Tag Archives: disease

Use iodine against Bad breath, Bad beer, Flu, RSV, COVID, monkeypox….

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We’re surrounded by undesired bacteria, molds, and viruses. Some are annoying, making our feet smell, our teeth rot, and our wine sour. Others are killers, particularly for the middle aged and older. Despite little evidence, the US government keeps pushing masks and inoculations with semi-active vaccine that does nothing to stop the spread. Among the few things one can do to stop the spread of disease, and protect yourself, is to kill the bacteria, molds and viruses with iodine. Iodine is cheap, effective even at very low doses, 0.1% to 10 parts per million, and it lasts a lot longer than alcohol. Dilute iodine will not dye your skin, and it does not sting. A gargle of iodine will kill COVID and other germs (e.g. thrush) and it has even been shown to be a protective, stopping COVID 19 and flu even if used before exposure. On a more practical level. I also use it to cleanse my barrels before making beer — It’s cheaper than the Camden they sell in stores.

Iodine is effective when used on surfaces, and most viruses spread by surfaces. A sick person coughs. Droplets end up on door knobs, counters, or in your throat, leaving virus particles that do not die in air. You touch the surface, and transfer the virus to your eyes and nose. Here’s a video I made. A mask doesn’t help because you rub your eyes around the mask. But iodine kills the virus on the surface, and on your hands, and lasts there far longer than alcohol does. Vaccines always come with side-effects, but there are no negative side effects to sanitization with dilute iodine. Here is a video I did some years ago on the chemistry of iodine.

Robert Buxbaum, February 1, 2023. I don’t mean to say that all bacteria and fungi are bad, it’s just that most of them are smelly. Even the good ones that give us yogurt, beer, blue cheese, and sour kraut tend to be smelly. They have the annoying tendency to causing your wine to taste and smell like sour kraut or cheese, and they cause your breath and feet to smell the same. If you’re local, I’ll give you some free iodine solution. Otherwise, you’ll have to buy it through REB Research.

Hand washing and masks help, just not that much.

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There are two main routes for catching flu. One is via your hands and your eyes and nose. Your hands pick up germs from the surfaces you touch, and when you touch your eyes or nose passages, the germs infect you. This was thought to be the main route for infection, and I still think it is. I’d been pushing iodine hand sanitizer for some time, the stuff used in hospitals, saying that that the alcohol hand sanitizer doesn’t work well, that it evaporates.

The other route, the one touted by the press these days is via direct cough droplets, breathing them in or getting them in your eyes. Masks and face shields are the preferred protection from this route, and the claim is that masks will stop 63% of the spread. The 63% number has an interesting history, it comes from this test with infected hamsters. Hamsters are 63% less likely to infect other hamsters when they wear a mask. Of course, the comparison has some weaknesses: hamsters don’t put their fingers in their noses, nor do they rub their eyes with their hands, and hamsters can be forced to keep the mask barrier all the time — read the study to see how.

A more realistic study, or more relevant to people, in my opinion showed a far lower effect for masks, about 20%. During the HiNi flu pandemic of 2009 a group of 1437 college students at a single university were divided into three randomized groups, see the original report here. Students at a few chosen residence halls were instructed to wash their hands regularly, use sanitizer, and wear masks. Students at other halls were either told to wear masks only, or told to go on as they pleased. This was the largest group, the control. They included students of the the largest residence hall on campus. The main results appear as the graph below, Figure 1 of the report. It shows a difference of 6% or 20%, depending on how you look at things, with the mask plus hand-health group, MPHH, doing the best.

After 6 weeks of monitoring, approximately 36% of the control group had gotten the flu or some collection of flu symptoms. The remaining 64% of the residents remained symptom free. This is he darkest line above.

Of the FM Only group, the medium line above, those instructed to wear face masks only. 30% of this group showed flu symptoms, with 70% remaining symptom free. Clearly masks do help with humans, but far less than what you’d expect from the news reports.

Sweden kept the primary schools open and allows people to wear masks and social distance at they see fit. The death toll to August 1 is identical to Michigan, or slightly bette Sweden’s top virologist recommends that the US follow suit. Open up and trust people.

