Tag Archives: plague

Disease, atom bombs, and R-naught

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A key indicator of the speed and likelihood of a major disease outbreak is the number of people that each infected person is likely to infect. This infection number is called R-naught, or Ro; it is shown in the table below for several major plague diseases.

R-naught - communicability for several contagious diseases, CDC.

R-naught – infect-ability for several contagious diseases, CDC.

Of the diseases shown, measles is the most communicable, with an Ro of 12 to 18. In an unvaccinated population, one measles-infected person will infect 12- 18 others: his/her whole family and/ or most of his/her friends. After two weeks or so of incubation, each of the newly infected will infect another 12-18. Traveling this way, measles wiped out swaths of the American Indian population in just a few months. It was one of the major plagues that made America white.

While Measles is virtually gone today, Ebola, SARS, HIV, and Leprosy remain. They are far less communicable, and far less deadly, but there is no vaccine. Because they have a low Ro, outbreaks of these diseases move only slowly through a population with outbreaks that can last for years or decades.

To estimate of the total number of people infected, you can use R-naught and the incubation-transmission time as follows:

Ni = Row/wt

where Ni is the total number of people infected at any time after the initial outbreak, w is the number of weeks since the outbreak began, and wt is the average infection to transmission time in weeks.

For measles, wt is approximately 2 weeks. In the days before vaccine, Ro was about 15, as on the table, and

Ni = 15w/2.

In 2 weeks, there will be 15 measles infected people, in 4 weeks there will be 152, or 225, and in 6 generations, or 12 weeks, you’d expect to have 11.39 million. This is a real plague. The spread of measles would slow somewhat after a few weeks, as the infected more and more run into folks who are already infected or already immune. But even when the measles slowed, it still infected quite a lot faster than HIV, Leprosy, or SARS (SARS is a form of Influenza). Leprosy is particularly slow, having a low R-naught, and an infection-transmission time of about 20 years (10 years without symptoms!).

In America, more or less everyone is vaccinated for measles. Measles vaccine works, even if the benefits are oversold, mainly by reducing the effective value of Ro. The measles vaccine is claimed to be 93% effective, suggesting that only 7% of the people that an infected person meets are not immune. If the original value of Ro is 15, as above, the effect of immunization is to reduce the value Ro in the US today to effectively 15 x 0.07 = 1.05. We can still  have measles outbreaks, but only on a small-scale, with slow-moving outbreaks going through pockets of the less-immunized. The average measles-infected person will infect only one other person, if that. The expectation is that an outbreak will be captured by the CDC before it can do much harm.

Short of a vaccine, the best we can do to stop droplet-spread diseases, like SARS, Leprosy, or Ebola is by way of a face mask. Those are worn in Hong Kong and Singapore, but have yet to become acceptable in the USA. It is a low-tech way to reduce Ro to a value below 1.0, — if R-naught is below 1.0, the disease dies out on its own. With HIV, the main way the spread was stopped was by condoms — the same, low tech solution, applied to sexually transmitted disease.

Image from VCE Physics, https://sites.google.com/site/coyleysvcephysics/home/unit-2/optional-studies/26-how-do-fusion-and-fission-compare-as-viable-nuclear-energy-power-sources/fission-and-fusion---lesson-2/chain-reactions-with-dominoes

Progress of an Atom bomb going off. Image from VCE Physics, visit here

As it happens, the explosion of an atom bomb follows the same path as the spread of disease. One neutron appears out of somewhere, and splits a uranium or plutonium atom. Each atom produces two or three more neutrons, so that we might think that R-naught = 2.5, approximately. For a bomb, Ro is found to be a bit lower because we are only interested in fast-released neutrons, and because some neutrons are lost. For a well-designed bomb, it’s OK to say that Ro is about 2.

The progress of a bomb going off will follow the same math as above:

Nn = Rot/nt

where Nn is the total number of neutrons at any time, t is the average number of nanoseconds since the first neutron hit, and nt is the transmission time — the time it takes between when a neuron is given off and absorbed, in nanoseconds.

