One outcome of the pandemic is that I have not had any illness (knock on wood), nary a cold nor sniffle, in a year and a half. On the other hand, my skin has fallen apart. I am constantly inflamed and itchy. I have no proof that the two are connected but my working hypothesis is that my immune system is hypersensitive right now because it has had little to do since the spring of 2020. It now overreacts to every mold spore, pollen grain, and speck of dust it runs into. The immune system is extremely complex, perhaps as complex as the brain. Its job is extremely difficult. It needs to recognize threats and eliminate them while not attacking itself. The brain and the immune system are intricately linked. How many people have gotten ill immediately after a final exam or deadline? The immune system was informed by the brain to delay action until the task was completed. The brain probably takes cues form the immune system too. One hypothesis for why asthma and allergies have been on the rise recently is that modern living has eliminated much contact with parasites and infectious agents, making the immune system hypersensitive. I for one, always welcome vaccinations because it gives my immune system something to do. In fact, I think it would be a good idea to get inoculations of all types regularly. I would take a vaccine for tape worm in a heartbeat. We are now slowly exiting from a global experiment in depriving the immune system of stimulation. We have no idea what the consequences will be. That is not to say that quarantine and isolation was not a good idea. Being itchy is clearly better than being infected by a novel virus (or being dead). There can be long term effects of infection too. Long covid is likely to be due to a miscalibrated immune system induced by the infection. Unfortunately, we shall likely never disentangle all the effects of COVID-19. We will not ever truly know what the long term consequences of infection, isolation, and vaccination will be. Most people will come out of this fine but a small fraction will not and we will not know why.

# Covid-19

# RNA

I read an article recently about an anti-vaccination advocate exclaiming at a press conference with the governor of Florida that vaccines against SARS-CoV-2 “change your RNA!” This made me think that most people probably do not know much about RNA and that a little knowledge is a dangerous thing. Now ironically, contrary to what the newspapers say, this statement is kind of true although in a trivial way. The Moderna and Pfizer vaccines insert a little piece of RNA into your cells (or rather cells ingest them) and that RNA gets translated into a SARS-CoV-2 spike protein that gets expressed on the surface of the cells and thereby presented to the immune system. So, yes these particular vaccines (although not all) have changed your RNA by adding new RNA to your cells. However, I don’t think this is what the alarmist was worried about. To claim that something that changes is a bad thing implies that the something is fixed and stable to start with, which is profoundly untrue about RNA.

The central dogma of molecular biology is that genetic information flows from DNA to RNA to proteins. All of your genetic material starts as DNA organized in 23 pairs of chromosomes. Your cells will under various conditions transcribe this DNA into RNA, which is then translated into proteins. The biological machinery that does all of this is extremely complex and not fully understood and part of my research is trying to understand this better. What we do know is that transcription is an extremely noisy and imprecise process at all levels. The molecular steps that transcribe DNA to RNA are stochastic. High resolution images of genes in the process of transcription show that transcription occurs in random bursts. RNA is very short-lived, lasting between minutes to at most a few days. There is machinery in the cell dedicated to degrading RNA. RNA is spliced; it is cut up into pieces and reassembled all the time and this splicing happens more or less randomly. Less than 2% of your DNA codes for proteins but virtually all of the DNA including noncoding parts are continuously being transcribed into small RNA fragments. Your cell is constantly littered with random stray pieces of RNA, and only a small fraction of it gets translated into proteins. Your RNA changes. All. The. Time.

Now, a more plausible alarmist statement (although still untrue) would be to say that vaccines change your DNA, which could be a bad thing. Cancer after all involves DNA mutations. There are viruses (retroviruses) that insert a copy of its RNA code into the host’s DNA. HIV does this for example. In fact, a substantial fraction of the human genome is comprised of viral genetic material. Changing proteins can also be very bad. Prion diseases are basically due to misfolded proteins. So DNA changing is not good, protein changing is not good, but RNA changing? Nothing to see here.

# COVID, COVID, COVID

Even though Covid- is going to be with us forever, I actually think on the whole the pandemic turned out better than expected, and I mean that in the technical sense. If we were to rerun the pandemic over and over again, I think our universe will end up with fewer deaths than average. That is not to say we haven’t done anything wrong. We’ve botched up many things of course but given that the human default state is incompetence, we botched less than we could have.

