Why we may have less control over this virus than we think: Fixed prevalence with variable incidence for a ubiquitous disease = Farr’s Law
By Howard Stupak, MD
We hear so many conflicting opinions about what is going on with this illness that has been with us for nearly half a year now, but no single authority seems to give us a plausible concept of what is really happening. In reality, the illness has to be governed by the laws of physics and biology, instead of the politics, randomness, and wishful thinking that seems to prevail our discussion. The goal of this article is to provide a unified concept of how weather, overall health, populations, distancing and incidence curves fit together!
Let’s start with clarify some terms with regards to Covid-19 that have created confusion for many, including journalists: exposure (contact but not necessarily disease), severity (once exposed, disease results in minor symptoms or hospitalization and death), incidence (how many new cases per day or viral generation) and prevalence (how many cumulative cases in a region or group once a measurable time period is complete). By confusing these terms, or even using them interchangeably, headlines can create unnecessary fear or complacency where the reader reads something, but imagines another. For example, the “grim milestone” headlines, that are usually discussing world or national prevalence crossing a numeric threshold, and are simply cumulative cases, can cause a reader to panic un-necessarily. In other cases, a rising incidence (daily new cases) makes others concerned that the virus has entered “exponential growth” and will continue to increase unchecked until lockdowns can stop growth.
In reality, based in part on the work of a pre-germ theory scientist, William Farr, we know that pandemics and epidemics follow a roughly Bell-shaped curve in incidence. In other words, the incidence accelerates over time or host generation before peaking and then decelerating until it has decayed to an incidence of nearly zero! However, the number of people that are susceptible may not be variable, and may remain fixed. This was evident in many of the cases where there was a coronavirus outbreak on isolated ocean-going ships, and there was a very specific amount of those who tested positive, a much smaller percentage of those who developed severe disease needing hospitalization, and an even smaller percentage of those who died. In other words, not all of the population is even susceptible to disease, and those who are, may be only susceptible to a specific severity of disease (ie no symptoms, severe, or death). Thus, prevalence, or the total number of people that are susceptible to a given severity may be fixed in advance! So, only incidence, or how many new cases occur on a specific day, or in a host generation is variable per a specific population. The “flatten the curve” campaign illustrates this well; that epidemiologists knew the final prevalence or total number susceptible and eventually effected would remain fixed and inevitable. What does vary between communities like cities or even neighborhood are two factors. First, the proportion of a set of individuals who are susceptible to specific severity outcomes depends on the long-term state of the health of the respiratory system and its associated “lining” or mucosa of that individual. Second, is the “reproduction number” (R0) for that given susceptible population for a pathogen, which depends on how efficiently a virus can be transmitted from person to person taking into account viral characteristics, concentration (crowding or distancing) of the population, as well as how many are susceptible in the population in question.
Let’s look at these two variables separately. First, for a population, the susceptible proportion to a specific severity (ICU admission for example) for a specific town, city, or vessel, is determined by the respiratory health of the population at the time of the exposure. For example, in a city with a high population of elderly, obese, sedentary individuals, during the cold, dry winter season, and limited prior exposure, there will be a relatively high proportion of this level of severity relative to a comparable city with the opposite characteristics. These are variables that are not changeable overnight, and are thus fixed for the season in question. On the mechanistic level, these characteristics are related to the health of the lining of the respiratory tract, (the mucosa layer of the throat or airway) or the barrier preventing entry by respiratory pathogens. This protective layer that lines the throat and airway consists of a water-proof layer of cells that clears invading pathogens by trapping them in produced mucous, or brushing them away with tiny cellular bristles called cilia. Dry air, or low humidity causes these mechanisms to fail due to dehydration of these functions. Also, chronic tone problems, like obesity, or advanced age causes variations on sleep apnea, which is manifested audibly as loud snoring, but is actually the buildup of vacuum forces in the throat and airway from mouth-breathing. These negative pressures with nightly breathing damage the mucosa, and particularly when combined with the dry heated air during low humidity winter months, can cause breaches in the protective layer of the throat, permitting deep viral invasion. This is why some reports correlate low outdoor temperature with disease, while this is just a surrogate marker of the heating systems that cause indoor humidity to drop and cause severe disease, even when outdoor humidity is adequate. To summarize, the fixed proportion of individuals who will experience severe disease in a given season is dependent partially on the dryness of the air (humidity), and the overall tone (Age/obesity) and other immunity factors of sectors of the population. Together, these variables can be called “mucosal immunity”, and rather than being assessed with antibody tests, (B-cell humoral immunity), mucosal immunity may be moderated by T-cell, non-antibody immunity, thus, not detectable with widespread testing.
