THE MATH OF UNCONTROLLED EPIDEMICS

The math of uncontrolled epidemics

By Christopher Monckton of Brenchley

As Willis Eschenbach has pointed out here, the epidemic curve is S-shaped. At first the curve is exponential, but eventually the rate at which new infections occur will decline for any combination of the following four reasons:

1. Decisive public-health measures control its transmission. China and South Korea are good examples.

2. There are no more susceptible people to infect.

3. A vaccine is found. Even when a vaccine is found, testing it for safety takes a year.

4. The population acquires immunity.

The most important step, where a new pathogen is spreading and is proving fatal to some, is that the public authorities should act determinedly and at the very earliest possible moment to hinder the transmission of the pathogen.

Here is why. In the early stages of an epidemic, transmission follows an approximately exponential curve. We now have enough data from the past 52 days of transmission outside China to derive the exponent. We begin with the curve of the daily cumulative case count:

Fig. 1. Cases of COVID-19 from January 22 to March 13, 2020 (worldometers.info)

From the cumulative cases C1 = 9 at January 22 and C52 = 64,659 at March 13, the daily growth rate g over 52 days d is given by (1):

From the shape of the curve at Fig. 1, it is evident that the epidemic is still in its early stages. In particular, public health measures adopted to date by most countries have been ineffective in preventing what appears to be the exponential rise in cases that one would expect from a standard epidemic curve in the absence of determined preventative action.

Every day, on average, the number of cases has been increasing by a little over 19% compared with the previous day.

Next, we verify using the Mk. 1 eyeball that the curve of actual reported cases from all around the world (Fig. 1) follows at all points an exponential curve calculated from the exponent derived via (1). The equation of the curve is plotted using (2).

Fig. 2 shows the graph thus derived:

Fig. 2. Cases of COVID-19 from January 22 to March 13, 2020 (calculated)

Figs. 1 and 2 are scaled and drawn to the same aspect ratio. The blue borders of the graphs will align neatly with the edges of a 16 x 9 PowerPoint slide. Copy the two figures from this article, place them on successive PowerPoint slides and align the borders with the edges of the slides.

Now we use a technique originally developed by astronomers to find moving satellites or planets in successive images of a field of fixed stars: the blink comparator. PowerPoint is a superlative blink comparator. Go to display mode and flick rapidly between the two slides.

You will at once see just how very close the actual data plotted in Fig. 1 are to the idealized exponential-growth curve calculated and plotted in Fig. 2. It is information presented like this

that is useful when trying to persuade governments that the predicted rate of transmission in the absence of more stringent government policies is not merely speculative.

Try the blink comparator for yourself. The two curves are near-perfectly coincident. Therefore, we may legitimately deduce that the daily rate at which the total cases will grow is likely to continue on the exponential-growth curve unless one of the reasons 1-4 listed at the beginning of this article comes into play.

Note again that this is not speculation. The epidemic curve has been well studied, and its characteristics are sufficiently understood. With more than 50 days of data one can derive the growth factor, as we have done, and one can use it to give a quite reliable indication of how fast the infection is likely to continue to be transmitted if existing policies are continued.

Why does this work? The reason is that each infected person will, roughly speaking, pass the infection on to the same number of uninfected people, who will, roughly speaking, acquire or resist the infection to the same degree.

For policymakers at government level, the question is when one should make determined efforts to contain the transmission of the infection, and how determined those control measures should be.

Should governments adopt the Trump approach of declaring a national emergency and engaging the public and private sectors at once to curtail transmission, or the Johnson approach of mumbling about the desirability of enough people contracting the infection to acquire what his Chief Medical Officer has contemptuously described as “herd immumity”?

To begin answering that question, we make Friday, 13 March 2020 Day 1 and calculate how the infection would spread over another 58 days using the established growth factor for COVID-19 that would prevail if the world continued with the talk-a-lot-but-do-too-little policy that continues to prevail in all but a few countries. Results are in Table 1:

Table 1. Cumulative COVID-19 cases from March 14 to May 10 on present policies

From mid-May on, but not until then, enough people will already be infected, as a percentage of global population, to reduce the exponential-growth factor. But how realistic is this table? Will there really be 1 million cases by 19 April, 100 million by 12 May, 500 million by 4 May, 1 billion by 8 May and 1.5 billion by 10 May?

