What are suggestions for reducing the transmission rate of the current epidemics?

In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the following rule for the general populace not directly involved in necessary services:

If members of your household go to public services on a certain day, the whole household should not use any public service for 2 weeks after. That way you still can get infected but cannot infect without knowing it.

Do you have some suggestions? Good models to look at? No predictions please, just advices what to do.

Edit: In more detail:

Model (the simplest version to make things as clear as possible): There are several categories of people $C_i$ that constitute portion $p_i$ of the population and have a certain matrix $A$ of interactions per day. Then, if $x_i(t)$ is the number of ever infected people in category $C_i$ by the time $t$, the driving ODE is $$ \dot x(t)=\alpha A[x(t)-x(t-\tau)] $$ where $\tau$ is the ("typical") time after which the sick person is removed from the population and $\alpha$ is the transmission probability. In this model the exponential growth is unsustainable if $\alpha\lambda(A)\tau<1$ where $\lambda$ is the largest eigenvalue of $A$. We do not know $\alpha$ (though we can try to make suggestions how to reduce it, most such suggestions are already made by the government). The government can modify $A$ by issuing orders. Some orders merely reduce $a_{ij}$ to $0$, but the government cannot shut essential public services completely this way.

Questions: What is $A$, which entries $a_{ij}$ are most important to reduce, how to issue a sensible order that will modify them, and by how much they will reduce the eigenvalue?

First suggestion for these 4 answers: There are two categories of people: ordinary population that only goes to public services and
public servants that both provide services and go to them. There is only one ("averaged") type of service involving a dangerous client-server interaction and all infection goes there. The portion of public servants in the population is $p$. The server sees $M$ clients a day. Then the current social interaction matrix (say, for the grocery store I've seen yesterday) is $A=\begin{bmatrix}0 & M(1-p)\\ Mp & 2Mp\end{bmatrix}$ (ordinary population does not transmit to ordinary population, servers transmit to clients who can be both servers and clients, clients transmits to servers. The largest eigenvalue is $M(p+\sqrt p)$. The lion's share comes from $\sqrt p$, which is driven by the off-diagonal entries, The order should be issued as above, the effect that ordinary people never come to the service infected, which will remove the left bottom corner and drop the largest eigenvalue to $2Mp$. Assuming $p=1/9$ (not too unrealistic), the drop will be two-fold even if you leave the service organization as it is.

That ends the solution I propose in mathematical language. In layman terms, the public will completely do this part (you cannot ask for more) and still have some life, and we can concentrate on the models of how servers should be organized.


Time to remove the non-relevant part and add some relevant thoughts about what else we can help with plus the response to JCK.

First of all, It is very hard to formulate the orders correctly. The stay home rule really means "avoid all close contacts outside your household except the necessary interactions with public servants providing vital services to you" (and even that version is, probably flawed). It is not about dogs, etc., as the Ohio version reads now. If everybody understood and implemented that meaning, my suggestion could be formulated as I said. However the intended meaning really is

When going to public services, minimize the probability that you can infect others as much as feasible and consider it to be $0$ if in the last two weeks nobody from your household had a contact with a stranger and nobody in the household had any symptoms.

Now it is more to the point, but also more complicated. And if a professional mathematician like myself is so inept, imagine the difficulties of other people.

So, within that model, what would be the best formulation of the order to give?

Second, the set of questions I asked is clearly incomplete. One has to add for instance "What assumption can be wrong and what effect that will have on the outcome under the condition that the order is given in the currently stated form. I have never seen a book that teaches the influence of the order formulation on the possible model behavior and that may be a crucial thing now. The interaction between the formal logic and differential equations within a given scenario is a non-existing science (or am I just ignorant of something? That reference would really be useful).

Third, if we have a particular question (say, how much to reduce and how organize the public transportation, which is NY and Tel Aviv headache now), what would be a good mathematical model for just that and what would be the corresponding order statement under this model?

The questions like that are endless and if there were ready answers in textbooks, the governments would just implement them already instead of having 7-hour meetings. So I can fairly safely conclude that they are not there.

What I tried with my model example was, in particular, to show that there may be some non-trivial moves in even seemingly optimal situations (strict stay home order and running only the absolutely vital services at the minimal rate that still allows to serve the population) that also make common sense and can be used by everyone right now and right here. Finding such moves can really help now. The main real life question now is "What can I (as government, business, or individual) do to reduce the largest eigenvalue of the social interaction matrix?" Now show me the textbook that teaches that and I'll stop the "ballspitting" and apologize for the wasted time of the people reading all this.