The main mistake in my opinion was the rhetoric on masks in March of 2020. Most of the major Western health agencies recommended against wearing masks at that time because they 1) there was already a shortage of N95 masks for health care workers and 2) they thought that cloth and surgical masks were not effective in keeping one from being infected. Right there is a perfect example of Western solipsism; masks were only thought of as tools for self-protection, rather than as barriers for transmission. If only it was made clear early on that the reason we wear masks is not to *protect me from you* but to *protect you from me*. (Although there is evidence that masks do protect the wearer too, see here). This would have changed the rhetoric over masks we are having right now. The anti-maskers would be defending their right to harm others rather than the right to not protect themselves from harm.

The thing we got right was in producing effective vaccines. That was simply astonishing. There had never been a successful mRNA-based drug of any type until the BioNTech and Moderna vaccines. Many things had to go right for the vaccines to work. We needed a genetic sequence (Chinese scientists made it public in January), from that sequence we needed a target (the coronavirus spike protein), we needed to be able to stabilize the spike (research that came out of the NIH vaccine center), we needed to make mRNA less inflammatory (years of work especially at Penn), we needed a way to package that mRNA (work out of MIT), and we needed a sense of urgency to get it done (Western governments). Vaccines don’t always work but we managed to get one in less than a year. So many things had to go right for that to happen. The previous US administration should be taking a victory lap because it was developed under their watch, instead of bashing it.

As I’ve said before, I am skeptical we can predict what will happen next but I am going to predict now that there will not be a variant in the next year that will escape from our current vaccines. We may need booster shots and minor tweaks but the vaccines will continue to work. Part of my belief stems from the work of JC Phillips who argues that the SARS-CoV-2 spike protein is already highly optimized and thus there is not much room for it to change and to become infectious. The virus may mutate to replicate faster within the body but the spike will be relatively stable and thus remain a target for the vaccines. The delta variant wave we’re seeing now is a pandemic of the unvaccinated. I have no idea if those against vaccinations will have a change of heart but at some point everyone will be infected and have some immune protection. (I just hope they approve the vaccine for children before winter). SARS-CoV-2 will continue to circulate just like the way the flu strain from the 1918 pandemic still circulates but it won’t be the danger and menace it is now.

# The final stretch

The end of the Covid-19 pandemic is within reach. The vaccines have been a roaring success and former Bell Labs physicist J.C. Phillips predicted it (see here). He argued that the spike protein, which is the business end of the SARS-CoV-2 virus, has been optimized to such a degree in SARS-CoV-2 that even a small perturbation from a vaccine can disrupt it. While the new variants perturb the spike slightly and seem to spread faster, they will not significantly evade the vaccine. However, just because the end is within sight doesn’t mean we should not still be vigilante and not mess this up. Europe has basically scored multiple own goals these past few months with their vaccine rollout (or lack thereof) that is a combination of both gross incompetence and excessive conservatism. The Astra-Zeneca vaccine fiasco was a self-inflicted wound by all parties involved. The vaccine is perfectly fine and any side effects are either not related to the vaccine or of such low probability that it should not be a consideration for halting its use. By artificially slowing vaccine distribution, there is a chance that some new mutation could arise that will evade the vaccine. Europe needs to get its act in gear. The US has steadily ramped up vaccinations and is on course to have all willing adults vaccinated by start of summer. Although there has been a plateauing and even slight rise recently because of relaxation from social distancing in some areas, cases and deaths will drop for good by June everywhere in the US. North America will largely be back to normal by mid-summer. However, it is imperative that we press forward and vaccinate the entire world. We will also all need to get booster shots next fall when we get our flu shots.

# ICCAI talk

I gave a talk at the International Conference on Complex Acute Illness (ICCAI) with the title Forecasting COVID-19. I talked about some recent work with FDA collaborators on scoring a large number of publicly available epidemic COVID-19 projection models and show that they are unable to reliably forecast COVID-19 beyond a few weeks. The slides are here.

# Why it is so hard to forecast COVID-19

I’ve been actively engaged in trying to model the COVID-19 pandemic since April and after 5 months I am pretty confident that models can estimate what is happening at this moment such as the number of people who are currently infected but not counted as a case. Back at the end of April our model predicted that the case ascertainment ratio ( total cases/total infected) was on the order of 1 in 10 that varied drastically between regions and that number has gone up with the advent of more testing so that it may now be on the order of 1 in 4 or possibly higher in some regions. These numbers more or less the anti-body test data.

