Physical Modelling of a Pandemic

These are strange times, the likes of which we probably see once in a generation, and it is amazing that it is happening across the world simultaneously, and in the age of internet. As physicists, as much as one would like to carry on , it is hard to ignore what is going on around you, and try to understand it, the same way we understand any other physical system. So here it goes:

A Growth+Diffusion Model for Early Epidemic

As we approach any other physics problem, we start from first principles that describe the system of interest. For the purpose of this note, I will start by describing how how a contagion is transmitted within a community and spreads across large geographical locations. What bothers me about the standard treatments (e.g., the SIR equation) is that, by reducing the system to ordinary differential equations, it ignores this geographical diversity and thus (as we shall see below) might miss important phenomenological features of the evolution.

Instead, I will use an inhomogeneous linear growth+diffusion equation to describe the early phase of an epidemic, prior to large-scale immunity or social intervention:

Here n(x,t) is the number density of infected individuals as a function of space and time. R(x) is the local linear growth rate (= ln(2) divided by the doubling time), while D(x) is a diffusion coefficient, quantifying how infected individuals can move around. Since the coefficients don’t have explicit time-dependence, we can decompose the solution into modes with exponential growth (or decay):

where \lambda_m‘s and u_m({\bf x})‘s are eigenvalues and eingenstates of the elliptic operator R+\nabla\cdot D \nabla:

For constant D, this is the same equation as energy states of a particle in 2D that satisfies time-independent Schrodinger Equation:

For example, one result that we can now use is that for generic random potentials (or disorder) in 2D the eigenstates are localized, otherwise known as Anderson localization. One can think of these different localized eigenstates as localized communities with different doubling times for the growth of the epidemic.

The next physics result we shall use is the probability distribution of spacing of energy states of a random potential, known as Wigner surmise:

which we shall use to quantify the distribution of the largest eigenvalue (equivalent to the energy of the most energy bound state). Plugging this into Equation (2) yields:

Now, for large \lambda and t, we would like to use saddle approximation compute this integral. To do this, we have to expand -\ln A(\lambda) to second order in \lambda, i.e. -\ln A(\lambda) = a_0 +a_1 \lambda + a_2 \lambda^2 + \dots where we assume a_2 >-C, otherwise the integral would diverge (or we need to include more terms). The saddle-point approximation for large times yields:

In other words, we predict a super-exponential growth at early times, which is contrast to the exponential expectation from simple uniform models, such as the SIR equation. The reason for this is clearly that we are dealing with a distribution of growth rates, and the later times will be dominated by populations with faster growth rates (smallest doubling time), no matter how small they start.

One clear limitation of Equation (8), apart that it only applies to the linear regime and no interventions, is that at some point we will be dominated by the community with the largest eigenvalue, at which point the exponent should become linear time. This will be followed by further nonlinear effects. I will include these effects, by terms that are higher order in time, i.e.

Comparison to Data

We shall next try to fit this to fatality data for different countries, up until March 29th, 2020 (using data provided by https://ourworldindata.org/coronavirus). We find the best fit parameters:

b_2= (7.7 \pm 1.4) \times 10^{-3}~ {\rm day}^2 and b_3= (8.7 \pm 3.6) \times 10^{-5}~ {\rm day}^3. However, these parameters are highly degenerate, as can be seen in the 1 and 2\sigma contours

Best fit parameters for US (blue), Canada (red) data, and Italy (green) as of March 29, 2020. There is just over 2\sigma evidence for the cubic term in US data, which means that t\exp(b_2 t^2) can describe most of US data. The Canadian parameters are more poorly constrained, but appear perfectly consistent with those of the US. The Italian parameters, are considerably better constrained but still consistent with the two other countries (but see below).

In fact, including the cubic terms only became necessary in the last 3 days to fit the US data. Furthermore, the Canadian parameters are perfectly consistent with those of US, but with larger errors.

Some comments on fitting the data: The most rigorous way to fit the data (assuming that you have a perfect model) is to fit daily values using Poisson statistics, as they are independent. Using this, our best fit to US, Canada, and Italy data gives \chi^2_{\rm red} =1.9, 0.9, and 12 respectively. I interpret this as data not being completely uncorrelated, as expected, since infection happens in clusters. As such, for parameter estimation I divide the Poisson log-likelihood by \chi^2_{\rm red} to reflect this sample variance.

The next figure shows the logarithmic derivative \dot{N}/N for US, Canada, Italy, and South Korea. While for the US and Canada, t=1 is the report of first death, the Italy data is shifted forward by 25 days from the day of first death. The best curve fit is to the US data, but clearly fits both Canada and Italy (after the shift), as the parameters are consistent at 95% confidence level, per the figure above.