The group that did best was FMHH, the group who both wore facemarks and used hand health, regular hand washing plus hand sanitizer. This group reported an average of 3.5 hours per day of mask use above the control group average. This is about as good or better than I see in Michigan. Adding the hand health provided an additional 1% improvement, or a 3% improvement, depending on how you look at these things. The press claims hand health is wasted effort, but I’m not so sure. I argue that the effect was significant, and that the hand sanitizer was bad. I argue that iodine hand wash would have done better at far less social cost.

I also note that doing nothing was not that much worse than mask use. This matches with the observation of COVID-19 in Sweden. With no enforced social distancing, Sweden did about the same as Michigan — slightly better, despite Michigan closing the schools and restaurants, and imposing some of the toughest requirements for social distancing and mask use.

Other things that affect how likely you are to get flu symptoms. I find these rustles more interesting than the main face-mask result.

There were other observations from the university study that i found isignificant. There are racial differences and social differences. The authors didn’t highlight these, but they are at least as large as the effect of mask use. Asians got the flu only 70% as often as others, while black students got it 8% more often. This matches what has been seen in the US with COVID-19. Also interesting, those with a recent flu shot got flu more often; those with physical activity 13% more often. Smokers got the flu less than non-smokers and women got it 22% more often than men. The last two are the reverse with COVID-19. I could speculate on the reasons, but clearly there is a lot going on.

Why did Asians do better than others? Perhaps Asians have had prior exposure to some similar virus, and are thus slightly immune, or perhaps they used the masks more, being more socially acceptable. Why were smokers protected? It’s likely that smoke kills germs; was that the cause. These are speculations, and as for the rest I don’t know.

I am not that bothered that the students probably didn’t wear their masks 100% of the time. Better would be better, but even with mask use 100% of the time, there are other known routes that are almost impossible to remove: clothing, food, touching your face. I still think there is a big advantage to iodine hand wash, and I suspect we would be better off opening up a bit too.

Robert Buxbaum August 7, 2020.

COVID-19 is worse than SARS, especially for China.

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The corona virus, COVID-19 is already a lot worse than SARS, and it’s likely to get even worse. As of today, there are 78,993 known cases and 2,444 deaths. By comparison, from the first appearance of SARS about December 1 2002, there have been a total of 8439 cases and 813 deaths. It seems the first COVID-19 patient was also about December 1, but the COVID-19 infection moved much faster. Both are viral infections, but it seems the COVID virus is infectious for more days, including days when the patient is asymptomatic. Quarantine is being used to stop COVID-19; it was successful with SARS. As shown below, by July 2003 SARS had stopped, essentially. I don’t think COVID-19 will stop so easily.

The process of SARS, worldwide; a dramatic rise and it’s over by July 2003. Source: Int J Health Geogr. 2004; 3: 2. Published online 2004 Jan 28. doi: 10.1186/1476-072X-3-2.

We see that COVID-19 started in November, like SARS, but we already have 10 times more cases than the SARS total, and 150 times more than we had at this time during the SARS epidemic. If the disease stops in July, as with SARS, we should expect to see about a total of 150 times the current number of cases: about 12 million cases by July 2020. Assuming a death rate of 2.5%, that suggests 1/4 million dead. This is a best case scenario, and it’s not good. It’s about as bad as the Hong Kong flu pandemic of 1968-69, a pandemic that killed 60,000 approximately in the US, and which remains with us, somewhat today. By the summer of 69, the spreading rate R° (R-naught) fell below 1 for and the disease began to die out, a process I discussed previously regarding measles and the atom bomb, but the disease re-emerged, less infectious the next winter and the next. A good quarantine is essential to make this best option happen, but I don’t believe the Chinese have a good-enough quarantine.

