Laser Physics Helps to Understand the Corona Virus Crisis
Posted on 2020-03-16 as a part of the Photonics Spotlight (available as e-mail newsletter!)
Permanent link: https://www.rp-photonics.com/spotlight_2020_03_16.html
Abstract: Because laser operation involves some similar aspects as the spread of an epidemic, in particular exponential growth and threshold behavior, one can in fact develop a clearer picture of what kind of measures has a real chance to avoid a catastrophic development of the current corona virus (SARS-CoV-2 / COVID-19) crisis. Clearly, stopping exponential growth of infections is absolutely vital, while other suggested solutions are bound to fail completely.
The current corona virus (SARS-CoV-2) crisis (COVID-19 pandemic) impacts the photonics industry in various ways – for example by blocking part of the workforce, disrupting supply chains and enforcing the cancellation or postponement of trade shows, effectively removing one important method for photonics marketing. Another relation of the corona problem to our business is that there are some noteworthy analogies between the virus spread and laser operation. In particular, various aspects of exponential growth, with which any expert in laser physics is well familiar, also play an important role in the current crisis, but are poorly understood by many. Some of the essential thoughts are actually quite simple, but nevertheless it can be hard for lay persons to fully grasp them. Curiously, even some medical experts appear to have substantial difficulties; it is not only the stupidity of politicians which leads to incompetent and potentially disastrous action (or non-action), but also the failure of some scientific advisers. Well, it will hardly help those to get it explained via laser physics, but I do hope that many peers in photonics will appreciate to get some clearer thoughts on some quite relevant issues, which will sooner or later affect all of us. In my view, this is a good justification to go well beyond photonics with this article.
The Analogy Between Virus Spread and Lasing
From lasers, we know the following. Light can circulate in a laser resonator, where it experiences gain (amplification) and also some losses. The rate of change can be described with a simple differential equation:
where P is the circulating intracavity power, g the gain coefficient, l denotes the losses per round-trip and Trt is the round-trip time (often a few nanoseconds or even less). The gain depends on the amount of deposited excitation energy (e.g. through optical pumping) in the gain medium. As long as we have g < l, i.e., the net round-trip gain is negative, the laser is essentially off: although there is constantly some fluorescence light in the gain medium, it cannot be amplified to any significant level. If the gain exceeds the losses, however, i.e., we get a positive round-trip gain, the intracavity power will rise exponentially (proportional to exp((g − l) t / Trt)) until it starts saturating the gain, eventually limiting the further growth. Depending on the laser parameters, there may be some relaxation oscillations until a steady state can be reached (assuming continuous pumping).
The behavior of virus spread is somewhat similar. Infections are basically always caused by already infected people (apart from the initial infection, e.g. from some animal – usually a rare event). Therefore, the growth rate of the number of infected people is roughly proportional to that number itself. Although various details are substantially more complicated than those in lasers, we have a growth rate growing with the reached infection level, and thus potentially rapid growth over orders of magnitude. Also, we have a kind of threshold behavior similar to that in lasers: if the average number of people infected by one infected person (called the reproduction factor or R factor) is below 1, the infection will die out; if that number is larger, we will get exponential growth. So the reproduction factor in the epidemic exactly corresponds to the pump parameter of the laser.
So a crucial insight is that it makes a huge difference whether the reproduction factor is e.g. 1.1 or 0.9. While 1.1 means growth, potentially until the infection runs out of not-yet-infected people, 0.9 means a steady decline, never reaching a high level.
In detail, virus spread is really far more complicated to understand than lasers. For example, there is some time delay after infection where the infected person starts spreading viruses for some time. Also, the probability of further infections is not constant, but depends substantially on the region (e.g. the lifestyle of people) and on time (hopefully getting lower as people acknowledge the problem and learn to react properly). Nevertheless, the aspects of threshold behavior and exponential growth fully apply.
By the way, there are of course many other examples for exponential growth and threshold behavior in physics and technology. For example, a nuclear chain reaction essentially behaves like that. Here, exponential growth quickly leads to a nuclear explosion, releasing vast amounts of energy, if exponential growth is not stopped in time. On the other hand, the chain reaction in a nuclear reactor quickly dies out if it is just below criticality – similar to the vanishing epidemic when the reproduction factor gets below unity.
Understanding Exponential Growth
Many people have difficulties grasping the power of exponential growth. It may initially seem slow (and indeed be slow!), but because the growth rate grows together with the growing quantity, it is getting faster and faster, and can soon make that growth dangerous or disastrous. People watched that epidemic over many weeks and felt that it is still far away, then got surprised how rapidly it becomes serious eventually.