Assuming an average neutron speed of 13 million m/s, and an average travel distance for neutrons of about 0.1 m, the time between interactions comes out to about 8 billionths of a second — 8 ns. From this, we find the number of neutrons is:

Nn = 2t/8, where t is time measured in nanoseconds (billionths of a second). Since 1 kg of uranium contains about 2 x 1024 atoms, a well-designed A-bomb that contains 1 kg, should take about 83 generations (283 = 1024). If each generation is 8 ns, as above, the explosion should take about 0.664 milliseconds to consume 100% of the fuel. The fission power of each Uranium atom is about 210 MeV, suggesting that this 1 kg bomb could release 16 billion Kcal, or as much explosive energy as 16 kTons of TNT, about the explosive power of the Nagasaki bomb (There are about 38 x10-24 Kcal/eV).

As with disease, this calculation is a bit misleading about the ease of designing a working atomic bomb. Ro starts to get lower after a significant faction of the atoms are split. The atoms begin to move away from each other, and some of the atoms become immune. Once split, the daughter nuclei continue to absorb neutrons without giving off either neutrons or energy. The net result is that an increased fraction of neutrons that are lost to space, and the explosion dies off long before the full power is released.

Computers are very helpful in the analysis of bombs and plagues, as are smart people. The Manhattan project scientists got it right on the first try. They had only rudimentary computers but lots of smart people. Even so, they seem to have gotten an efficiency of about 15%. The North Koreans, with better computers and fewer smart people took 5 tries to reach this level of competence (analyzed here). They are now in the process of developing germ-warfare — directed plagues. As a warning to them, just as it’s very hard to get things right with A-bombs, it’s very hard to get it right with disease; people might start wearing masks, or drinking bottled water, or the CDC could develop a vaccine. The danger, if you get it wrong is the same as with atom bombs: the US will not take this sort of attack lying down.

Robert Buxbaum, January 18, 2019. One of my favorite authors, Issac Asimov, died of AIDS; a slow-moving plague that he contacted from a transfusion. I benefitted vastly from Isaac Asimov’s science and science fiction, but he wrote on virtually every topic. My aim is essays that are sort-of like his, but more mathematical.

Measles, anti-vaxers, and the pious lies of the CDC.

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Measles is a horrible disease that contributed to the downfall that had been declared dead in the US, wiped out by immunization, but it has reappeared. A lot of the blame goes to folks who refuse to vaccinate: anti-vaxers in the popular press. The Center for Disease Control is doing its best to promote to stop the anti-vaxers, and promote vaccination for all, but in doing so, I find they present the risks of measles worse than they are. While I’m sympathetic to the goal, I’m not a fan of bending the truth. Lies hurt the people who speak them and the ones who believe them, and they can hurt the health of immune-compromized children who are pushed to vaccinate. You will see my arguments below.

The CDC’s most-used value for the mortality rate for measles is 0.3%. It appears, for example, in line two of the following table from Orenstein et al., 2004. This table also includes measles-caused complications, broken down by type and patient age; read the full article here.

Measles complications, death rates, US, 1987-2000, CDC.

Measles complications, death rates, US, 1987-2000, CDC, Orenstein et. al. 2004.

The 0.3% average mortality rate seems more in tune with the 1800s than today. Similarly, note that the risk of measles-associated encephalitis is given as 10.1%, higher than the risk of measles-diarrhea, 8.2%. Do 10.1% of measles cases today produce encephalitis, a horrible, brain-swelling disease that often causes death. Basically everyone in the 1950s and early 60s got measles (I got it twice), but there were only 1000 cases of encephalitis per year. None of my classmates got encephalitis, and none died. How is this possible; it was the era before antibiotics. Even Orenstein et. al comment that their measles mortality rates appear to be far higher today than in the 1940s and 50s. The article explains that the increase to 3 per thousand, “is most likely due to more complete reporting of measles as a cause of death, HIV infections, and a higher proportion of cases among preschool-aged children and adults.”

