Coronavirus IV: Hypothesis as to what process limits the numbers (continued)

As mentioned in the last post, the phenomenon discussed there is a way to explain why the portion of a population that is ultimately infected by a pathogen may be less than "almost everybody." It is noted that this approach does not inform as to what is going on day-to-day. Here is a model (again a model, not a description of a a biologic process) that results in epidemics petering out prior than they would otherwise be expected to.

First, we should consider how the quantity that was previously described as "exposure" changes as an epidemic progresses. It seems reasonable that the more people infected, the more exposure there is for everyone else. It may be that the amount of exposure increases exponentially with the number infected, linearly, logarithmically, and so on. It is important to note that we are not talking about exposure per time (which would be expected to be exponential if plotted against time) but exposure as a function of the percentage of the population infected. For purposes of explaining the concept the actual relationship does not matter, so we will assume that each infected person generated the same number of exposures, i.e. that the relationship is linear.

If we graph exposure versus percent of the population infected then, we get a line, and we also get a line if we instead graph corresponding percent of the population infected versus exposures. Now we have the same axes as Figure 1 from yesterdays chart.

Next, we want to graph the cumulative portion of the population from Figure 1, that is integral or sums of the total population as we go from left to right. This is the same process used to generate cumulative distribution functions in probability and statistics. We can plot this on the same chart and if we just retain these two new curves we get something like the following:

The red line is cumulative population distributed according to amount of exposure associated with infection, and the green line is the amount of exposure as a function of the population infected. Here, it is useful to think of exposure as an environmental variable. In reality, it is not likely to be evenly distributed, but this fact does not affect the principle to be illustrated.

The important thing to note is that on the left hand side of the chart, the red line is above the green line. We can use the fact that we have plotted the two curves on teh same chart to obseerve the following: for a given percentage of the population we can compare the amount of exposure associated with infection in the population to the amount of exposure associated with the percent of the population infected. We can observe various scenarios as illustrated:

Here we see that at a Percentage of population of 23% the amount of exposure is greater than the exposure associated with infection (or more accurately, the percentage of the population capable of transmitting the infection) in that portion of the population. There is therefore, excess exposure and the contagion will continue to grow. Conversely, if we move up to 52%. we see that the green line is above the red line and at that point the amount of exposure is less than the exposure associated with infection. Everyone whose risk of infection requires exposures greater than that produced by the current percentage of the population infected is less likely to become infected. Therefore the contagion will slow significantly, and as fewer people remain capable of transmitting the infection, will decrease. The vertical green line will begin moving back to the left as the percentage of the population capable of transmitting infection falls, but as it does so, it is moving back into a portion of the population that has already been infected. The epidemic will therefore drop off rapidly, creating the bell-shaped curve described by Farr.

The interesting point is that at which the red and green lines cross. This is the point where the epidemic peaks, and where it is located is a function of how susceptibility and exposure is distributed in a population. The point of cross-over may be at a very low percentage of the population in which case, the numbers observed for the Diamond Princess and San Marino may not be as exceptional as they seem. As mentioned in the previous post, these distributions can be affected by public health interventions, and are more likely to be significant as the two curves described above begin to cross.

These models make several assumptions. The first is that not everyone has the same susceptibility to infection, even at the same level of exposure. The second is that people do not remain capable of transmitting the infection indefinitely, but in a large number of infected people, the mean duration of infectivity is adequate to provide a useful model. The third is that the distributions of the exposures and risks are to some degree modifiable. The last, and most important, is an observation, more or less universal that applies to all types of systems: electrical, mechanical, chemical, economic, political, ecological, etc. That is that stable systems seek points of equilibrium. In the case of an epidemic, if that point exists at a number of infections greater than those currently existing under current conditions, the epidemic will grow until that point is reached. If the number exceeds the point of equilibrium, the epidemic will slow. If people remained infectious forever, and did not develop any immunity, the epidemic would find teh point of equilibrium and persist, but because people are only infectious for a limited time, and likely develop some degree of immunity once infected, the epidemic ends.