Note that at many places we are currently at a multiplication factor of well above 3 per week, so it evolves even substantially faster than in the diagram shown above.
As an example, if we have 30% growth per day (a bit less than recently calculated for Germany before stricter measures were implemented), this corresponds to a factor of 6.3 per week, and it takes only about two months to multiple the number of infected people by several millions!
From Italy, we are already getting news from a disaster in the hospitals: even though only a small part of the infected people needs intensive care, that number has quickly exceeded the existing capacities. The consequences are terrible – for example, they cannot treat all people and therefore let more and more of them die who could otherwise have survived. And this happens already at infection levels amounting to a tiny proportion of the population – orders of magnitude below what can be reached with exponential growth within a short time.
Many Germans (but decreasing in number every day) still (at the time of original publication of this article) do not recognize how close we all are to that same disastrous situation – just because our numbers are still substantially lower and we have higher capacities in the medical sector. However, it should be easy to see that with such a growth rate it is a matter of roughly one week to get to the same number of infections. Further, if you have 3 times more capacities, for example, but a multiplication factor of 3 per week, it takes only one week more to get into the same disaster (with 3 times more affected people). Fortunately, German politicians seem now to have understood this, and the now stricter measures should substantially reduce the growth rate. Whether that is already enough, remains to be seen. Curiously, that crucial question is not debated that much.
Addition of 2020-05-17: the German measures turned out to be sufficiently effective for stopping the exponential growth and even let the numbers decline significantly. Unfortunately, they were not sufficient to get down to really small infection numbers within a couple of weeks, because the reproduction factor did not get that far below unity. A couple of weeks with a larger R factor, as might result from a gradual return to normality, might well spoil that success.
In laser technology, we are familiar with exponential growth. We know that laser operation starts out with a few fluorescence photons, which even for a net gain of only a few percent per resonator round trip can evolve e.g. into a giant pulse within a tiny fraction of a second.
You may think that lasers are extremely different in the sense that they exhibit far higher growth rates. Yes, but a growth rate of more than a factor of 5 per week is also very fast for a society. It is not a catastrophe in slow motion, as an expert recently said, but rather a kind of explosion!
Strategies and Their Chances to Succeed
I often hear people judging the chances to get this under control, where they are apparently not even aware of the fact that there are several profoundly different types of strategies:
Tracking Infected People and Minimizing Seeds
As long as only few are infected in a society, it can be effective to diligently track the few infected person and those who may have infected by them recently. (Here, we have a method which I cannot compare with lasers!) If this is done well enough, one may contain the outbreak, strongly limiting the number of infected peopled. However, it requires rapid well coordinate action, reliable test kits and the like, and it quickly becomes impractical when the number of infected people gets too high. For many countries, including Germany and the United States, that chance has passed. In the United States, it has been minimal from the beginning, since the government essential dismantled the existing structures for pandemic action. A good example for how disastrous (and expensive!) it can be to save some money at the wrong place.
A particular difficulty in that case arises from some characteristics of the SARS-CoV-2 virus: there is a substantial time where someone infected is infectious without feeling any symptoms. Therefore, e.g. it helps little to monitor air passengers for fever: they can be infectious without having any fever yet.
Minimizing the influx of infected people from abroad (e.g. through travel restrictions for foreign visitors) helps only in that early phase, where the internal number of infections is still small. However, that measure becomes insignificant once you get too far. It is simple: it can help only as long as a substantial proportion of the infections is caused by foreigners. Anyway, exponential growth can surely not be stopped with that, once it has really started. Nobody would try to stop a laser pulse by shielding the laser against ambient light!
Fighting the Growth Rate
We can reduce the multiplication factor, mostly with measures of social distancing and with improved hygiene. The goal must obviously be to get that kind of death laser below threshold. Clearly, for that purpose we do not need to be perfect, suppressing any transmissions. However, we need to be sufficiently effective. Reaching 90 % of that will not solve 90 % of the problem, as some may think, because it still means exponential growth. How much would that actually help?
- Still, it would be very valuable because we would win substantial time; flattening the curve would also substantially reduce the required medical capacities, because recovered (or passed) patients free beds for new patients. Over longer times, we can treat more people with the same capacities.
- However, we would still need to accept that a large percentage of the population gets infected sooner or later – which inevitably implies huge numbers of fatalities (unless we soon find a good cure, see below).