However, I do not really trust my model to forecast what will happen a month from now much less six months. There are several reasons. One is that while the pandemic is global the dynamics are local and it is difficult if not impossible to get enough data for a detailed fine grained model that captures all the interactions between people. Another is that the data we do have is not completely reliable. Different regions define cases and deaths differently. There is no universally accepted definition for what constitutes a case or a death and the definition can change over time even for the same region. Thus, differences in death rates between regions or months could be due to differences in the biology of the virus, medical care, or how deaths are defined and when they are recorded. Depending on the region or time, a person with a SARS-CoV-2 infection who dies of a cardiac arrest may or may not be counted as a COVID-19 death. Deaths are sometimes not officially recorded for a week or two, particularly if the physician is overwhelmed with cases.

However, the most important reason models have difficulty forecasting the future is that modeling COVID-19 is as much if not more about modeling the behavior of people and government policy than modeling the biology of disease transmission and we are just not very good at predicting what people will do. This was pointed out by economist John Cochrane months ago, which I blogged about (see here). You can see why getting behavior correct is crucial to modeling a pandemic from the classic SIR model

where and are the infected and susceptible fractions of the initial population, respectively. Behavior greatly affects the rate of infection and small errors in amplify exponentially. Suppression and mitigation measures such as social distancing, mask wearing, and vaccines reduce , while super-spreading events increase . The amplification of error is readily apparent near the onset of the pandemic where grows like . If you change by , then the will grow like and thus the ratio is growing (or decaying) exponentially like . The infection rate also appears in the initial reproduction number . From a previous post, I derived approximate expressions for how long a pandemic would last and show that it scales as and thus errors in will produce errors , which could result in errors in how long the pandemic will last, which could be very large if is near one.

The infection rate is different everywhere and constantly changing and while it may be possible to get an estimate of it from the existing data there is no guarantee that previous trends can be extrapolated into the future. So while some of the COVID-19 models do a pretty good job at forecasting out a month or even 6 weeks (e.g. see here), I doubt any will be able to give us a good sense of what things will be like in January.

# There is no herd immunity

In order for an infectious disease (e.g. COVID-19) to spread, the infectious agent (e.g. SARS-CoV-2) must jump from one person to another. The rate of this happening depends on the rate that an infectious person will come into contact with a susceptible person multiplied by the rate of the virus making the jump when the two people are nearby. The reproduction number R is obtained from the rate of infection spread times the length of time a person is infectious. If R is above one then a single person will infect more than one person on average and thus the pandemic will grow. If it is below one, then the pandemic will diminish. Herd immunity happens when enough people have been infected that the rate of finding a susceptible person becomes low enough that R drops below one. You can find the math behind this here.

However, a major assumption behind herd immunity is that once a person is infected they can never be infected again and this is not true for many infectious diseases such as other corona-viruses and the flu. There are reports that people can be reinfected by SARS-CoV-2. This is not fully validated but my money is on there being no lasting immunity to SARS-CoV-2 and this means that there is *never* any herd immunity. COVID-19 will just wax and wane forever.

This doesn’t necessarily mean it will be deadly forever. In all likelihood, each time you are infected your immune response will be more measured and perhaps SARS-CoV-2 will eventually be no worse than the common cold or the seasonal flu. But the fatality rate for first time infection will still be high, especially for the elderly and vulnerable. Those people will need to remain vigilante until there is a vaccine, and there is still no guarantee that a vaccine will work in the field. If we’re lucky and we get a working vaccine, it is likely that vaccine will not have lasting effect and just like the flu we will need to be vaccinated annually or even semi-annually.

# Another Covid-19 plateau

The world seems to be in another Covid-19 plateau for new cases. The nations leading the last surge, namely the US, Russia, India, and Brazil are now stabilizing or declining, while some regions in Europe and in particular Spain are trending back up. If the pattern repeats, we will be in this new plateau for a month or two and then trend back up again, just in time for flu season to begin.

# Why we need a national response

It seems quite clear now that we do not do a very good job of projecting COVID-19 progression. There are many reasons. One is that it is hard to predict how people and governments will behave. A fraction of the population will practice social distancing and withdraw from usual activity in the absence of any governmental mandates, another fraction will not do anything different no matter any official policy and the rest are in between. I for one get more scared of this thing the more I learn about it. Who knows what the long term consequences will be particularly for autoimmune diseases. The virus is triggering a massive immune response everywhere in the body and it could easily develop a memory response to your own cells in addition to the virus.