A far more likely explanation is that the CDC value is wrong. That the measles cases that were reported and certified as such are the ones that are the most severe. There were about 450 measles deaths per year in the 1940s and 1950s, and 408 in 1962, the last year before the MMR vaccine was developed and by Dr. Hilleman of Merck (a great man of science, forgotten). In the last two decades there were some 2000 measles cases reported US cases but only one measles death. A significant decline in cases, but the ratio does not support the CDC’s death rate. For a better estimate, I propose to divide the total number of measles deaths in 1962 by the average birth rate in the late 1950s. That is to say, I propose to divide 408 by the 4.3 million births per year. From this, I calculate a mortality rate just under 0.01% in 1962, That’s 1/30th the CDC number, and medicine has improved since 1962.

I suspect that the CDC inflates the mortality numbers, in part by cherry-picking its years. It inflates them further by treating “reported measles cases.” as if they were all measles cases. I suspect that the reported cases in these years were mainly the very severe ones. Mild case measles clears up before being reported or certified as measles. This seems the only normal explanation for why 10.1% of cases include encephalitis, and only 8.2% diarrhea. It’s why the CDC’s mortality numbers suggest that, despite antibiotics, our death rate has gone up by a factor of 30 since 1962.

Consider the experience of people who lived in the early 60s. Most children of my era went to public elementary schools with some 1000 other students, all of whom got measles. By the CDC’s mortality number, we should have seen three measles deaths per school, and 101 cases of encephalitis. In reality, if there had been one death in my school it would have been big news, and it’s impossible that 10% of my classmates got encephalitis. Instead, in those years, only 48,000 people were hospitalized per year for measles, and 1,000 of these suffered encephalitis (CDC numbers, reported here).

To see if vaccination is a good idea, lets now consider the risk of vaccination. The CDC reports their vaccine “is virtually risk free”, but what does risk-free mean? A British study finds vaccination-caused neurological damage in 1/365,000 MMR vaccinations, a rate of 0.00027%, with a small fraction leading to death. These problems are mostly found in immunocompromised patients. I will now estimate the neurological risk for actual measles based on the ratio of encephalitis to births, as before using the average birth rate as my estimate for measles cases; 1000/4,300,000 = 0.023%. This is far lower than the risk the CDC reports, and more in line with experience.

The risk for neurological damage from measles that I calculate is 86 times higher risk than the neurological risk from vaccination, suggesting vaccination is a very good thing, on average: The vast majority of people should get vaccinated. But for people with a weakened immune system, my calculations suggest it is worthwhile to not immunize at 12 months as doctors recommend. The main cause of vaccination death is encephalitis, but this only happens in patients with weakened immune systems. If your child’s immune system is weakened, even by a cold, I’d suggest you wait 1-3 months, and would hope that your doctor would concur. If your child has AIDS, ALS, Lupus, or any other, long-term immune problem, you should not vaccinate at all. Not vaccinating your immune-weakened child will weaken the herd immunity, but will protect your child.

We live in a country with significant herd immunity: Even if there were a measles outbreak, it is unlikely there would be 500 cases at one time, and your child’s chance of running into one of them in the next month is very small assuming that you don’t take your child to Disneyland, or to visit relatives from abroad. Also, don’t hang out with anti-vaxers if you are not vaccinated. Associating with anti-vaxers will dramatically increase your child’s risk of infection.

As for autism: there appears to be no autism advantage to pushing off vaccination. Signs of autism typically appear around 12 months, the same age that most children receive their first-stage MMR shot, so some people came to associate the two. Parents who push-off vaccination do not push-off the child’s chance of developing autism, they just increase the chance their child will get measles, and that their child will infect others. Schools are right to bar such children, IMHO.

I’ve noticed that, with health care in, particular, there is a tendency for researchers to mangle statistics so that good things seem better than they are. Health food: is not necessarily so healthy as they say; nor is weight loss. Bicycle helmets: ditto. Sometimes this bleeds over to outright lies. Generic modified grains were branded as cancer-causing based on outright lies and  missionary zeal. I feel that I help a bit, in part by countering individual white lies; in part by teaching folks how to better read statistic arguments. If you are a researcher, I strongly suggest you do not set up your research with a hypothesis so that only one outcome will be publishable or acceptable. Here’s how.