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    $\begingroup$ Hmm much as I understand the urge to ask this question, I would have preferred MO to be a "safe space" that I could come to in order to get away from the Coronavirus. And, unlike Gil Kalai's question, I don't see any mathematical connection here. $\endgroup$
    – Lucia
    Commented Mar 23, 2020 at 4:49
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    $\begingroup$ @Lucia I addressed your doubts and concerns the best I could at 5AM, now back to sleep. You are right, I should have said more in the question formulation. My only excuse is that I don't ask questions too often :-) $\endgroup$
    – fedja
    Commented Mar 23, 2020 at 9:11
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    $\begingroup$ I would upvote a version of this that asked: “here is a model; what variants would be mathematically tractable and lead to good ideas?” I am downvoting the current version, which is not focused on mathematical research. $\endgroup$
    – user44143
    Commented Mar 23, 2020 at 10:14
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    $\begingroup$ My immediate instinct as an economist is that any useful model is going to have to incorporate maximizing behavior: People will voluntarily change their behavior in response to the transmission probabilities (to protect themselves, and also, perhaps, because they care about protecting others). So the parameter $M$, for example, should be determined within the model, not taken as a constant. $\endgroup$ Commented Mar 23, 2020 at 14:23
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    $\begingroup$ With respect, there's an enormous literature out there on mathematical epidemiology, and I think digging into that is probably a better course of action than spitballing on MO. For example, "Mathematical Epidemiology", Fred Brauer et al, Springer Lecture Notes vol 1945. Or "An Introduction to Mathematical Epidemiology", Martchava. $\endgroup$
    – JCK
    Commented Mar 23, 2020 at 22:58

6 Answers 6


This is just a slight expansion of my comment.

When the environment changes, behavioral parameters (that is, parameters like $M$ that describe people's behavior --- in this case the behavior of public servants deciding how many clients to serve each day) are going to change. Therefore such parameters should not be taken as constants; they should be determined within the model.

This means we need to be able to predict how $M$ will change in an unprecedented circumstance. Fortunately, we have a lot of relevant data. For example, consider the function $f(p)$ that tells you how much income a person is willing to forgo in order to avoid a probability $p$ of death. At least in the United States, we know (somewhat roughly and with various caveats) that when $p$ is small, $f(p)\approx \$10,000,000\times p$. (Theory predicts, and evidence seems to confirm, that $f(p)$ is linear for small $p$.) We infer this, for example, from the premiums you have to pay people in order to get them to take on dangerous jobs, or from the amount people are willing to pay for safety devices. I'm not sure whether $p\approx 1\%$ counts as small for this purpose, but there are data available that will help decide that.

(This, incidentally, is precisely what economists mean when they say that "In the United States, the value of a life is about $10,000,000".)

The best way to account for all this is to assume that people are maximizing some functionn $U$ which takes as arguments things like income, social interaction, time spent being sick, and probability of death. Try to estimate the function $U$ by observing the choices people make in a great variety of ordinary circumstances. In other words, observe their behavioral parameters in ordinary times, assume that those behavioral parameters are the solutions to some maximization problem, and try to infer what's being maximized.

Now when the pandemic comes along, you've got to assume that something is unchanged; otherwise you have no basis to make any predictions whatsoever. The idea is to assume that what's unchanged is the maximand $U$, and that the pandemic represents a change in the constraints subject to which people are trying to maximize. Having estimated $U$, having assumed it's fixed, and writing down the new constraints, you can calculate the new behavioral parameters that result from the new maximization problem.

Now your model is essentially a fixed point problem: Behavioral parameters (that is, the solutions to the maximization problem) cause changes in behavior, which cause changes in the way the pandemic spreads (that is, the constraints on the maximization problem), which cause changes in behavioral parameters. The solution to the model is a fixed point of that process.

You can also estimate how the fixed point will change if you add additional constraints, such as penalties for going outside, prohibitions on meetings, etc.

This sort of modeling is what economists try to do all the time. I expect, but do not know, that one could say the same of epidemiologists. No model is perfect, but economists have learned the painful lesson that some models are a lot less perfect than others, and that fixed behavioral parameters are generally a hallmark of such models.

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    $\begingroup$ OK, but as long as the infection curve goes up, $M$ won't be voluntarily increased by anyone, and since all entries of $A$ are nonnegative, the current $M$ whatever it is is the worst case scenario, so reducing $\lambda(A)$ is the first priority. But you made an excellent point that when we reach a descending part, the people may want to increase $M$ and ruin the game. One solution is to issue a decree to not increase the operation rate (which is what $M$ is) for any business, but if you can solve this with incentives rather than regulations, that would be an excellent thing. $\endgroup$
    – fedja
    Commented Mar 23, 2020 at 18:28
  • $\begingroup$ I expanded my comment to you in a separate thread :-) $\endgroup$
    – fedja
    Commented Mar 26, 2020 at 15:42

Here is another model from Gil Kalai last idea:

We have the moving compressible fluid in which the particles can teleportate, diffuse, and organize the motion, while contamination can only diffuse. Each particle can be contaminated or not. You see contamination levels but not individual contamination. How to move?