The virus also spreads in local clusters that may reach local saturation before infecting new clusters but the cross-cluster transmission events are low probability and hard to detect. The virus reached American shores in early January and maybe even earlier but most of those early events died out. This is because the transmission rate is highly varied. A mean reproduction number of 3 could mean everyone has R=3 or that most people transmit with R less than 1 while a small number (or events) transmit with very high R. (Nassim Nicholas Taleb has written copiously on the hazards of highly variable (fat tailed) distributions. For those with mathematical backgrounds, I highly recommend reading his technical volumes: The Technical Incerto. Even if you don’t believe most of what he says, you can still learn a lot.) Thus it is hard to predict when an event will start a local epidemic, although large gatherings of people (i.e. weddings, conventions, etc.) are a good place to start. Once the epidemic starts, it grows exponentially and then starts to saturate either by running out of people in the locality to infect or people changing their behavior or more likely both. Parts of New York may be above the herd immunity threshold now.

Thus at this point, I think we need to take a page out of Taleb’s book (like literally as my daughter would say), and don’t worry too much about forecasting. We can use it as a guide but we have enough information to know that most people are susceptible, about a third will be asymptomatic if infected (which doesn’t mean they won’t have long term consequences), about a fifth to a tenth will be counted as a case, and a few percent of those will die, which strongly depends on age and pre-existing conditions. We can wait around for a vaccine or herd immunity and in the process let many more people die, ( I don’t know how many but I do know that total number of deaths is a nondecreasing quantity), or we can act now everywhere to shut this down and impose a strict quarantine on anyone entering the country until they have been tested negative 3 times with a high specificity PCR test (and maybe 8 out of 17 times with a low specificity and sensitivity antigen test).

Acting now everywhere means, either 1) shutting everything down for at least two weeks. No Amazon or Grubhub or Doordash deliveries, no going to Costco and Walmart, not even going to the super market. It means paying everyone in the country without an income some substantial fraction of their salary. It means distributing two weeks supply of food to everyone. It means truly essential workers, like people keeping electricity going and hospital workers, live in a quarantine bubble hotel, like the NBA and NHL or 2) Testing everyone everyday who wants to leave their house and paying them to quarantine at home or in a hotel if they test positive. Both plans require national coordination and a lot of effort. The CARES act package has run out and we are heading for economic disaster while the pandemic rages on. As a recent president once said, “What have you got to lose?”

# How to make a fast but bad COVID-19 test good

Among the myriad of problems we are having with the COVID-19 pandemic, faster testing is one we could actually improve. The standard test for the presence of SARS-CoV-2 virus uses PCR (polymerase chain reaction), which amplifies targeted viral RNA. It is accurate (high specificity) but requires relatively expensive equipment and reagents that are currently in short supply. There are reports of wait times of over a week, which renders a test useless for contact tracing.

An alternative to PCR is an antigen test that tests for the presence of protein fragments associated with COVID-19. These tests can in principle be very cheap and fast, and could even be administered on paper strips. They are generally much more unreliable than PCR and thus have not been widely adopted. However, as I show below by applying the test multiple times, the noise can be suppressed and a poor test can be made arbitrarily good.

The performance of binary tests are usually gauged by two quantities – sensitivity and specificity. Sensitivity is the probability that you test positive (i.e are infected) given that you actually are positive (true positive rate). Specificity is the probability that you test negative if you actually are negative (true negative rate). For a pandemic, sensitivity is more important than specificity because missing someone who is infected means you could put lots of people at risk while a false positive just means the person falsely testing positive is inconvenienced (provided they cooperatively self-isolate). Current PCR tests have very high specificity but relatively low sensitivity (as low as 0.7) and since we don’t have enough capability to retest, a lot of tested infected people could be escaping detection.

The way to make any test have arbitrarily high sensitivity and specificity is to apply it multiple times and take some sort of average. However, you want to do this with the fewest number of applications. Suppose we administer tests on the same subject, the probability of getting more than positive tests if the person is positive is , where is the cumulative distribution function of the Binomial distribution (i.e. probability that the number of Binomial distributed events is less than or equal to ). If the person is negative then the probability ofÂ or fewer positives is . We thus want to find the minimal given a desired sensitivity and specificity, and . This means that we need to solve the constrained optimization problem: find the minimal under the constraint that , and . decreases and increases with increasing and *vice versa* for . We can easily solve this problem by sequentially increasing and scanning through until the two constraints are met. I’ve included the Julia code to do this below.Â For example, starting with a test with sensitivity .7 and specificity 1 (like a PCR test), you can create a new test with greater than .95 sensitivity and specificity, by administering the test 3 times and looking for a single positive test. However, if the specificity drops to .7 then you would need to find more than 8 positives out of 17 applications to be 95% sure you have COVID-19.

using Distributions

function Q(k,n,q)

d = Binomial(n,q)

return 1 – cdf(d,k)

endfunction R(k,n,r)

d = Binomial(n,1-r)

return cdf(d,k)

endfunction optimizetest(q,r,qp=.95,rp=.95)

nout = 0

kout = 0for n in 1:100

for k in 0:n-1

println(R(k,n,r),” “,Q(k,n,q))

if R(k,n,r) >= rp && Q(k,n,q) >= qp

kout=k

nout=n

break

end

end

if nout > 0

break

end

endreturn nout, kout

end

# Slides for Covid-19 talk

Here are my slides for my recent COVID-19 talk at the Centre for Applied Mathematics in BioScience and Medicine (CAMBAM). It’s an updated version of the one I gave to the FDA.

