New paper on Wilson-Cowan Model

I forgot to post that my fellow Yahya and I have recently published a review paper on the history and possible future of the Wilson-Cowan Model in the Journal of Neurophysiology tribute issue to Jack Cowan. Thanks to the patient editors for organizing this.

Before and beyond the Wilson–Cowan equations


The Wilson–Cowan equations represent a landmark in the history of computational neuroscience. Along with the insights Wilson and Cowan offered for neuroscience, they crystallized an approach to modeling neural dynamics and brain function. Although their iconic equations are used in various guises today, the ideas that led to their formulation and the relationship to other approaches are not well known. Here, we give a little context to some of the biological and theoretical concepts that lead to the Wilson–Cowan equations and discuss how to extend beyond them.


PDF here.

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

1 -\frac{e^{-R_0}}{1- R_0e^{-R_0}}

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

\frac{ds}{dt} = -\beta sl

\frac{dl}{dt} = \beta sl -\sigma l

where s is the fraction of the population susceptible to SARS-CoV-2 infection and l 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

\frac{dl}{ds} = \frac{\sigma l -\beta sl}{\beta sl},

which integrates to l =  \frac{\sigma}{\beta} \ln s - s + C. If we suppose that initially s=1 and l = l_0<<1 then we get

l =  \frac{1}{R_0} \ln s + 1 - s + l_0 (*)

where R_0 = \beta/\sigma is the reproduction number. The total number infected will be 1-s for l=0. Rearranging gives

s = e^{-R_0(1+l_0+s)}

If we assume that R_0 s <<1 and ignore l_0 we can expand the exponential and solve for s to get

s \approx \frac{e^{-R_0}}{1- R_0e^{-R_0}}

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 R_0 increases. So reducing R_0 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 s to get

\frac{ds}{dt} = -\beta(\frac{1}{R_0} \ln s + 1 - s + l_0) s

We compute the time to peak, T, by separating variables and integrating both sides. The peak is reached when s = 1/R_0.  We must thus compute

T= \int_0^T dt =\int_{1/R_0}^1 \frac{ds}{ \beta(\frac{1}{R_0} \ln s + 1 - s +l_0) s}

We can’t do this integral but if we set s = 1- z and z<< 1, then we can expand \ln s = -\epsilon and obtain

T= \int_0^T dt =\int_0^{l_p} \frac{dz}{ \beta(-\frac{1}{R_0}z  + z +l_0) (1-z)}

where l_p = 1-1/R_0. This can be re-expressed as

T=\frac{1}{ \beta (l_0+l_p)}\int_0^{l_p} (\frac{1}{1-z} + \frac{l_p}{l_p z + l_0}) dz

which is integrated to

T= \frac{1}{ \beta (l_0+l_p)} (-\ln(1-l_p) + \ln (l_p^2 + l_0)-\ln l_0)

If we assume that l_0<< l_p, then we get an expression

T \approx \sigma \frac{\ln (R_0l_p^2/l_0)}{ R_0 -1}

So, T is proportional to the recovery time \sigma and inversely related to R_0 as expected but if l_0 is very small (say 0.00001) compared to R_0 (say 3) then \ln R_0/l_0 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 R_0 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

s = \frac{1}{{R_{0}}^2} ( e^{R_0} - R_0 - \sqrt{(e^{R_0}-R_0)^2 - 2{R_0}^2})

However, for R_0 near 1, a better expansion is to substitute z = 1-s into equation (*) and obtain

l =  \frac{1}{R_0} \ln 1-z + z + l_0

Set l=0, after rearranging and exponentiating,  obtain

1 - z = e^{-R_0(l_0+z)}, which can be expanded to yield

1- z = e^{-R_0 l_0}(1 - R_0z + R_0^2 z^2/2

Solving for z gives the total fraction infected to be

z = (R_0 -e^{R_0l_0} + \sqrt{(R_0-e^{R_0l_0})^2 - 2 R_0^2(1-e^{R_0l_0})})/R_0^2

This took me much longer than it should have.


The formal logic of legal decisions

The US Supreme Court ruled today that the ban on sex-based discrimination in Title VII of the 1964 Civil Rights Act protects employees from discrimination based on sexual orientation or gender identity. Justice Gorsuch, who is a textualist (i.e. believes that laws should only be interpreted according to the written words alone without taking into any consideration the intent of the writers), writes “An employer who fires an individual for being homosexual or transgender fires that person for traits or actions it would not have questioned in members of a different sex. Sex plays a necessary and undisguisable role in the decision, exactly what Title VII forbids.” I find this to be an interesting exercise in logic. For example, consider the case of a “man married to a man”. According to Gorsuch’s logic, this cannot be an excuse to fire someone because if you replace “man” with “woman” in the first instance then you end up with the the phrase “woman married to a man”, and since this is not sufficient for firing, the reason is not sex invariant. Dissenting legal scholars disagree. They argue that the correct logic is to replace all instances of “man” with “woman”, in which case you would end up with “woman married to a woman” and thus the reason for firing would be sex invariant. I think in this case Gorsuch is correct because in order to have complete sex invariance, the rule must apply for any form of exchange. The law should be applied equally in all the possible ways that one gender is replaced by the other.

The depressing lack of American imagination

Democratic presidential candidate Andrew Yang made universal basic income a respectable topic for debate. I think this is a good thing because I’m a major proponent of UBI but my reasons are different. Yang is a technology dystopian who sees a future where robots take all of our jobs and the UBI as a way to alleviate the resulting pain and suffering. I think a UBI (and universal health care) would lead to less resentment of the welfare system and let people take more entrepreneurial risks. I believe human level AI is possible but I do not accept that this necessarily implies an economic apocalypse. To believe such a thing is to believe that the only way society can be structured is that a small number of tech companies owns all the robots and everyone else is at their mercy. That to me is a depressing lack of imagination. The society we live in is a human construct. There is no law of nature that says we must live by any specific set of rules or economic system. There is no law that says tech companies must have monopolies. There is no reason we could not live in a society where each person has her own robot who works for her. There is no law that says we could not live in a society where robots do all the mundane work while we garden and bake bread.

I think we lack imagination in every sector of our life. We do not need to settle for the narrow set of choices we are presented. I for one do not accept that elite colleges must wield so much influence in determining the path of one’s life. There is no reason that the US meritocracy needs to be a zero sum game, where one student being accepted to Harvard means another is not or that going to Harvard should even make so much difference in one’s life. There is no reason that higher education needs to cost so much. There is no reason students need to take loans out to pay exorbitant tuition. That fact that this occurs is because we as a society have chosen such.

I do not accept that irresponsible banks and financial institutions need to be bailed out whenever they fail, which seems to be quite often. We could just let them fail and restart. There is no reason that the access to capital needs to be controlled by a small number of financial firms. It used to be that banks would take in deposits and lend out to homeowners and businesses directly. They would evaluate the risk of each loan. Now they purchase complex financial products that evaluate the risk according to some mathematical model. There is no reason we need to subsidize such activity.

I do not accept that professional sports teams cannot be community owned. There is no law that says sports leagues need to be organized as monopolies with majority owners. There is no reason that communities cannot simply start their own teams and play each other. There is no law that says we need to build stadiums for privately owned teams. We only choose to do so.

The society we live in is the way it is because we have chosen to live this way. Even an autocrat needs a large fraction of the population to enforce his rule. The number of different ways we could organize (or not organize) is infinite. We do not have to be limited to the narrow set of choices we are presented. What we need is more imagination.

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.