I have also added South Korean data, but shifted by 40 days. While the US best-fit, extrapolated forward can fit well the first 20 days of S Korean dat, it clearly misses the second half. However, it could well be that higher order terms in the exponent of Equation (9) will become important at this point.

Logarithmic derivative of the total fatality \frac{d\ln N}{dt} \simeq \frac{N(t)-N(t-1)}{N(t-1)} for US (blue), Canada (red), Italy (green), and South Korea (black), as of March 29th. For US and Canada, t=1 is at the report of first death, while the data for Italy and South Korea are shifted forward by 25 and 40 days, respectively. The curve shows the best fit to US data.

This plot is of course very suggestive. Could there be a universal \dot{N}/N as a function of time (analogous to Hubble constant H(t), or Friedmann equation, in cosmology)? This may suggest the intriguing possibility that the true onset of the Italian (South Korean) epidemic might have been 25 (40) days before the first death was attributed to the Covid-19 epidemic. Of course, the alternative is that there is no universality, and the different behaviors are dictated by environmental and social factors.

Final Note

Finally, let me end on a somber note: The best-fit model (to US, Canada, and Italy) predicts that Canadian mortality will pass 1000 around April 9th, while the US mortality will pass 10,000 around April 5th. I will not dare extrapolate beyond that point, and I sincerely hope that the model is wrong.

So, Stay Home! Stay Safe!

Congratulations to Qingwen Wang on becoming a 2019 Institute of Particle Physics (IPP) Early Career Fellow

Congratulations to Qingwen Wang on becoming a 2019 Institute of Particle Physics (IPP) Early Career Fellow.

Qingwen travels to Pasadena, California to work with Yanbei Chen at Caltech for three months.

“We hope to amend the effective one body formalism in the numerical relativity with our Boltzmann boundary conditions for quantum black holes, so that we can find predictions for signatures from quantum gravity, using both physical reflectivities and initial conditions.”

Read more at:

https://particlephysics.ca/research-activities/ipp-early-career-theory-fellows/?lang=en

Prof of Old School

[Intro]
High in the halls of the grads who are gone
The prof would write with their ghosts
The ones he had lost and the ones he had found
And the ones who had loved him the most

[Verse 1]
The ones who’d been gone for so very long
He couldn’t remember their names
They spun him around on the damp old school
Spun away all his sorrow and pain

[Chorus]
And he never wanted to leave, never wanted to leave
Never wanted to leave, never wanted to leave

[Verse 2]
They wrote through the day
And into the night through the snow that swept through the hall
From winter to summer and winter again
‘Til the walls did crumble and fall

[Chorus]
And he never wanted to leave, never wanted to leave
Never wanted to leave, never wanted to leave
And he never wanted to leave, never wanted to leave
Never wanted to leave, never wanted to leave

[Outro]
High in the halls of the grads who are gone
The prof would write with their ghosts
The ones he had lost and the ones he had found
And the ones who had loved him the most

#JennyofOldstones

Farewell to Dr Chiamaka Okoli

Chiamaka’s PhD defence on Dec. 4th, 2018 (image courtesy of Natacha Altamirano)

It is with great sadness that I share here the news of passing of my former PhD student, Chiamaka Okoli. Chiamaka successfully defended her PhD with the title “Dark Matter and Neutrinos in the Foggy universe” last December. Her PhD convocation at the University of Waterloo was scheduled for last week, on June 13th.

Post-defence celebration with cake at PI bistro; From left to right: James Taylor (co-supervisor), Chiamaka, Larry Widrow (Chiamaka’s PhD external examiner), me and my son Juyah, and Mike Hudson (images courtesy of Natacha Altamirano)

However, that was not meant to be. Chiamaka’s life was cut short on June 6th in McMaster hospital in Hamilton. You can read more about Chiamaka on Perimeter Institute’s website. If you wish to contribute to The Chiamaka Okoli Trust Fund, to help her family with funeral costs (which involves moving her body back to her native Nigeria), please contact Jamie Cooper as soon as possible.

But then, here is my story …

I first met Chiamaka in Fall 2012, when she joined the then recently established Perimeter Scholar International program. She had just finished the diploma program with Ravi Sheth at the International Centre for Theoretical Physics in Trieste, and was eager to work more on cosmological structure formation. We started working on understanding the profiles of dark matter haloes, establishing a novel paradigm to predict their concentration based on energy conservation. This led to Chiamaka’s first paper.