# Remember the ventilator

According to our model, the global death rate due to Covid-19 is around 1 percent for all infected (including unreported). However, if it were not for modern medicine and in particular the ventilator, the death rate would be much higher. Additionally, the pandemic first raged in the developed world and is only recently engulfing parts of the world where medical care is not as ubiquitous although this may be mitigated by a younger populace in those places. The delay between the appearance of a Covid-19 case and deaths is also fairly long; our model predicts a mean of over 50 days. So the lower US death rate compared to April could change in a month or two when the effects of the recent surges in the US south and west are finally felt.

# How long and how high for Covid-19

Cases of Covid-19 are trending back up globally and in the US. The world has nearly reached 10 million cases with over 2.3 million in the US. There is still a lot we don’t understand about SARS-CoV-2 transmission but I am confident we are no where near herd immunity. Our model is consistently showing that the case ascertainment ratio, that is the ratio of official Covid-19 cases to total SARS-CoV-2 infections, is between 5 and 10. That means that the US has less than 25 million infections while the world is less than 100 million.

Herd immunity means that for any fixed reproduction number, R0, the number of active infections will trend downward if the fraction of the initially susceptible population falls below 1/R0, or the total number infected is higher than 1- 1/R0. Thus, for an R0 of 4, three quarters of the population needs to be infected to reach herd immunity. However, the total number that will eventually be infected, as I will show below, will be

which is considerably higher. Thus, mitigation efforts to reduce R0 will reduce the total number infected. (2020-06-27: This expression is not accurate when R0 is near 1. For a formula in that regime, see Addendum.)

Some regions in Western Europe, East Asia, and even the US have managed to suppress R0 below 1 and cases are trending downward. In the absence of reintroduction of SARS-CoV-2 carriers, Covid-19 can be eliminated in these regions. However, as the recent spikes in China, South Korea, and Australia have shown, this requires continual vigilance. As long as any person remains infected in the world, there is always a chance of re-emergence. As long as new cases are increasing or plateauing, R0 remains above 1. As I mentioned before, plateauing is not a natural feature of the epidemic prediction models, which generally either go up or go down. Plateauing requires either continuous adjustment of R0 through feedback or propagation through the population as a wave front, like a lawn mower cutting grass. The latter is what is actually going on from what we are seeing. Regions seem to rise and fall in succession. As one region reaches a peak and goes down either through mitigation or rapid spread of SARS-CoV-2, Covid-19 takes hold in another. We saw China and East Asia rise and fall, then Italy, then the rest of Western Europe, then New York and New Jersey, and so forth in series, not in parallel. Now it is spreading throughout the rest of the USA, South America, and Eastern Europe. Africa has been spared so far but it is probably next as it is beginning to explode in South Africa.

A reduction in R0 also delays the time to reach the peak. As a simple example, consider the standard SIR model

where is the fraction of the population susceptible to SARS-CoV-2 infection and is the fraction of the population actively infectious. Below are simulations of the pandemic progression for R0 = 4 and 2.

We see that halving R0, basically doubles the time to reach the peak but much more than doubles the number of people that never get infected. We can see why this is true by analyzing the equations. Dividing the two SIR equations gives

,

which integrates to . If we suppose that initially and then we get

(*)

where is the reproduction number. The total number infected will be for . Rearranging gives

If we assume that and ignore we can expand the exponential and solve for to get

This is the fraction of the population that never gets infected, which is also the probability that *you* won’t be infected. It gets smaller as increases. So reducing can exponentially reduce your chances of being infected.

To figure out how long it takes to reach the peak, we substitute equation (*) into the SIR equation for to get

We compute the time to peak, , by separating variables and integrating both sides. The peak is reached when .Â We must thus compute

We can’t do this integral but if we set and , then we can expand and obtain

where . This can be re-expressed as

which is integrated to

If we assume that , then we get an expression

So, is proportional to the recovery time and inversely related to as expected but if is very small (say 0.00001) compared to (say 3) then can be big (around 10), which may explain why it takes so long for the pandemic to get started in a region. If the infection rate is very low in a region, the time it takes a for a super-spreader event to make an impact could be much longer than expected (10 times the infection clearance time (which could be two weeks or more)).