I find it appalling that this crucial difference is so often overlooked not only in statements of politicians, but also even of medical experts. That may lead us to doing not quite enough, with the consequence of millions of additional deaths. Aren't they aware of that responsibility?! Or are they more afraid of inducing widespread panic? So far, I think we are far from panic, and much closer to not realizing the real danger.
An interesting aspect is that in contrast to various others approaches, fighting the growth rate with social distancing still works after many have been infected. Also, the above mentioned nasty virus characteristics do not matter much in this context. So it is not too late for this to work.
Unfortunately, while a single country has a good chance of getting the crisis under control, viruses will not be completely wiped out, e.g. coming back from other countries, from which we can never completely isolate ourselves. Therefore, a negative growth rate may have to be maintained over longer times – not just a few weeks, which would be sufficient if we did that effectively worldwide.
Increasing Health Care Capacities
A crucial risk in the corona crisis is that we sooner or later exceed the capacities of our medical systems, and that by a very large margin. So a natural reaction seems to be to increase those capacities. But wait a moment – how much can that help?
Imagine what happens if we don't stop the exponential growth and e.g. millions of people in the United State will require intensive care (e.g. 1 % of the population – we may easily top that)? It is easy to see that this would completely exceed the hospital capacities very soon. You may see 100 patients competing for a single hospital bed. But if you would manage to instantly expand those capacities by a factor of ten (which you won't), it would just take roughly 9 days more to reach that higher limit, if we have a growth rate of 30% per day (a bit less than recently calculated for Germany). Obviously, the expansion of capacities can never be enough to more or less solve the problem and avert catastrophic developments. Even if we could still ensure proper care for everyone affected – which we surely cannot –, numerous people would die despite intensive care. The clear conclusion: the expansion of hospital capacities makes sense only in addition to other, really effective measures.
Nevertheless, some public statements create that entirely false impression to those who are not used to think themselves. I find that unbelievably irresponsible.
Developing Herd Immunity
The British government has recently explained its new strategy: letting the infections spread in the population, except for elderly and sick persons who should be isolated from all others. The idea is that once the younger ones, most of whom would get through the infection without major problems, would afterwards create safety for the whole society in the form of a “herd immunity”.
While the theoretical idea is at least not logically defect, it is obviously not workable, since one cannot isolate all the high-risk persons so well from all the others that their risk to get infected among numerous infected younger persons remains tolerable. For example, where to draw the line – for example on which side of it should all those be placed who care for the ill and elderly? Or should the ill and elderly largely care for themselves for a couple of months, until more and more immune younger ones become available? And how about the large number of younger ones – not in relative, but in absolute terms – who don't cope that well with the virus?
Well, a herd immunity strategy could in principle also be implemented in a softer form: try to control the epidemic in such a way that you always have a substantial number of (mostly younger) sick people, but always within the limits of the health system – hoping that this way you build up herd immunity within a reasonable time. However, this obviously also requires stopping the exponential growth – only slightly later – and therefore about the same strict measures (lock-down). Moreover, this state would need to be maintained for a long time. For example, if Germany would constantly have 400,000 sick people, with roughly 20,000 of those getting intensive care (after a moderate expansion of our capacities), each one for two weeks (probably a bit too optimistic), the system would get through about 800,000 per month, and it would take about 4 years to get half of the population immunized! So we would have to multiply our health care capacities even to get this done within one year. Therefore, it is much wiser to press the infection rate further down – with about the same strictness of measures – and do the immunization later on with a vaccine. By the way, controlling the epidemic so precisely would be another challenge.
For such reasons, this plan is harshly criticized by most experts, including the WHO. I think it was just another insane and dangerous idea of a government formed by irresponsible clowns. It is easy to predict that either they will abandon their plan pretty soon or quickly lead their country into a massive disaster, with the NHS breaking down under the load of seriously ill young and old people. (Addition of 2020-03-17: Indeed, they changed course while I was writing this.)
My estimate are of course based on a rather crude model. However, I have found a much more refined model from Swiss epidemiology experts, which fully confirms the essential results, only in that case for Switzerland. There, you see interesting curves, showing how the epidemic is estimated to progress with different levels of suppressing infections: getting below threshold or not, which of course makes a huge difference. However, they don't tell how much needs to be done to achieve what result. Another interesting study, just published by a group at Imperial College London, goes much further in that direction.
Addition of 2020-03-19: The Dutch government now also goes for herd immunity, but the they seem to be aware that they also will soon have to switch to lock-down as well. Addition on the same evening: Indeed, they also gave up that concept today.