The second variable in the “fixed prevalence” model is the efficiency of exposure. Again, assuming that the virus, like most respiratory pathogens is ubiquitous or omnipresent, the number of people eventually exposed is not variable, but only the efficiency by which with they will be exposed. This characteristic is represented as the “reproduction number” (R0) for that given susceptible population (determined above) for a pathogen, which depends on how efficiently a virus can be transmitted from person to person taking into account viral characteristics as well as concentration (crowding or distancing) of the population in question. So, measures like social distancing, mask-wearing, or having a very spread-out or concentrated population will effect this viral reproduction efficiency (R0), and exposure can be rapid, or very slow, but will not affect the eventual number of total cases, assuming no vaccine intervention is available.
So, then why does the disease incidence (daily cases) over time seem to first go up nearly exponentially, then peak, then decline exponentially? Because of the interplay of the two factors we just discussed (transmission efficiency and fixed prevalence of severity populations). This can be demonstrated mathematically, using Farr’s Law, describing incidence decay in a vulnerable population.
A bell-shaped curve results when one uses the following equation: Incidence of disease of subsequent generation, or I(t+1) = Incidence of current generation I (t) x viral reproduction efficiency for a given population(R0) x Incidence decay factor which is: (Total Prevalence — Total incidence so far or “sum of all incidence to date”)/ Total expected prevalence.
I(t+1) = I (t) x R0 x ([Total Expected Prevalence — Incidence to date] /Total Expected prevalence)
For example, let’s imagine that in a city of 1,000,000 individuals, in a season of excessive dry air, there are 10,000 individuals that are susceptible to having severe disease due to their tone and age (Total expected prevalence). Because the city in densely populated, (large apartment buildings with poor ventilation), the R0 will be 2, where each infected susceptible person can transmit the disease to 2 new susceptible people. Thus, in the first generation, there is an incidence of 1 individual, with an R0 of 2, and an incidence decay factor of 0.9999, and the next generation will have 1.9998 (nearly 2) individuals with severe disease. In the next generation, these nearly 2 individuals will be slightly less efficient at transmitting the disease due to a slightly decreased incidence decay factor to 3.9988 individuals (nearly 4). The incidence continues to rise until peak incidence of 1875.109 individuals in generation 13, with an incidence decay factor of 0.683. The declining incidence occurs in subsequent generations (the down-hill part of the curve), where the decay incidence factor is below 0.5 and thus more than half of the susceptible population has been effected. This example is graphically represented by:
On the other extreme, we can take a city with the same size population (1,000,000), with a similar susceptible population to severity (10,000), but spread out by vast land mass, or using extensive social distancing measures. Thus, the R0 drops to 1.1 (from 2 in the crowded city), where it is harder to find people to infect quickly. Now, it takes 54 generations to peak, at a peak incidence of 44.76 people per generation, and 142 generations before the incidence approaches zero. But, the total prevalence remains the same! So, the second city, has a delayed peak, even though their prevalence remains exactly the same. If the second city also has its extreme humidity peak or depression, causing excessively dry mucosa of the throat in a period that corresponds to this, it may also show a shift for that season to a higher prevalence of susceptible population to severe disease than expected, because the health of the mucosal immunity is reduced by environmental factors. Here is the graphic representation of the second city.
So how does this translate to the real world situation that we see unfolding before us with Covid-19? Each city or region that is a geographical and socio-political unit will have its own characteristic fixed prevalence proportion of the population that is susceptible to disease. This will vary from season to season, where climate conditions can vary from the least favorable to mucosa, cold dry winters exacerbated by low-quality indoor heat, or hot, dry summers, or even overly humid swamp-like conditions, where mucosa can be injured by excessive humidity.
This is why northern, crowded cities (New York City for example) with cold, dry winter climates peak early and rapidly, and then have a precipitous decline, due to both incidence day as prevalence is approached and the season changes to a more favorable one to mucosal health, decreasing the proportion of susceptible individuals for the subsequent season. In contrast, less dense cities like Phoenix will have a decreased R0 or transmission efficiency, and has a lowest humidity period in the summer. Thus, Phoenix will experience a flatter more prolonged curve with lower daily incidences, and can expect maximal prevalence in its hot, dry summer. So, looking at a snapshot on the day of this writing, June 25, 2020, one will see incidence rates in New York City approaching zero, as the prevalence has been exhausted, after high incidence in the late winter early spring, and in Phoenix, where prevalence has not been exhausted, we will see incidences on the rise until they hit their delayed peak in their dry, hot desert summer.