The answer is that unorganized teleportation should be excluded (uncontrolled private cars), diffusion should be reduced (Israeli 100 meters from home rule for walking out), the movement of all particles should be towards the higher contamination density until the maximum allowed density of the fluid is reached there (meaning sending goods only from warehouses in less contaminated areas to the shops in more contaminated areas, but never the other way around, sending the hospital cases to the hospitals in the highest contamination areas until full capacity is reached there, the doctors working in the nearest to home hospitals whenever the switch is possible, etc, etc. Has this been implemented anywhere?

Now the next question: how to let China return to the normal mode and provide goods for the rest of the world? That would require more than hemotaxis in the model: we need a flow. It cannot be maintained forever, of course, but it is possible for quite a long time. What would that mean in practical terms? First, no one can enter China (they believe their cases now are just "imported", but why do we need any import? BTW, no question is rhetorical now). No shipments of goods should be sent to China; all hospital cases should be moved to the closest hospital to where they get most cases at the moment (by military helicopters or planes if it is not 911 and they can afford now to test every symptomatic case and move it), and the areas that should be opened to the relaxed life should be furthest from the areas with cases. In other words, if the cases are imported, they should stay at the border, the doctors treating them should stay at the borders, and somewhere along the borderline a deserted area should be created, so China will split itself into productive inland and defensive outskirts. In the fluid terminology, in addition to hemotaxis on the patch boundary towards the contamination, there should be hemotaxis away from the boundary inside until the patch separates into the outer layer and the inner bulk with minimal density in between (transit transport only, no real inhabitance). How much of that is feasible is another question. But does it make sense?


Response to Steven Landsburg:

The main point Steven made was that M is at least as important as $A$. How to reduce M? (the interaction between public servants and ordinary people)? That, as Steven correctly pointed out, is in the hands of people. The servants just serve what you need. If you reduce what you need, you can fill your supplies to the household capacity less frequently and thus reduce $M$ locally in the matrix $A$ (at last I have enough preliminary explanations to speak in the language in which I think). Example: I eat mainly grain now (bought a few bags at Costco some time ago). That produces zero garbage emission, so my garbage bin fills slowly and I can put it out, probably, 5 times less frequently, so most of the time the truck bypasses my house and I am not afraid of contaminated garbage been 5 times longer than an average person, so I have an egoistic incentive to do it as well. Besides, I just have no supply of required garbage bags :-) ). That potentially reduces M five times for the garbage pickup part of $A$. And that is where the virus teleports: the garbage trucks go everywhere. But it teleports at the speed of car and we can teleport ideas at the speed of light (Internet). The proposed modus operandi for propagation of thought is exactly what I posted in the beginning: share your thoughts on what to do in the widest network available to you where you have enough reputation to be listened to. We'll fight the car teleportaion of virus with the light speed teleportation of ideas. That is not even restricted to country: explain to your friends everywhere what is going on, what's the global strategy, why you are doing what you are doing to support it, and talk to them in the common language between you and them trying to keep it as simple as possible, but not simpler than that. Everyone is free to implement the rules that are not weaker than the current regulations in his household (I follow max(Israel,Ohio)) and think how to advance them more. That's why the government does not impose the draconian regulations itself. They hope that graduate reduction will be met by gradual self-restriction and we'll bargain not at the strict line but on a curve allowing freedoms where they are needed and using freedoms only if they are needed. That's the advantage of democracy, used properly. The advantage of dictatorship is the speed of bargaining. What we'll get depends on how we act.

If you get lost, please don't downwote, just ask questions :-)


Here is a suggestion that may get downvoted. I make it not because it is a good idea, but because some modification may lead to a good idea.

Get everyone sick to get better.

If you hunt down my WordPress blog (grpaseman) you will see a mild expansion of this idea. The crux is to interrupt the replication of SARS-CoV-2 by introducing a different virus whose systemic effects are known , are not fatal, but use the same resources that SARS-CoV-2 would use to replicate. This might delay replication inside a host long enough for the immune system to manufacture a response.

The idea has problems and needs thinking through. If it suggests a better path to interrupt replication at a different level, it will be worth all the down votes.

Gerhard "Getting Downvotes To Inspire Others" Paseman, 2020.03.26.