Several things suggest that the Chinese will not be able to stop this disease, and thus that the spread of COVID-19 will be worse than that of the HK flu and much worse than SARS. For one, both those disease centered in Hong Kong, a free, modern country, with resources to spend, and a willingness to trust its citizens. In fighting SARS, HK passed out germ masks — as many as anyone needed, and posted maps of infection showing places where you can go safely and where you should only go with caution. China is a closed, autocratic country, and it has not treated quarantine this way. Little information is available, and there are not enough masks. The few good masks in China go to the police. Health workers are dying. China has rounded up anyone who talks about the disease, or who they think may have the disease. These infected people are locked up with the uninfected in giant dorms, see below. In rooms like this, most of the uninfected will become infected. And, since the disease is deadly, many people try to hide their exposure to avoid being rounded up. In over 80% of COVID cases the symptoms are mild, and somewhat over 1% are asymptomatic, so a lot of people will be able to hide. The more people do this, the poorer the chance that the quarantine will work. Given this, I believe that over 10% of Hubei province is already infected, some 1.5 million people, not the 79,000 that China reports.

Wuhan quarantine “living room”. It’s guaranteed to spread the disease as much as it protects the neighbors.

Also making me think that quarantine will not work as well here as with SARS, there is a big difference in R°, the transmission rate. SARS infected some 2000 people over the first 120 days, Dec. 1 to April 1. Assuming a typical infection time of 15 days, that’s 8 cycles. We calculate R° for this stage as the 8th root of 2000, 8√2000 = 2.58. This is, more or less the number in the literature, and it is not that far above 1. To be successful, the SARS quarantine had to reduce the person’s contacts by a factor of 3. With COVID-19, it’s clear that the transmission rate is higher. Assuming the first case was December 1, we see that there were 73,437 cases in only 80. R° is calculated as the 5 1/3 root of 73,437. Based on this, R° = 8.17. It will take a far higher level of quarantine to decrease R° below 1. The only good news here is that COVID-19 appears to be less deadly than SARS. Based on Chinese numbers the death rate appears to be about 2000/73,437, or about 3%, varying with age (see table), but these numbers are overly high. I believe there are a lot more cases. Meanwhile the death rate for SARS was over 9%. For most people infected with COVID-19, the symptoms are mild, like a cold; for another 18% it’s like the flu. A better estimate for the death rate of COVID-19 is 0.5-1%, less deadly than the Spanish flu of 1918. The death rate on the Diamond Princess was 3/600 = 0.5%, with 24% infected.

The elderly are particularly vulnerable. It’s not clear why.

Backing up my value of R°, consider the case of the first Briton to contact the disease. As reported by CNN, he got it at conference in Singapore in late January. He left the conference, asymptomatic on January 24, and spent the next 4 days at a French ski resort where he infected one person, a child. On January 28, he flew to England where he infected 8 more before checking himself into a hospital with mild symptoms. That’s nine people infected over 3 weeks. We can expect that schools, factories, and prisons will be even more hospitable to transmission since everyone sits together and eats together. As a worst case extrapolation, assume that 20% of the world population gets this disease. That’s 1.5 billion people including 70 million Americans. A 1% death rate suggests we’ll see 700,000 US deaths, and 15 million world-wide this year. That’s almost as bad as the Spanish flu of 1918. I don’t think things will be that bad, but it might be. The again, it could be worse.

If COVID-19 follows the 1918 flu model, the disease will go into semi-remission in the summer, and will re-emerge in the fall to kill another few hundred thousand Americans in the next fall and winter, and the next after that. Woodrow Wilson got the Spanish Flu in the fall of 1918, after it had passed through much of the US, and it nearly killed him. COVID-19 could continue to rampage every year until a sufficient fraction of the population is immune or a vaccine is developed. In this scenario, quarantine will have no long-term effect. My sense is that quarantine and vaccine will work enough in the US to reduce the effect of COVID-19 to that of the Hong Kong flu (1968), so that the death rate will be only 0.1 – 0.2%. In this scenario, the one I think most likely, the US will experience some 100,000 deaths, that is 0.15% of 20% of the population, mostly among the elderly. Without good quarantine or vaccines, China will lose at least 1% of 20% = about 3 million people. In terms of economics, I expect a slowdown in the US and a major problem in China, North Korea, and related closed societies.

Robert Buxbaum, February 18, 2020. (Updated, Feb. 23, I raised the US death totals, and lowered the totals for China).

