Time-Varying Parameters in Virus Transmission Models

In Modeling the Incidence of COVID-19 Over Time, I mentioned some of the models I follow for the purpose of tracking the spread of the SARS-CoV-2 virus. These models are particularly useful in projecting the incidence of infection, recovery, death, etc., given a set of parameter values describing characteristics such a the average number of contacts per person per unit time, the probability of a contact resulting in an infection, and the typical duration of an infection.

In practice, most governments have introduced public policy measures designed to slow or reverse the spread of the virus, including social distancing, testing, and contact tracing. 

From the perspective of a transmission model, these non-pharmaceutical interventions are intended primarily to reduce the reproduction number, R0 of the pandemic. And to the extent these interventions are successful, we need to reflect that in our models.

For example, the graph below shows two infection curves for a hypothetical virus, in a population the size of the UK (67.9 million people). The blue curve assumes:

1. the average person has 20 contacts per day
2. the probability of a contact resulting in an infection is 0.01
3. the duration of the infection is 28 days.

With these three assumptions, the reproduction number, R0, is 5.6. The maximum number of infections under this scenario is 34.9 million people (51.4% of the poplulation), reached 114 days into the epidemic. Infections then decrease as individuals move from being infected to being removed (ie, immune or deceased).

The blue curve shows the trajectory of infections when the reproduction number, R0, is held constant at 5.6. The orange line shows the alternative trajectory in the event measures are introduced to reduce R0 to 0.7 -- in this case when the number of infected people first exceeds 10 million.

Now let's imagine that public policy interventions are introduced as the number of infections first reaches 10 million. In particular, let's assume these interventions reduce the average number of contacts per person per day from 20 to 5 and that the probability of a contact resulting in an infection declines from 0.01 to 0.005.

In this case, the reproduction number decreases from 5.6 to 0.7, with the result that that trajectory of the virus changes dramatically, from a rapid rate of increase to a moderate rate of decline, as illustrated by the orange line in the graph.

Now consider a country like the UK or the US, for which public policy measures have been changing nearly every day in recent weeks. These continual changes in policy measures create a situation in which R0 can't be assumed to be constant over the period of analysis one might otherwise use to estimate the parameters of the model.

As the experience of Imperial's MRC Centre illustrates, a model with inappropriate parameter values may actually be worse than useless in the event it supports policy measures that do more harm than good. So in that case, what good are these models -- particularly for financial analysts wishing to consider implications for the macroeconomy and financial markets?

From my perspective, these sorts of models offer two types of benefits. First, they help us understand the reasons that viral infection curves observed in practice are characterized by a period of near-exponential growth, followed by an inflection point, followed by a maximum, followed by a steady decline. Second, they provide a framework we can use to produce first-order approximations of viral transmission under various public policy initiatives. These projections needed to be treated with caution, due to the combination of model error, parameter estimation error, and data errors. (For example, who really knows the actual current number of infectious persons in the UK?)

And speaking of infection curves observed in practice, it's time to turn to some actual data...