Figure 5 from Chiamaka’s first paper, comparing her prediction for concentration of dark matter haloes as a function their mass, with previous work.

Chiamaka then started her PhD program at the University of Waterloo, working with me and James Taylor.

Chiamaka’s second paper studied a novel possibility for tracing the fingerprints of cosmic neutrinos by how they could slow down the motion of dark matter haloes through dynamical friction. We predicted that, with proper modeling, this effect could be detected in current and future galaxy surveys.

A figure from Chiamaka’s second paper , showing how wakes of neutrinos can slow down clustered dark matter haloes, through their gravitational pull

And there was much more:

  • In her third and final publication, Chiamaka established the range of theoretical uncertainties in predictions for annihilation signal from dark matter haloes
  • She was working with me and Ue-Li Pen to test her theoretical predictions for dynamical friction due to neutrinos using TianNu simulations
  • She was working with Natacha Altamirano, Utkarsh Giri, and I to measure and understand the thermal Sunyaev-Zel’dovich effect from groups of galaxies, as seen by Planck satellite

So long, farewell …

Chiamaka’s final two years on Earth were embodiments of perseverance in the face of adversity. In August 2017, Chiamaka gave birth to her baby boy, Munachi. The same month, her mother passed away back home in Nigeria. In November 2017, while still on maternity leave, she submitted two papers to arXiv, and then started applying for postdoc positions. In January 2018, she came back to work. In February 2018, she was hit with her first near-fatal cerebral aneurysm, which she talked about in a facebook post, in its one-year anniversary. It took a few surgeries, as well as weeks and months of recovery in the McMaster hospital, as well as Grand River hospital and Rehab center for her to get back on her feet. Meanwhile, she was being contacted for postdoc interviews and offers. In September 2018, Chiamaka again came back to work, determined to wrap up her PhD thesis, which she managed to do by mid-October. She defended her PhD in December 2018. In March 2019, Chiamaka had a final cranioplasty surgery in Hamilton, so that she’d fully recover from her first aneurysm episode. Alas, it struck again in May, and ended her life.

Even though Chiamaka had postdoc offers, she ended up turning them down, and instead started looking for data science jobs in the area. The exact reason remains a mystery to me, but I would like to think the ordeal that she had gone through had given her a new perspective on life. I also would like to think that I understood all her struggles, and helped her as best as I could, even though that might just be wishful thinking.

Like all academics, Chiamaka’s legacy now propagates through those who read and study her work. They are all those open-ended questions and ongoing projects that will permeate through journals and workshops, along with the dreams of what she could have done with them, only if universe treated her more kindly.

And finally, her legacy continues through her 22-month old son, Munachi, who is still too young to know what is happening. I do hope I get to tell him at some point about his mother’s enthusiasm for cosmology, zest, determination and sense of humor (even when she couldn’t talk and had to write her thoughts, while in a hospital bed).

Farewell Chiamaka.

Image may contain: 1 person, sitting

HC SVNT DRACONES!

I gave a talk about dragons!

Here be Dragons: Ancient cartographers often used illustrations of dragons to show treacherous and uncharted territories. Physicists, Astronomers and Cosmologists have managed to vanquish these dragons out to the farthest reaches of the cosmos, highest temperatures imaginable, and deepest holes in the galaxies. Beyond these boundaries lie our new age dragons. I will retell the tales of these creatures and our battles to slay them.

(Physics 10, University of Waterloo, September 2018; keynotePDF)

got_dragons_hed

Congratulations to Dr. Mansour Karami!

Congratulations to Dr. Mansour Karami, who has just successfully defended his PhD thesis, entitled:

“Probing the dark universe with gravitational lensing”

Mansour’s PhD focused on different ways in which gravitational lensing teaches us about dark objects in the universe:

Here is a picture of Mansour, with his proud co-supervisors, and a cake that features pictures from his thesis:

Mansour
From Left to Right: Me marveling at the cake, the freshly minted doctor, and his other proud co-supervisor, Avery Broderick (photo courtesy of Jorge Preciado)

And here is a picture of the said cake, which you can understand better by reading Mansour’s thesis!

2018-09-07 16.09.20
Mansours’ PhD cake!

We wish Mansour all the best in his future adventures in the world of quantitative finance 👏😲😉

Congratulations to Dr. Natacha Altamirano!