Kindness and Cholera in California

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California likely leads the nation in socially activist government kindness. It also leads the nation in homelessness, chronic homelessness, and homeless veterans. The US Council on Homelessnesses estimates that, on any given day, 129,972 Californians are homeless, including 6,702 family households, and 10,836 veterans; 34,332 people are listed among “the chronic homeless”. That is, Californians with a disability who have been continuously homeless for one year or cumulatively homeless for 12 months in the past three years. No other state comes close to these numbers. The vast majority of these homeless are in the richer areas of two rich California cities: Los Angeles and San Francisco (mostly Los Angeles). Along with the homeless in these cities, there’s been a rise in 3rd world diseases: cholera, typhoid, typhus, etc. I’d like to explore the relationship between the policies of these cities and the rise of homelessness and disease. And I’d like to suggest a few cures, mostly involving sanitation. 

A homeless encampment in LosAngeles

Most of the US homeless do not live in camps or on the streets. The better off US homelessness find it is a temporary situation. They survive living in hotels or homeless shelters, or they “couch-serf,” with family or friends. They tend to take part time jobs, or collect unemployment, and they eventually find a permanent residence. For the chronic homeless things are a lot grimmer, especially in California. The chronic unemployed do not get unemployment insurance, and California’s work rules tend to mean there are no part time jobs, and there is not even a viable can and bottle return system in California, so the homeless are denied even this source of income*. There is welfare and SSI, but you have to be somewhat stable to sign up and collect. The result is that California’s chronic homeless tend to live in squalor strewn tent cities, supported by food handouts.

Californians provide generous food handouts, but there is inadequate sewage, or trash collection, and limited access to clean water. Many of the chronic homeless are drug-dependent or mentally ill, and though they might  benefit from religion-based missions, Los Angeles has pushed the missions to the edges of the cities, away from the homeless. The excess food and lack of trash collection tends to breed rats and disease, and as in the middle ages, the rats help spread the diseases. 

Total homelessness by state, 2018; California leads the nation. The better off among these individuals do not live on the streets, but in hotels or homeless shelters. For most, this is a short term situation. The rest, about 20%, are chronically homeless. About half of these live on the streets without adequate sewage and water. Many are drug-dependent.

The first major outbreaks of the homeless camps appeared in Los Angeles in August and September of 2017. They reappeared in 2018, and by late summer, rates were roughly double 2017’s. This year, 2019, looks like it could be a real disaster. The first case of a typhoid infected police officer showed up in May. By June there were six police officers with typhoid, and that suggests record numbers are brewing among the homeless.

To see why sanitation is an important part of the cure, it’s worth noting that typhoid is a disease of unclean hands, and a relative of botulism. It is spread by people who go to the bathroom and then handle food without washing their hands first. The homeless camps do not, by and large, have hand washing stations. and forced hygiene is prohibited. Los Angeles has set up porta-potties, with no easy hand washing. The result is typhoid epidemic that’s even affecting the police (six policemen in June!).

rate od disease spread.
R-naught, reproduction number for some diseases, CDC.

Historically, the worst outbreaks of typhoid were spread by food workers. This was the case with “typhoid Mary of the early 20th century.” My guess is that some of the police who got typhoid, got it while trying to feed the needy. If so, this fellow could become another Typhoid Mary. Ideally, you’d want shelters and washing stations where the homeless are. You’d also want to pickup the dirtier among the homeless for forced washing and an occasional night in a homeless shelter. This is considered inhumane in Los Angeles, but they do things like this in New York, or they did.

Typhus is another major disease of the California homeless camps. It is related to typhoid but spread by rodents and their fleas. Infected rodents are attracted to the homeless camps by the excess food. When the rodents die, their infected fleas jump to the nearest warm body. Sometimes that’s a person, sometimes another animal. In a nastier city, like New York, the police come by and take away old food, dead animals, and dirty clothing; in Los Angeles they don’t. They believe the homeless have significant squatters rights. California’s kindness here results in typhus.

Reproduction number and generation time for some diseases.

The last of the major diseases of the homeless camps is cholera. It’s different from the others in that it is not dependent on squalor, just poor health. Cholera is an airborne disease, spread by coughing and sneezing. In California’s camps, the crazy and sick dwell close to each other and close to healthy tourists. Cholera outbreaks are a predictable result. And they can easily spread beyond the camps to your home town, and if that happens a national plague could spread really fast.