Addendum 2020-06-26: Fixed typos in equations and added clarifying text to last paragraph

Addendum 2020-06-27: The approximation for total infected is not very good when is near 1, a better one can be obtained by expanding the exponential to quadratic order in which case you get the new formula for the

However, for near 1, a better expansion is to substitute into equation (*) and obtain

Set , after rearranging and exponentiating,Â obtain

, which can be expanded to yield

Solving for gives the total fraction infected to be

This took me much longer than it should have.

# The fatal flaw of the American Covid-19 response

The United States has surpassed 2 million official Covid-19 cases and a 115 thousand deaths. After three months of lockdown, the country has had enough and is reopening. Although it has achieved its initial goal of slowing the growth of the pandemic so that hospitals would not be overwhelmed, the battle has not been won. We’re not at the beginning of the end; we may not even be at the end of the beginning. If everyone in the world could go into complete isolation, the pandemic would be over in two weeks. Instead, it is passed from one person to the next in a tragic relay race. As long as a single person is shedding the SARS-CoV-2 virus and comes in contact with another person, the pandemic will continue. The pandemic in the US is not heading for extinction. We are not near herd immunity and R0 is not below one. By the most optimistic yet plausible scenario, 30 million people have already been infected and 200 million will never get it either by having some innate immunity or by avoiding it through sheltering or luck. However, that still leaves over 100 million who are susceptible of which about a million will die if they all catch it.

However, the lack of effectiveness of the response is not the fatal flaw. No, the fatal flaw is that the US Covid-19 response asks one set of citizens to sacrifice for the benefit of another set. The Covid-19 pandemic is a story of three groups of people. The fortunate third can work from home, and the lockdown is mostly just an inconvenience. They still get paychecks while supplies and food can be delivered to their homes. Sure it has been stressful and many of have forgone essential medical care but they can basically ride this out for as long as it takes. The second group who own or work in shuttered businesses have lost their income. The federal rescue package is keeping some of them afloat but that runs out in August. The choice they have is to reopen and risk getting infected or be hungry and homeless. Finally, the third group is working to allow the first group to remain in their homes. They are working on farms, food processing plants, and grocery stores. They are cutting lawns, fixing leaking pipes, and delivering goods. They are working in hospitals and nursing homes and taking care of the sick and the children of those who must work. They are also the ones who are most likely to get infected and spread it to their families or the people they are trying to take care of. They are dying so others may live.

A lockdown can only work in a society if the essential workers are adequately protected and those without incomes are supported. Each worker should have an N100 mask, be trained how to wear it and be tested weekly. People in nursing homes should be wearing hazmat suits. Everyone who loses income should be fully compensated. In a fair society, everyone should share the risks and the pain equally.

# Covid-19 continues to spread

The global plateau turned out to just be a pause and the growth in new cases continues. The rise seems to be mostly driven by increases in Brazil, India, and until very recently Russia with plateauing in the US and European countries as they relax their mitigation policies. The pandemic is not over by a long shot. There will most certainly be further growth in the near future.

# How much Covid-19 testing do we need?

There is a simple way to estimate how much SARS-CoV-2 PCR testing we need to start diminishing the COVID-19 pandemic. Suppose we test everyone at a rate , with a PCR test with 100% sensitivity, which means we do not miss anyone who is positive but we could have false positives. The number of positives we will find is , whereÂ is the prevalence of infectious individuals in a given population. If positive individuals are isolated from the rest of the population until they are no longer infectious with probability , then the rate of reduction in prevalence is . To reduce the pandemic, this number needs to be higher than the rate of pandemic growth, which is given by , where is the fraction of the population susceptible to SARS-CoV-2 infection and is the rate of transmission from an infected individual to a susceptible upon contact. Thus, to reduce the pandemic, we need to test at a rate higher than .

In the initial stages of the pandemic is one and , where is the mean reproduction number, which is probably around 3.7 and is the mean rate of becoming noninfectious, which is probably around 10 to 20 days. This gives an estimate ofÂ to be somewhere around 0.3 per day. Thus, in the early stages of the pandemic, we would need to test everyone at least two or three times per week, provided positives are isolated. However, if people wear masks and avoid crowds then could be reduced. If we can get it smaller then we can test less frequently. Currently, the global average of is around one, so that would mean we need to test every two or three weeks. If positives don’t isolate with high probability, we need to test at a higher rate to compensate. This threshold rate will also go down as goes down.

