Empirical Infection Curves

In Time-Varying Parameters in Virus Transmission Models, I illustrated the effect that non-pharmaceutical interventions could have on the infection curve in a basic SIR virus transmission model. In particular, it's difficult to estimate the parameters of these models when there's reason to believe they're varying over time. And for that reason, it's useful to consider the empirical curves, when available.

For example, the graph below shows confirmed cases, recovered cases, infected cases, and deceased cases in recent months in China.

Source: John Hopkins

Note that none of the curves appear as simple as our basic SIR model would predict. Of course, there are numerous reasons for this. For example, the basic SIR may be too simple to capture dynamics in a real-world epidemic. In addition, Chinese authorities intentionally changed the basis on which some of the data was reported in mid-February. And important for our discussion: interventions on the part of authorities in China almost surely changed the dynamics of viral transmission within China. In particular, lowering the reproduction number, R0, is an important tactic in all efforts to restrict viral transmission.

Notwithstanding, the basic characteristics of viral transmission are clear in these reported figures. The 'infected' curve starts with near-exponential growth; it then reaches an inflection point followed by a peak, after which the number of people infected with the virus declines.

The next graph shows similar curves for South Korea.

Source: John Hopkins

Here again, note the initial growth, followed by an inflection point, followed by a peak, followed by a decline. South Korea appears to have rigorously followed WHO advice for people to isolate and for authorities to engage in intensive testing and contact tracing. And these curves give the impression that the South Koreans have made notable progress in recovering from the infection.

In contrast, the figures for Italy are shown in the next graph.

Source: John Hopkins

The 'infected' curve for Italy is no longer growing at a near-exponential rate, but neither has it reached an inflection point. As a result, it appears too early to forecast when Italians might experience the peak and the number of people who will be infected at the peak.

The same is true of Spain.

Source: John Hopkins

The reported transmission curves for the UK are shown in the next graph below.

Source: John Hopkins

Unfortunately, the number of reported infections in the UK is still exhibiting near-exponential growth. More specifically, simply fitting an exponential curve to the 'infected' curve in the UK still provides quite a good fit, as seen in the graph below.

Source: John Hopkins; author. The blue line is the actual number of active COVID-19 infections reported daily in the UK since these first topped 100, on 5-Mar-20. The dotted red line is the exponential curve of best fit through these points. The fact that this exponential curve still fits the data quite closely suggests the UK infection is still in its early stages of transmission. 

How about the US? The confirmed, infected, recovered, and deceased curves are shown in the graph below.

From this graph, it's difficult to tell whether the US infection is still at the initial near-exponential stage or whether it's beginning to bend toward an inflection point. The next graph below, with a fitted exponential curve, confirms that the US still appears to be at the near-exponential stage of growth. As with the UK, there's no basis in this data for predicting the timing or severity of peak infection in the US.

Source: John Hopkins; author. The blue line is the actual number of active COVID-19 infections reported daily in the US since these first topped 100, on 3-Mar-20. The dotted red line is the exponential curve of best fit through these points. The fact that this exponential curve still fits the data quite closely suggests the US infection is still in its early stages of transmission.

Even when circumstances are changing and the parameters governing the evolution of the infection are changing, it's useful to analyze the empirical data, when available -- particularly to look for

• the period of near-exponential growth in the infection curve
• the subsequent inflection point, when the rate of growth stops increasing
• the peak of the infection curve
• the subsequent decline in the incidence of infection.

Market implications

Of course, our interest isn't in epidemiology per se. We're ultimately interested in assessing the likely effect of the pandemic on economic growth, inflation, employment, and the financial markets.

With that in mind, what might we have learned from this exercise? A few things:

• China and South Korea appear to have turned the tide on their epidemics.
• The virus still appears to be spreading rapidly in Europe -- eg, in Italy and Spain, where the rate of infection is still increasing, though not at a near-exponential rate.
• In the US and the UK, the rate of infection still appears to be in the initial, near-exponential stage.

Equity markets have reacted favorably to news of fiscal stimulus in the US, the UK, and elsewhere -- and perhaps also to President Trump's insistence that US workers should return to work en masse by Easter (April 12). But given the still rapid rate at which the virus appears to be spreading in the US, it seems more likely that more American workers will be in isolation in mid-April than there are now. 

In fact, if the number of infections were to continue to grow at the current exponential rate between now and Easter, the total number of reported infections in the US would be on the order of 15 million.

Of course, it's quite possible that the infection curve will begin to move away from this near-exponential growth between now and then. It's even possible that the infection curve in the US will have reached an inflection point between now and Easter, in which case the number of reported infections may be well below 15 million.

But to put this in context, the peak number of active infections in China was reported to be about 58,000. Even if the number of infections in the US peaked at, say, 5 million, the number of infected would be roughly 86 times the peak number of infected people in China.

In Italy, 12% of all detected COVID-19 cases were admitted to the intensive care unit. If that percentage were observed in the US, and if the peak number of infections in the US were 'only' 5 million, that would still point to a simultaneous demand for 600,000 ICU beds in the US -- far outstripping the available supply, estimated to be on the order of 100,000.

I should stress that these figures are all just attempts to produce first-order approximations of the eventual numbers. But they're broadly consistent with figures produced elsewhere. For example, The Harvard Global Health Institute produced an online report, in which ICU beds in the US were estimated at just over 87,000. The number of patients requiring hospitalization for COVID-19 over time was predicted to be 10 million, and the number of patients requiring an ICU bed was predicted to be just over 2 million. These Harvard figures aren't for a single point in time, as were my attempts to predict peak demand for ICU beds. But these sorts of figures at least put the problem into broad perspective.

All in all, what are the conclusions from analyses of this sort?

1. The rate of infection in the US is still increasing at a near-exponential rate.
2. Peak infection is still well in the future in the US, with the peak infection rate many times greater than the infection rate currently reported.
3. The situation at Easter will almost surely be far worse than it is currently.
4. Under these circumstances, it is difficult to see droves of US employees returning to their workplaces, admonitions by the President notwithstanding. More likely, an even larger percentage of the US population will be under strict lock-down orders at Easter.

The rally in the S&P yesterday and today, from a low of 2174 (June S&P futures) at the start of the week to 2498 now (up by nearly 15%) is impressive. But my sense is that with peak infection well ahead of us, it's quite unlikely that we've seen the lows in the June S&P futures -- particularly if the number of peak infections is even close to the estimates consistent with the analysis discussed here.