The answer is that the table is a realistic portrayal of what would happen if governments continued to fail to take determined steps to prevent transmission. Since most governments are not wicked, they will realize in due course that they need to raise their game. Therefore, this table, based on the current do-little option, is a benchmark against which one can measure henceforward the effectiveness (or ineffectiveness) of public-health containment measures.

In reality, governments will now begin to take the threat more seriously than most of them have done so far. One can also hope that, at least in the northern extratropics, the warmer weather of spring and early summer will inhibit transmission.

But the value of the present exercise is to give readers of WUWT a handy primer that they can show to their own elected representatives, and request that they should immediately ensure that more decisive containment measures are taken from now on.

To this end, the PowerPoint blink comparator showing how the formula derived from the known cases to date reliably follows the curve of those cases will help to convince your elected representatives that, given the known characteristics of the transmission of epidemics, the formula will just as reliably predict future cases based on current public-health policies worldwide – in short, that the table above represents approximately what would happen unless tough decisions are taken immediately.

In particular, it is in the nature of exponential curves that the sooner one intervenes to prevent the curve from continuing unabated the more effective the control measures will prove to be.

What will be the cost in lives if the do-little option continues? It is notoriously difficult, early in an epidemic, to establish the true death rate. One cannot simply divide the number of deaths to date by the number of infections to date, because the number of infections is rising very steeply, and the deaths lag by about a week.

A more reliable method – though not definitive at this early stage and far less reliable than the future cases predicted in the table above – is to express the death rate as the percentage of closed cases – i.e., of cases whose outcome is known. People either recover or they die (or they are not yet a closed case). At present, the death rate appears to be about 7%. I had hoped that 6% would prove to be the asymptote, but in the last few days the death rate has increased to 7%.

Thus, those who suffer frank symptoms have a 1:14 chance of dying of the infection. In reality, however, anyone under 50 is at negligible risk. Small-sample studies of hospitalized patients suggest nosocomial death rates of 4-15%. Age-related mortality rates are as follows: over-80s 22% in confirmed cases, 15% in all cases; 70-79 8% in all cases; 60-69 3.6%; 50-59 1.3%; 40-49 0.4%; 10-39 0.2%; children 0-9 zero.

Fig. 3. Outcome of closed COVID19 cases outside China (worldometers.info)

For comparison, the death rate from other recent new infections is as follows: SARS 9.6% against an original World Health Organization prediction of 2%; MERS 34%; Swine Flu 0.02%; COVID-19 (based on closed cases to date) 7% against an original WHO prediction of 2% (since revised to 3.4%).

Based on Chinese data, the overall death rate for males is 4.7% in confirmed cases and 2.8% in all cases; and in females 2.8% in confirmed cases and 1.7% in all cases.

Death rates for patients with relevant co-morbidities – chronic illnesses that increase the mortality rate from COVID-19 – are as follows: cardiovascular disease 13.2% in confirmed cases, 10.5% in all cases; diabetes 9.2% and 7.3%; chronic respiratory diseases 8.0% and 6.3%; hypertension 8.4% and 6.0%; cancer 7.6% and 5.6%; no pre-existing conditions 0.9%.

If 1.6 billion people become infected by May 10, up to 54 million (at a 3.4% death rate) or 110 million (at 7%) may have died worldwide by a month or two later. This is one more reason why governments would do well to act sooner than later. It is Mr Trump who is right and Mr Johnson who is wrong.

What can the individual citizen do? The following does not constitute medical advice. It is precautionary, and the precautions may not be sufficient. Your own public health authorities will have their own advice online. At present, however, it is not likely to be as detailed as what follows.

