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.
Edit:
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.
Best Answer
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.