly included.
Among many of the things the current covid-19 pandemic has shown is the difficulty in predicting whether an outbreak of an infectious disease will grow into an epidemic and what might be the potential impact of subsequent interventions. Using models we can build up a picture of what this uncertainty might be and factor in some elements that we don’t know about the disease such as if individuals can be asymptomatic carriers.
Another recent point of debate that’s been particularly exemplified in the UK is around whether it is best to build up herd immunity or to impose strict controls on movement early in the epidemic as has been done in Singapore, Hong Kong, and Taiwan. The apps below will allow you to explore the impact of both intervention both early and later in the epidemic.
Outbreak control
We have to begin by defining the most important number in infectious disease epidemiology, the $R_0$. It’s full definition is,
The average number of secondary cases for every primary case in a completely susceptible population
This is a little technical, so let’s break down each part of the definition. A completely susceptible population is where all individuals are able to be infected by the virus, where no one has any prior immunity. A secondary case following a primary case is the number of individuals who are infected by one individual. The average number is also important here. For example if an infection has an $R_0$ of 2, on average an infected individual would infect 2 others, but this could potentially be more or less.
At the start of an epidemic, many random infection events can make it incredibly difficult to predict how many cases we would expect even a week later. To explore this, the simulation below shows a series of infection events following one infected individual (the technical name for this type of model is a branching process). Using the slider you can change the reproduction number $R_0$ and simulate three generations ahead (for this purpose we can assume a generation is 7 days, so we are simulating three weeks ahead).
Early estimates of the $R_0$ for covid-19 are around 2.5 although there is a large amount of uncertainty around this. Even if we did knew the $R_0$ exactly due to the randomness with how infection events transpire we still see some scenarios where there are a large number of cases after three generations and so where there aren’t any.
Now let’s consider this process happening several times, where we keep simulating what would transpire if one individual is infected. This builds up a probability of an outbreak occurring and what would be the size of the outbreak. Using the app below, you can change the initial $R_0$ and observe what the final number of cases after three generations. As more simulations are run a pattern begins to build up that describes the distribution of all possible infection scenarios for that particular $R_0$.
Although I mentioned above that current estimates of the $R_0$ for covid-19 are around 2.5, this is dependent on there being no intervention to its spread. Lots of factors may help to limit the spread and reduce the $R_0$ including social distancing, self-quarantining and contact tracing. Try reducing the $R_0$ above and see how it impacts the probability of an outbreak occurring. You’ll notice that if the $R_0$ is below 1 then there is a zero probability of the epidemic taking off.
Later epidemic case management
Many countries are now observing sustained community-based transmission where the majority of new cases are from individuals becoming infected in their own community and not individuals who had recently travelled abroad. In this situation outbreak control is no longer feasible and so other measures must be used including encouraging social distancing and hand washing. In more extreme cases countries, such as Italy have begun to impose national quarantines for a given period.
Both the timing and duration of these interventions can be incredibly important for controlling the total number of individuals who become infected, but also location and height of the epidemic peak when the most individuals are infected in a given week. The idea is to “flatten the curve” so as to not overwhelm a nation’s healthcare services and give them more time to respond.
The simulator below lets you explore the consequences of an intervention event where the risk of transmission is reduced. The top-left graph shows the curve of the epidemic where there is intervention and the counterfactual scenario where no intervention occurs. The top-right shows the total number of infected individuals at the end of an epidemic and those that had remained susceptible. The bottom graph below shows the effective $R_0$ at a point in time, this is the average number of individuals infected from a case in that moment. Try moving the $R_0$ slider below to see how this impacts the size of the epidemic and its peak.
The sliders above control how much the intervention reduces the spread of the infection, where the start of the intervention occurs and its total duration. For this simulation as soon as an intervention stops the $R_0$ returns to its initial value, which depending on when the intervention starts can lead to the epidemic being delayed or creating a double-peak epidemic. Also if the intervention begins too late after the peak then there is little impact on the overall epidemic.
Conclusions
As Thomas House from Manchester University has also commented on, initial interventions that lower the effective reproduction may only be delaying those individuals from becoming infected, however does reduce the peak of the epidemic. Many factors impact the overall epidemiology of a virus and how that translates into the total cases infected. This is especially problematic in the current pandemic where estimates of infectivity, incubation period, and recovery time all have large uncertainty. Even more, it is not clear how much these current or future interventions will impact the ability for covid-19 to spread.