In fact, you can just test randomly at rate and monitor the positive rate. If the positive rate trends downward then you are testing enough. If it is going up then test more. In any case, we may need less testing capability than we originally thought, but we do need to test the entire population and not just suspected cases.

# The Covid-19 plateau

For the past five weeks, the appearance rate of Covid-19 cases has plateaued at about a hundred thousand new cases per day. Just click on the Daily Cases Tab on the JHU site to see for yourself. This is quite puzzling because while individual nations and regions are rising, falling, and plateauing independently, the global total is flat as a pancake. A simple resolution to this seeming paradox was proposed by economist John Cochrane (see his post here). The solution is rather simple but the implications as I will go into more detail below are far reaching. The short answer is that if the world (either through behavior or policy) reacts to the severity of Covid-19 incrementally then a plateau will arise. When cases go up, people socially distance, and the number goes down, when cases go down, they relax a little and it goes back up again.

This can be made more precise with the now-famous SIR model. For the uninitiated, SIR stands for Susceptible Infected Recovered model. It is a simple dynamical model of disease propagation that has been in use for almost a century. The basic premise of an SIR model is that at any given time, the proportion of the population is either infected with the virus I, susceptible to infection S, or recovered from infection and no longer susceptible R. Each time an S comes across an I, it has a chance of being infected and becoming another I. An I will recover (or die) with some rate and become an R. The simplest way to implement an SIR model is to assume that people interact completely randomly and uniformly across the population and the rate of transmission and recovery is uniform as well. This is of course a gross simplification and ignores the complexity and diversity of social interactions, the mechanisms of actual viral transmission, and the progression of disease within individuals. However, even though it misses all of these nuances, it captures many of the important dynamics of epidemics. In differential equation form, the SIR model is written as

Â Â (SIR model)

where is the total number of people in the population of interest. Here, and are in units of number of people.Â The left hand sides of these equations are derivatives with respect to time, or rates.Â They have dimensions or units of people per unit time, say day. From this we can infer that and must have units of inverse day (per day) since , , and all have units of numbers of people. Thus is the infection rate per day and is the recovery/death rate per day. The equation assumes that the probability of an meeting an is . If there was one infected person in a population of a hundred, then if you were to interact completely randomly with everyone then the chance you would run into an infected person is 1/100. Actually, it would be 1/99 but in a large population, the one becomes insignificant and you can round up. Right away, we can see a problem with this assumption. I interact regularly with perhaps a few hundred people per week or month but the chance of me meeting a person that had just come from Australia in a typical week is extremely low. Thus, it is not at all clear what we should use for in the model. The local population, the regional population, the national population?

The model assumes that once an has run into an , the rate of transmission of the virus is . The total rate of decrease of is the product of and . The rate of change of is given by the increase due to interactions with and the decrease due to recovery/death . These terms all have units of person per day. Once you understand the basic principles of constructing differential equations, you can model anything, which is what I like to do. For example, I modeled the temperature dynamics of my house this winter and used it to optimize my thermostat settings. In a post from a long time ago, I used it to model how best to boil water.

Given the SIR model, you can solve them to get how and will change in time. The SIR model is a system of nonlinear differential equations that do not have what is called a closed-form solution, meaning you can’t write down that is some nice function like or . However, you can solve them numerically on a computer or infer properties of the dynamics directly without actually solving them. For example, if is initially greater than , then $dI/dt$ is positive and thus $I$ will increase with time. On the other hand, since is always negative (rate of change is negative), it will decrease in time. As increases and decreases, since is decreasing at a faster rate than is increasing because is slowing the growth of , then at some point will equal and . This is a stationary point of .Â However, it is only a momentary stationary point becauseÂ keeps decreasing and this will make start to decrease too and thus this stationary point is a maximum point. In the SIR model, the stationary point is given by the condition

(Stationary condition)

which you can solve to get either or . The point is not a peak but just reflects the fact that there is no epidemic if there are no infections. The other condition gives the peak:

where is the now-famous R naught or initial reproduction number. It is the average number of people infected by a single person since is the infection rate and is the infection disappearance rate, the ratio is a number. The stationary condition gives the herd immunity threshold. When the fraction of reaches then the pandemic will begin to decline.Â This is usually expressed as the fraction of those infected and no longer susceptible, . The 70% number you have heard is because is approximately 70% for , the presumed value for Covid-19.