First and foremost, if you are over 60, and particularly if you are male and have pre-existing co-morbidities, protect yourself by isolating yourself at home for the time being. A couple of weeks ago, when I did the math summarized here, I canceled two holidays in the north of

England, an important business meeting in Yorkshire and a dental appointment. I have been at home ever since.

If you must go out, travel by car or motorcycle. Avoid all forms of public transport. In particular, do not use public washrooms: go before you go.

In any public place, wear motorcycle gloves and a motorcycle helmet (a lot more effective than a face-mask). Modern helmets are quite lightweight. Also, wear leather knee-boots and, if possible, leather breeks and a leather jacket rather than any kind of fabric clothes, gloves or boots: the virus can endure on fabric for up to 12 hours, whereas you can wash down leather as often as you like.

Experts in Taiwan have come up with a simple self-diagnosis method that anyone can do every morning. Take a deep breath and hold it for at least 10 seconds. If you can do this without coughing, discomfort, stiffness or tightness, there is little or no fibrosis in the lungs. This does not mean you are not infected: but, if you are infected, the infection is not yet at the dangerous phase.

Japanese doctors give the following advice:

Take a few sips of water at least every 15 minutes. Warm water is best: avoid iced water. Drinking works because, if the virus gets into your mouth, drinking will wash the virions into your stomach, where the digestive acids will dissolve the lipid membrane encasing them, rendering them harmless. Regular drinking also prevents the virus from entering the windpipe and lungs.

Sidenote: I use bottled water, because the tap-water is fluoridated and there is peer-reviewed medico-scientifcic evidence that, in the United States alone, about 1 million deaths of fluoride-induced cancers have occurred as a direct result of fluoridation.

The Japanese doctors also advise us a runny nose and sputum are symptoms of the common cold. Coronavirus gives a dry cough without a runny nose (unless you have a cold as well).

The new virus appears to be unable to endure an ambient temperature above 26-27 Celsius. If so, the summer will help a lot. But there won’t be much summer by May 10.

If anyone who is infected sneezes, the virions will travel up to 10 ft (3 m) before reaching the ground. Therefore, try to keep away from other people in public places by at least 15 feet (4.5 meters), and more if you are downwind.

Do not touch any surface in any public place unless you are wearing motorcycle gloves. Virions on a metal surface can survive for at least 12 hours.

The symptoms of COVID-19 are as follows:

The throat is typically infected first, with a sore throat lasting three to four days. The virus then blends into a nasal fluid that enters the trachea and then the lungs, inducing pneumonia. The nasal fluid is not as normal: it feels as though the patient is drowning. This process takes

5-6 days after the sore-throat phase. With the pneumonia comes high fever and breathing difficulties. At this point, telephone your doctor’s office or health provider, but do not visit them.

The moral of this tale is this. Like it or not, every epidemic spreads exponentially during its early stages. It has here been demonstrated – graphically in both senses of the term – that COVID-19 is no exception to this rule.

Exponential transmission at the now-known rate will diminish, as with any infection, only in response to (1) public-health measures, (2) infection of most of the susceptible population or, eventually, (3) discovery of a vaccine or (4) acquisition of population-wide immunity.

The sooner we act the better. As Table 1 shows, unlike global warming the coronavirus is a real emergency. Thirty years after IPCC’s First Assessment Report, CO2 emissions continue to exceed IPCC’s then business-as-usual case, and yet the world has warmed at only half the 0.33 K/decade that IPCC had then confidently predicted. Thirty days after you read this, if your governments have not taken heed, and if the exponential growth rate therefore continues, 12 million people worldwide will be infected with COVID-19, of whom close to a million will die.

Like it or not, the now-known exponential growth-rate of COVID-19 will continue unless and until effective control measures are taken not only by governments but also by you and me. Tell your governments and tell your friends. If they say you are merely speculating, show them the blink comparator.

Be safe, and do not be afraid to be careful. Yes, a motorcycle helmet looks ridiculous except when on a motorcycle, but ridiculous is better than dead.

Like all things, this thing will pass, but it will do much damage on the way.

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