I’d discussed R-naught as a measure of contagiousness some months ago, comparing it to the reproductive number of an atom bomb design, but there is more to understanding a disease outbreak. R-naught refers merely to the number of people that each infected person will infect before getting cured or dying. An R-naught greater than one means the disease will spread, but to understand the rate of spread you also need the generation time. That’s the average time between when the host becomes infected, and when he or she infects others. The chart above shows that, for cholera, r-naught is about 10, and the latency period is short, about 9 days. Without a serious change in California’s treatment of the homeless, each cholera case in June will result in over 100 cases in July, and well over 10,000 in August. Cholera is somewhat contained in the camps, but once an outbreak leaves the camps, we could have a pandemic. Cholera is currently 80% curable by antibiotics, so a pandemic would be deadly.

Hygiene is the normal way to prevent all these outbreaks. To stop typhoid, make bathrooms available, with washing stations, and temporary shelters, ideally these should be run by the religious groups: the Salvation Army, the Catholic Church, “Loaveser and Fishes”, etc. To prevent typhus, clean the encampments on a regular basis, removing food, clothing, feces and moving squatters. For cholera, provide healthcare and temporary shelters where people will get clean water, clean food, and a bed. Allow the homeless to work at menial jobs by relaxing worker hiring and pay requirements. A high minimum wage is a killer that nearly destroyed Detroit. Allow a business to hire the homeless to sweep the street for $2/hour or for a sandwich, but make a condition that they wash their hands, and throw out the leftovers. I suspect that a lot of the problems of Puerto Rico are caused by a too-high minimum wage by the way. There will always be poor among you, says the Bible, but there doesn’t have to be typhoid among the poor, says Dr. Robert Buxbaum.

*California has a very strict can and bottle return law where — everything is supposed to be recycled– but there are very few recycling centers, and most stores refuse to take returns. This is a problem in big government states: it’s so much easier to mandate things than to achieve them.

July 30, 2019. I ran for water commissioner in Oakland county, Michigan, 2016. If there is interest, I’ll run again. One of my big issues is clean water. Oakland could use some help in this regard.

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.

Zombie invasion model for surviving plagues

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Imagine a highly infectious, people-borne plague for which there is no immunization or ready cure, e.g. leprosy or small pox in the 1800s, or bubonic plague in the 1500s assuming that the carrier was fleas on people (there is a good argument that people-fleas were the carrier, not rat-fleas). We’ll call these plagues zombie invasions to highlight understanding that there is no way to cure these diseases or protect from them aside from quarantining the infected or killing them. Classical leprosy was treated by quarantine.

I propose to model the progress of these plagues to know how to survive one, if it should arise. I will follow a recent paper out of Cornell that highlighted a fact, perhaps forgotten in the 21 century, that population density makes a tremendous difference in the rate of plague-spread. In medieval Europe plagues spread fastest in the cities because a city dweller interacted with far more people per day. I’ll attempt to simplify the mathematics of that paper without losing any of the key insights. As often happens when I try this, I’ve found a new insight.

Assume that the density of zombies per square mile is Z, and the density of susceptible people is S in the same units, susceptible population per square mile. We define a bite transmission likelihood, ß so that dS/dt = -ßSZ. The total rate of susceptibles becoming zombies is proportional to the product of the density of zombies and of susceptibles. Assume, for now, that the plague moves fast enough that we can ignore natural death, immunity, or the birth rate of new susceptibles. I’ll relax this assumption at the end of the essay.

The rate of zombie increase will be less than the rate of susceptible population decrease because some zombies will be killed or rounded up. Classically, zombies are killed by shot-gun fire to the head, by flame-throwers, or removed to leper colonies. However zombies are removed, the process requires people. We can say that, dR/dt = kSZ where R is the density per square mile of removed zombies, and k is the rate factor for killing or quarantining them. From the above, dZ/dt = (ß-k) SZ.

We now have three, non-linear, indefinite differential equations. As a first step to solving them, we set the derivates to zero and calculate the end result of the plague: what happens at t –> ∞. Using just equation 1 and setting dS/dt= 0 we see that, since ß≠0, the end result is SZ =0. Thus, there are only two possible end-outcomes: either S=0 and we’ve all become zombies or Z=0, and all the zombies are all dead or rounded up. Zombie plagues can never end in mixed live-and-let-live situations. Worse yet, rounded up zombies are dangerous.