Addition of 2020-03-24, amended 2020-04-30 and 2020-05-17: Donald Trump initially seemed to consider the same herd immunity strategy, which would have turned the United States into the area of a mass experiment with predictably historic impact. However, he was eventually persuaded not to dare that, and is on a meandering course since then – spreading propaganda against the lock-down, while not actively enforcing its end, and promoting silly and dangerous miracle cures. Anyway, the US lock-down became no less severe than elsewhere, although it came too late to prevent enormous fatality numbers. It also demonstrates that it is not just a choice between a health emergency and economic disaster – with suitable political decisions, you can have both combined:
- Tens of thousands of additional fatalities (and enormous suffering by many more who finally survived) have been caused by the government failure to act swiftly and effectively. This is a tremendous responsibility. Most persons would be shattered by having caused the loss of a single human live; other don't seem to care much about having caused tens of thousands of death cases with their neglect.
- Eventually, pressing that R factor below unity must succeed if a catastrophe is to be avoided, and the measures to achieve that at the high infection level are no less severe than if one had done that early on.
- The economic advantage of having started the lockdown a couple of weeks later is probably not sufficient to compensate the consequences. We already see that loss of jobs in the US is terrible – certainly more severe than in Germany, for example.
Addition of 2020-03-27, amended 2020-05-17: The Swedish case appears to be less clear. There, the growth of infections seems to have developed significantly more slowly than expected for exponential growth. Maybe their strategy for reducing infection rates is more effective than one would expect, e.g. based on schools continuing to operate more or less normally at least for the younger pupils. While the idea to solve the problem via herd immunity, created by infections rather than vaccination, is as crazy there as it is anywhere, the reproduction factor is not staying approximately constant as it would for a “let it go” strategy, and that at least gives them more time e.g. to correct their strategy if necessary. Maybe with some further adjustments they can get it below unity, if they turn out to be right with the assumption that young people do not often infect the elder ones. Note also that some conditions in Sweden appear to be favorable, e.g. that the contact between people is traditionally not that intense. However, it does not seem to be quite enough; the government had to admit that the isolation of elderly and sick people against the pandemic hasn't really worked, resulting in much more fatalities than in otherwise comparable countries like Norway. In any case I am sure that a fundamental fact holds for Sweden as for any other country: either you stop exponential growth in time or you ride into a disaster. And doing that exercise somewhat later may in the end well not be cheaper.
Let us turn back to lasers for a moment. Image that you have a Q-switched laser, where you want to lengthen the emitted pulse by a factor of 10 in order to limit its peak power. For that, you need to reduce the round-trip gain (measured in decibels) by a factor of 10, e.g. by inserting an additional loss which compensates for 90% of the gain. If you do a little more – e.g. compensating for 101% of the gain – you can totally suppress the pulse. So in terms of required measures there is no substantial difference – translated to the virus crisis, between controlling the epidemic for creating herd immunity and largely suppressing it.
We heard the argument against a lock-down strategy that the epidemic will come back as soon as you stop it. That's indeed a big concern: if we can't suppress the epidemic worldwide, which we probably won't achieve due to a lack of international cooperation, it will sooner or later come back, and we have to start a new lock-down. However, quite obviously the herd immunity strategy is not immune against that problem, since exponential growth needs to be stopped in any case to avert a catastrophic development.
Research for Vaccinations and Medication
Experts tell us that it is likely to take at least a year from now, if not substantially more, to obtain a widely usable vaccination which could end this crisis. So it is great that various people continue with that, but don't bet on it for 2020.
There seem to be higher chances to get some medication developed which helps patients substantially – maybe even with some already existing drugs which only need to be tested and proven to be effective and sufficiently safe. I imagine that we might have something like that until summer or so (very rough estimate, very uncertain). However, if this works, it will still not avoid a catastrophe if we get many millions of patients until then. Only perhaps if it were a wonder cure – throw in a few pills, and the patient will stand up and happily walk away! Unfortunately, such miracle cures have so far only been advertised, but never convincingly demonstrated.
Cooperation is Essential
It is essential to foster intense cooperation on many levels – at the level of communities, administrations, governments and also on the international level. We need this for many reasons, including the faster data collection, learning on mechanisms, effectivity of measures etc., and for avoiding that viruses will be coming back from regions which are left behind. Nationalistic and populist approaches are bound to fail here just as in many other areas. I suppose that most people working in photonics are aware of that …
Conclusions and Concluding Remarks
I shortly discuss the resulting conclusions:
- It is absolutely vital to stop exponential growth of infections; even the boldest increases of hospital capacities could otherwise not avoid catastrophic consequences. Some flattening the curve alone is not enough: it would only get us to a terrible catastrophe instead of an even more terrible one. The plateau needs to be reached very soon indeed.