  • $\begingroup$ Right now it is exactly what is going on but in a different setting: the humanity, as a whole, uses the internet to spread the information about the existence of virus and enforcing the long-distance behavior of communities (cells in biological language). The main problem is that the body (community) reacts only if it experiences pain (anxiety) in the immediate physical (geographical) vicinity. If we could make pain (anxiety) felt everywhere at once but at the safe level, the body (society) and pass the blueprint, the body (society) would react way faster. $\endgroup$
    – fedja
    Commented Mar 26, 2020 at 21:29
  • $\begingroup$ So, if we could just transfer the pain (generically understood) and the virus blueprint from lungs to other organs as well and start the production of antibodies everywhere at the same time that might help. Just my crazy 2 cents. $\endgroup$
    – fedja
    Commented Mar 26, 2020 at 21:31
  • $\begingroup$ The modification is exactly that: you do not need to or should use the same resources. All you really need is to get the antibodies that fight both. If you use the same resources, you will just kill more cells in the lungs and there will be zero advantage. Am I making sense now? $\endgroup$
    – fedja
    Commented Mar 26, 2020 at 21:38
  • $\begingroup$ The exact reformulation of the OP idea that corresponds to what I wrote: Get everybody who is better to feel sick. Alas, that is exactly the idea of the vaccination but vaccination uses the same blood network as the virus. Can we use the neural network somehow in the body to magnify the pain in the lungs? That would not lead to cure but it will lead to quick and sure detection and the result should be different from cough, of course. Some symptom like you have for no common illness. Now the suggestion is complete and I wrote the whole train of thought. We can just inject a self-test this way. $\endgroup$
    – fedja
    Commented Mar 26, 2020 at 21:48
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    $\begingroup$ This post and especially the comments is firmly in "not even wrong" territory. $\endgroup$
    – JCK
    Commented Mar 27, 2020 at 0:40

Mathematically speaking it looks that if we split the categories $C_i$ into subcategories with zero (or small) cross transmission edges then this suppress the exponential growth at some scale. (And this is also the case for other geometric limitation on the graph of possible transmission, e.g if there are no edges for people who arge geographically remote). In technical terms one need to make the transmission graph (and every large subgraph) a very "bad" expander.

  • $\begingroup$ You can also separate in time and turn the blocks on and off one after another. If you could do it with 16 blocks (i.e., only some particular 1/16th of the population is out running the services and the rest stays at homes (especially strictly on their days 2-5 from their day to be out when they can be stealth virus carriers) .on every given day and can only call for medical emergency response), we would be done. I mean, make $A(t)$ depending on $t$ turning on one block a time on any given day and forcing the rest to be identity. That would sort of raise $A$ to the power $1/16$. $\endgroup$
    – fedja
    Commented Mar 25, 2020 at 16:08
  • $\begingroup$ Moreover, the time splitting may work no matter what the actual dynamics is. Misha Sodin told me that the Israeli army is already doing something like that, apparently, letting people in group $C_i$ out to do the work on day $i$ and educating the rest how to run necessary services when they are locked in (if I understood him right). In principle, this model can be applied to the whole population. $\endgroup$
    – fedja
    Commented Mar 25, 2020 at 16:16
  • $\begingroup$ The time splitting is also better because we have no guarantee that each particular block $C_i$ has a smaller largest eigenvalue than the whole matrix, and every geographical block is infected by now. That would be "the" solution, if only it were feasible... Thanks for the response, BTW :-) $\endgroup$
    – fedja
    Commented Mar 25, 2020 at 16:22
  • $\begingroup$ Dear fedja, this may mean that not allowing (non essential) people to use cars is essential. (In some cases public transformation is stopped but there is no limitation on private cars.) $\endgroup$
    – Gil Kalai
    Commented Mar 25, 2020 at 19:57
  • $\begingroup$ Dear Gil, I tried to bring your idea about cars to its logical extreme. See if it makes sense. $\endgroup$
    – fedja
    Commented Mar 26, 2020 at 12:56

I have one rather practical answer, which does not answer the question in the body, but possibly in the title.

I was very surprised to learn that a (small) hospital in my home town was doing their scheduling ("rostering" might be the correct term, but I don't know) by hand.

They were clever enough to ask for help just in time. Again surprisingly: informally via a mathematician they knew privately. Thus, they now have a professional university team developing a plan that fits their new needs - minimal service, minimizing the probability that a large portion of their staff simultaneously drops out.

I guess that such a strategy might also make sense for other community services, who might not even be aware of this fact.

(Disclaimer: I am not helping with the implementation, but I was lucky to know someone who knew someone who was a professional in this field, who luckily agreed to do it.)


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