A plateau in the number of new cases per day is an indication that we are at a stationary point in . This is because only a fraction of the total infected are counted as cases and if we assume that the case detection rate is uniform across all , then the number of new cases per day is proportional to . Thus, a plateau in cases means we are at a stationary point in , which we saw above only occurs at a single instance in time. One resolution to this paradox would be if the peak is broad so it looks like a plateau. We can compute how broad the peak is from the second derivative, which gives the rate of change of the rate of change. This is the curvature of the peak. Taking the second derivative of the I equation in the SIR model gives

Using and the formula for at the peak, the curvature is

It is negative because at a peak the slope is decreasing. (A hill is easier to climb as you round the top.)Â There could be an apparent plateau if the curvature is very small, which is true if is small compared to . However, this would also mean we are already at the herd immunity threshold, which our paper and recent anti-body surveys predict to be unlikely given what we know about .

If a broad peak at the herd immunity threshold does not explain the plateau in new global daily cases then what does? Cochrane’s theory is that depends on .Â He postulated that ,where is the initial infectivity rate, but any decreasing function will do. When goes up, goes down. Cochrane attributes this to human behavior but it could also be a result of policy and government mandate. If you plug this into the stationary condition you get

or

and the effective reproduction number Â is one.

However, this is still only a quasi-stationary state because if is a constant , then will decrease as , which has solution

Â Â Â Â (S)

Plugging this into the equation forÂ gives

which means that is not really plateaued but is decreasing slowly as

We can establish perfect conditions for a plateau if we work backwards. Suppose again that has plateaued at .Â Then, is given by equation (S).Â Substituting this into the (Stationary Condition) above then gives or

which means that the global plateau is due to us first reducing to near , which halted the spread locally, and then gradually relaxing pandemic mitigation measures so that is creeping upwards back to it’s original value.

The Covid-19 plateau is both good news and bad news. It is good news because we are not seeing exponential growth of the pandemic. Globally, it is more or less contained. The bad news is that by increasing at a rate of a hundred thousand cases per day, it will take a long time before we reach herd immunity. If we make the naive assumption that we won’t reach herd immunity until 5 billion people are infected then this pandemic could blunder along for days! In other words, the pandemic will keep circling the world forever since over that time span, babies will be born and grow up. Most likely, it will become less virulent and will just join the panoply of diseases we currently live with like the various varieties of the common cold (which are also corona viruses) and the flu.

# A Covid-19 Manhattan Project

Right now there are hundreds if not thousands of Covid-19 models floating out there. Some are better than others and some have much more influence than others and the ones that have the most influence are not necessarily the best. There must be a better way of doing this. The world’s greatest minds convened in Los Alamos in WWII and built two atomic bombs. Metaphors get thrown around with reckless abandon but if there ever was a time for a Manhattan project, we need one now. Currently, the world’s scientific community has mobilized to come up with models to predict the spread and effectiveness of mitigation efforts, to produce new therapeutics and to develop new vaccines. But this is mostly going on independently.

Would it be better if we were to coordinate *all* of this activity. Right now at the NIH, there is an intense effort to compile all the research that is being done in the NIH Intramural Program and to develop a system where people can share reagents and materials. There are other coordination efforts going on worldwide as well.Â This website contains a list of open source computational resources.Â This link gives a list of data scientists who have banded together. But I think we need a world wide plan if we are ever to return to normal activity. Even if some nation has eliminated the virus completely within its borders there is always a chance of reinfection from outside.

In terms of models, they seem to have very weak predictive ability. This is probably because they are all over fit. We don’t really understand all the mechanisms of SARS-CoV-2 propagation. The case or death curves are pretty simple and as Von Neumann or Ulam or someone once said, “give me 4 parameters and I can fit an elephant, give me 5 and I can make it’s tail wiggle.” Almost any model can fit the curve but to make a projection into the future, you need to get the dynamics correct and this I claim, we have not done. What I am thinking of proposing is to have the equivalent of FDA approval for predictive models. However, instead of a binary decision of approval non-approval, people could submit there models for a predictive score based on some cross validation scheme or prediction on a held out set. You could also submit as many times as you wish to have your score updated. We could then pool all the models and produce a global Bayesian model averaged prediction and see if that does better. Let me know if you wish to participate or ideas on how to do this better.

# Covid-19 talk

Here are the slides for my webinar at FDA today .

# Covid-19 paper link

Here is the medrXiv link to our Covid-19 paper.Â We actually have more updated versions with nicer graphs, which we will upload shortly.