- It has been proven e.g. by China, South Korea and Singapore that exponential growth can be stopped with appropriate measures, and this was largely social distancing and hygiene (maybe apart from Singapore, where they acted so quickly that tracking was also fairly effective). We all need the same, because it is the only chance of containing this problem on a manageable level.
- Travel restrictions are helpful in some cases, but become largely ineffective once the disease has spread by some amount. I suppose we have already passed that point at many places. Well, we may need to come back to that after successfully reducing infections in a country.
- Trying clearly ineffective things instead of sufficiently strong social distancing and hygiene measures can waste more valuable time and also increase the final cost.
- Thinking about the economy is of course very important, but getting this problem under control is a prerequisite to limit economical damage as well. So far, too little has obviously been done, and as a consequence the economical damage will be greater than it would have had to be.
By the way, we see again that (a) it is essential to follow competent scientific insight (in this case, of course, not primarily of laser experts!) and (b) always to be aware that single scientists often fail, while the conclusions of the scientific community as a whole are the safest advise available on Earth. Unfortunately, we see another big wave of fake news – some call it an infodemic – , discrediting science and scientists, sometimes even started or spread by medical doctors. Nothing seems to be too stupid to spread well – for example baseless conspiracy theories and the denial of a pandemic. Even in mid May 2020, some still deny the pandemic against overwhelming evidence.
Another aspect: remarkably, the normal mode of operation of our modern societies – involving a lot of exchange, including far distance traveling – allows for exponential growth of any new infectious disease (e.g. create by a spontaneous mutation of a virus or by a new transition from animals to humans) providing only that it is sufficiently contagious. That latter is given in some cases, but fortunately not that often or at least not often in combination with other bad properties, such as a high mortality. One of my most severe worries: what if we once get a super-bug which has the combination of easy and fast transmission, high mortality and long incubation period (making it difficult to identify infections in time)? We have seen all such things separately, but I cannot see why that combination should be impossible or at least extremely unlikely. The fact that we have never seen such a thing before doesn't tell us that much; after all, the current international high-exchange mode has been tried out only for a couple of decades.
This is at least a positive aspect of the corona crisis: it may help us to prepare for a much more severe challenge to come in the future. The most important insight of the current crisis will probably be that voting for populist who show contempt for science and reason can have terrible consequences hitting basically everyone. Bolstering populists obviously do not have the power e.g. to stop an epidemic and minimize the damage for their country; viruses don't give in, however loud a politician may shout. In the end, their voters and everyone else pay a very high price. So wherever a society is not destroyed to the point where sound reasoning and democratic choice has no chance any more, such people will lose power, and the country can get back to solving problems.
Hopefully, at least one crucial thing will be learned at least by most health experts and politicians: that the the consequences of fast exponential growth are far more severe than one may feel. The failure to fully understand this is apparently at the heart of many costly mistakes done in this crisis.
Unfortunately, we have so far seen severe failure, which already made the problem much larger and more dangerous than it would have had to be – beginning with the early mishandling of the crisis in China, but continuing for many weeks until finally more and more governments realized what is at stake and what needs to be done. At least, there is hope justified by recent successes in China, Taiwan, South Korea and Singapore, for example. Let's see for how long it will take the most powerful country on Earth to get rid of a president who fails so spectacularly, also in this crises – first dismantling the carefully designed pandemic action resources, then playing down the problem as a Democratic hoax, taking ineffective measures, creating total confusion through media, stirring up people against necessary hygiene measures, damaging international cooperation with uncoordinated and ruthless action, etc., etc. You don't need to be a laser physicist to recognize that it would be hard for someone to be less suitable for that office, particularly in a crisis. (Later comment of 2020-10-27: In the meantime, far over 200,000 Americans died from Covid-19, and numerous others survived but may be permanently impaired. That tragedy is in no way a surprise based on the terrible mishandling of the crisis from start.)
At the same time, much is also to be improved in many other countries. I am particularly concerned about poor and developing countries.
You are welcome to enter your questions or comments below, and of course to spread the message to others who may also find it interesting!
This article is a posting of the Photonics Spotlight, authored by Dr. Rüdiger Paschotta. You may link to this page and cite it, because its location is permanent. See also the RP Photonics Encyclopedia.
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