If you start with a small fraction of infected people Z0/S0 <<1, the equations above suggest that the outcome depends entirely on k/ß. If zombies are killed/ rounded up faster than they infect/bite, all is well. Otherwise, all is zombies. A situation like this is shown in the diagram below for a population of 200 and k/ß = .6

FIG. 1. Example dynamics for progress of a normal disease and a zombie apocalypse for an initial population of 199 unin- fected and 1 infected. The S, Z, and R populations are shown in (blue, red, black respectively, with solid lines for the zombie apocalypse, and lighter lines for the normal plague. t= t Nß where N is the total popula- tion. For both models the k/ß = 0.6 to show similar evolutions. In the SZR case, the S population disap- pears, while the SIR is self limiting, and only a fraction of the population becomes infected.

Fig. 1, Dynamics of a normal plague (light lines) and a zombie apocalypse (dark) for 199 uninfected and 1 infected. The S and R populations are shown in blue and black respectively. Zombie and infected populations, Z and I , are shown in red; k/ß = 0.6 and τ = tNß. With zombies, the S population disappears. With normal infection, the infected die and some S survive.

Sorry to say, things get worse for higher initial ratios,  Z0/S0 >> 0. For these cases, you can kill zombies faster than they infect you, and the last susceptible person will still be infected before the last zombie is killed. To analyze this, we create a new parameter P = Z + (1 – k/ß)S and note that dP/dt = 0 for all S and Z; the path of possible outcomes will always be along a path of constant P. We already know that, for any zombies to survive, S∞ = 0. We now use algebra to show that the final concentration of zombies will be Z∞ = Z0 + (1-k/ß)S0. Free zombies survive so long as the following ratio is non zero: Z0/S0 + 1- k/ß. If Z0/S0 = 1, a situation that could arise if a small army of zombies breaks out of quarantine, you’ll need a high kill ratio, k/ß > 2 or the zombies take over. It’s seen to be harder to stop a zombie outbreak than to stop the original plague. This is a strong motivation to kill any infected people you’ve rounded up, a moral dilemma that appears some plague literature.

Figure 1, from the Cornell paper, gives a sense of the time necessary to reach the final state of S=0 or Z=0. For k/ß of .6, we see that it takes is a dimensionless time τ of 25 or to reach this final, steady state of all zombies. Here, τ= t Nß and N is the total population; it takes more real time to reach τ= 25 if N is high than if N is low. We find that the best course in a zombie invasion is to head for the country hoping to find a place where N is vanishingly low, or (better yet) where Z0 is zero. This was the main conclusion of the Cornell paper.

Figure 1 also shows the progress of a more normal disease, one where a significant fraction of the infected die on their own or develop a natural immunity and recover. As before, S is the density of the susceptible, R is the density of the removed + recovered, but here I is the density of those Infected by non-zombie disease. The time-scales are the same, but the outcome is different. As before, τ = 25 but now the infected are entirely killed off or isolated, I =0 though ß > k. Some non-infected, susceptible individuals survive as well.

From this observation, I now add a new conclusion, not from the Cornell paper. It seems clear that more immune people will be in the cities. I’ve also noted that τ = 25 will be reached faster in the cities, where N is large, than in the country where N is small. I conclude that, while you will be worse off in the city at the beginning of a plague, you’re likely better off there at the end. You may need to get through an intermediate zombie zone, and you will want to get the infected to bury their own, but my new insight is that you’ll want to return to the city at the end of the plague and look for the immune remnant. This is a typical zombie story-line; it should be the winning strategy if a plague strikes too. Good luck.

Robert Buxbaum, April 21, 2015. While everything I presented above was done with differential calculus, the original paper showed a more-complete, stochastic solution. I’ve noted before that difference calculus is better. Stochastic calculus shows that, if you start with only one or two zombies, there is still a chance to survive even if ß/k is high and there is no immunity. You’ve just got to kill all the zombies early on (gun ownership can help). Here’s my statistical way to look at this. James Sethna, lead author of the Cornell paper, was one of the brightest of my Princeton PhD chums.