In other words, as the epidemiologists have stated all along, distancing rules will not affect eventual prevalence of disease, which remains fixed, but it will lower the R0 or efficiency of transmission, prolonging the duration, but lowering the daily incidence. Also, as they have stated, this will prevent overwhelming hospitals and intensive care units, which did happen during New York City’s peak. On the other hand, the constant banter from supposed experts and all forms of media about which cities or politicians are “doing a good job”, and which are “failing”. Of course, if the goal is to stretch the disease course out as long as possible, even though the eventual number of people to get ill will remain the same, then the banter is correct. But, of course this illogical, as are the politicians that get credit for the downward sloping incidence, or the politicians or citizens that are blamed for the upward slope. Even worse, is the attempt to analyze total or aggregated and cumulative data for political entities like the entire United States, which is really many distinct geographic entities. By not looking at the individual geographic region’s curves temporally, (as fixed prevalences per season), and instead looking at overall data for the entire nation or world, (as most headlines reflect) only confusion and fear will result. Political leaders do deserve credit for their hard work, expansion of hospital capacity and deployment of the military to areas of the United States during their peak prevalence. However, it is pure hubris or excessive confidence in our species’ ability to conquer nature to believe that we either cause the peaks to occur, or cause incidences to fall dramatically in a region. We should do much less finger-pointing for “failure” and back-slapping for “success” respectively, when a process as natural as the rise and fall of the tides is underway in the form of a novel infectious disease. But, of course, applying politics to nature itself is of course a human characteristic.
So, assuming we cannot control the eventual fixed prevalence per season, and that our society has already figured out how to balance distancing with productivity to maintain an R0 that will prevent overwhelming of hospitals and morgues, what strategies can we use to protect our cities using basic, common sense in future seasons? A “second wave” can consist of either an incomplete prevalance curve that was initially stunted by distancing, but was eventually going to run its course, or can represent a newly susceptible group as a region enters a new climate season that worsens mucosal or airway health, if it was not exposed in the prior one. By improving the overall health of our airway immune systems, or effectively reducing the proportion of our regional populations that are susceptible to severe disease. This can be achieved by improving our habitats, decreasing dry heat in the winters and using humidity increasing devices. Preventing mouth-breathing and sleep apnea using devices that control these factors, as it is the pressure-related trauma from these conditions that injures our mucosal barriers and permits entry to our more vulnerable cells to viruses. These conditions can also be improved with improving our body tone, allowing our own muscle tone to support our jaws, tongues, and collapsing throats during the limp state of sleep. A city like New York City can achieve severity reduction by having policies that reduce heat and dryness to crowded apartments and require landlords to monitor and maintain indoor humidity. Parks should remain open always, permitting people to breathe fresh air all year long, and improve their own tone. Confining people to dry apartments lowers viral transmission efficiency, but increases severity due to reduced airway health with a sedentary indoor lifestyle. Public hospitals in the city should, in addition to obviously creating reserve intensive care unit capacity and stockpiling personal protective equipment for workers, create public wellness centers that address diet and exercise patterns instead of treating only the end-stage of poor health with acute care treatments. The public hospitals should invest further in easy-to-access (and revenue generating) sleep centers for the population that permits efficient treatment of sleep apnea, bariatric surgery, and nutrition and exercise counseling. Finally, a culture of rational discussion must be established between all parties, permitting efficient discourse and non-draconian safety measures to permit useful interaction of the state, the health-care system and the public.
About the author:
Howard Stupak is the director of Otolaryngology/Head and Neck Surgery at Jacobi Medical Center in the Bronx, and is an Associate Professor of Clinical Otolaryngology at Albert Einstein College of Medicine. He also maintains a private practice in Facial Plastic and Reconstructive Surgery in Westport, Connecticut (https://westportfacialplasticsurgery.com/) and is the author of “Rethinking Rhinoplasty and Facial Surgery”, published by Springer last month, where further information on this topic can be found. https://rd.springer.com/book/10.1007/978-3-030-44674-1
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