Robert E. Buxbaum, December 9, 2018.

A plague of combined sewers

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The major typhoid and cholera epidemics of the US, and the plague of the Al Qaeda camps, 2009 are understood to have been caused by bad sewage, in particular by the practice of combining sanitary + storm sewage. Medieval plagues too may have been caused by combined sewers.

Combined sewer system showing an rain-induced overflow, a CSO.

A combined sewer system showing an rain-induced overflow, a CSO.

A combined system is shows at right. Part of the problem is that the outfall is hard to contain, so they tend to spew sewage into the lakes and drinking water as shown. They are also more prone to back up during rain storms; separated systems can back up too, but far less often. When combined sewers back up, turds and other infected material flows into home basements. In a previous post, “follow the feces,” I showed the path that Oakland county’s combined sewers outfalls take when they drain (every other week) into Lake St. Clair just upstream of the water intake, and I detailed why, every few years we back up sewage into basements. I’d like to now talk a bit more about financial cost and what I’d like to do.

The combined sewer system shown above includes a small weir dam. During dry periods and small rain events, the dam keeps the sewage from the lake by redirecting it to the treatment plant. This protects the lakes so that sometimes the beaches are open, but there’s an operation cost: we end up treating a lot of rain water as if it were sewage. During larger rains, the dam overflows. This protects our basements (usually) but it does so at the expense of the drinking water, and of the water in Lake St. Clair and Lake Erie.

As a way to protect our lakes somewhat Oakland county has added a retention facility, the George W. Kuhn. This facility includes the weir shown above and a huge tank for sewage overflows. During dry periods, the weir holds back the flow of toilet and sink water so that it will flow down the pipe (collector) to the treatment plant in Detroit, and so it does not flow into the lake. Treatment in Detroit is expensive, but nonpolluting. During somewhat bigger rains the weir overflows to the holding tank. It is only during yet-bigger rains (currently every other week) that the mixture of rain and toilet sewage overwhelms the tank and is sent to the river and lake. The mess this makes of the lake is shown in the video following. During really big rains, like those of August 2014, the mixed sewage backs up in the pipes, and flows back into our basements. With either discharge, we run the risk of plague: Typhoid, Cholera, Legionnaires…

Some water-borne plagues are worse than others. With some plagues, you can have a carrier, a person who can infect many others without becoming deathly sick him or herself. Typhoid Mary was a famous carrier of the 1920s. She infected (and killed) hundreds in New York without herself becoming sick. A more recent drinking water plague, showed up in Milwaukee 25 years ago. Some 400,000 people were infected, and 70 or so died. Milwaukee disinfected its drinking water with chlorine and a bacteria that entered the system was chlorine tolerant. Milwaukee switched to ozone disinfection but Detroit still uses chlorine.

Combined sewers require much larger sewage treatment plants than you’d need for just sanitary sewage. Detroit’s plant is huge and its size will need to be doubled to meet new, stricter standards unless we bite the bullet and separate our sewage. Our current system doesn’t usually meet even the current, lower standards. The plant overflows and operation cost are high since you have to treat lots of rainwater. These operation costs will keep getter higher as pollution laws get tougher.

In Oakland county, MI we’ve started to build more and more big tanks to hold back and redirect the water so it doesn’t overwhelm the sewage plants. The GWK tank occupies 1 1/2 miles by about 100 feet beneath a golf course. It’s overwhelmed every other week. Just think of the tank you’d need to hold the water from 4″ of rain on 900 square mile area (Oakland county is 900 square miles). Oakland’s politicians seem happy to spend money on these tanks because it creates jobs and graft and because it suggests that something is being done. They blame politics when rain overwhelms the tank. I say it’s time to end the farce and separate our sewers. My preference is to separate the sewers through the use of French drains or bio-swales, and through the use of weir dams. I’m running for drain commissioner. Here’s something I’ve written on the chemistry of sewage, and on the joy of dams.

Dr. Robert E. Buxbaum, July 1-Sept 16, 2016.