Congratulations to the newly minted Doctor Natacha (Naty) Altamirano on successfully defending her PhD, entitled:

“The quantum and the gravity: Newtonian and Cosmological applications” 

In her thesis, Natacha looked into an innovative idea about the nature of gravity, where it is speculated that gravity is inherently a classical and not a quantum interaction.  She further studied the laboratory tests of this idea, as well is its potential cosmological applications.  While most of this work was with Natacha’s co-supervisor Robb Mann (and other collaborators), she also worked with me on various topics in cosmology, including holographic big bang, modified gravity, and Sunyaev-Zel’dovich (SZ) signal from galaxies (yet to appear).

Below are pictures of Natacha’s proud supervisors, as well as a celebratory cake, featuring some of her PhD work. The latter includes the only published picture of her SZ work, showing a mysterious SZ deficit for small galaxies! 😕 IMG_3950

IMG_3897

You can read more of Natacha’s broad, innovative and exciting research on arXiv. We wish her all the best in her future adventures 👏 🙂

Congratulations to Dr. Elizabeth Gould!

This news post is a bit late but for good reason!

First, let me start by congratulating Dr. Elizabeth (Beth) Gould on successfully defending her PhD thesis “New Views on the Cosmological Big Bang”, last September.

20170914_173008
Beth’s graduation cake, highlighting pictures from her PhD thesis (photo credit: Joel Gould)

Beth’s thesis focused on innovative ways in which physics could be different as we approach big bang. Could it be that Quantum Mechanics is “out of equilibrium”, or that spacetime really has one more or less dimension?  Or, could it be that history literally repeats itself, as our future is the same as our past (what we call periodic time cosmology)?  In fact, one of Beth’s papers was even cited by the late great Stephen Hawking, in his final publication.

20170914_173740
Beth, along with Auzaud and me, posing with the said cake! (photo credit: Joel Gould)

Beth is my 7th graduating PhD student. After a short postdoc at the University of Southampton this year, Beth will be joining the brand new Arthur B. McDonald Canadian Astroparticle Physics Research Institute, at Queen’s University as a postdoctoral fellow, in September 2018.

And here is that last bit of good news that came out this week:

“Every year, top graduate students from the Faculty of Science are nominated for the W.B. Pearson Medal, which is given to a Doctoral student from each department in recognition of their creative research …

The W.B. Pearson Medal in Physics & Astronomy has been awarded to Elizabeth Gould for her research on “New Views on the Cosmological Big Bang”, with Niayesh Afshordi.”

 

So, please join me in congratulating Dr. Gould on successfully finishing her PhD, starting a prestigious postdoctoral fellowship, and being recognized for her creativity by the W.B Pearson Medal.  

Echoes strike back!

DCny-VxUAAIcPmG
Black Hole echoes, as rendered by Kaća Bradonjić (@physartphil)

 

 

Invitation

In a post last year, I talked about our search for “echoes from the abyss” with Jahed Abedi and Hannah Dykaar in the Advanced LIGO gravitational wave observations, which are smoking guns for Planck-scale structure near black hole event horizons. Such structures are not expected in classical General Relativity, but may be motivated by various versions of the black hole information paradox such as the fuzzball models of black holes in string theory, or the infamous firewall paradox. Most surprisingly,  we found that the evidence seen for (a toy model of) echoes in first observational run of LIGO data can only be seen in <1% of random noise realizations.

 

 

 

While this was widely covered by science journalists (see here for some links), two of my favourites were by Zeeya Merali on Nature News and Sabine Hossenfelder @skdh on Aeon.

Headline from Sabine's article about "Echoes from the Abyss"

Controversy

And then, there was the response from the community. While the theorists were beside themselves with excitement (our two papers are cited close to a hundred times in 16 months), we got a long silence from observers/experimentalists with one exception. A group of of LIGO collaboration members in Albert Einstein Institute (AEI) in Hanover were quick to express (a very healthy and deserved) scepticism. We responded to their comments in our “Holiday Edition!”, but the real question was whether our results could be reproduced by people who had more experience with LIGO data.

The latter was a long time coming, partly because big collaborations work slowly, and partly because of other exciting discoveries (such as the first binary neutron star merger, seen by the LIGO collaboration). In the meantime, I also organized a workshop on “Quantum Black Holes in the Sky?”, along with Vitor Cardoso and Samir Mathur, where we discussed various observational and theoretical aspects of  black hole echoes, and more.

Untitled

The AEI group finally released their analysis last December (which was later updated with a 4th black hole merger event in February), and lo and behold, they found that the evidence seen for echoes (using the same dataset and model), is only seen in 2±1% of random noise models (i.e. within 1σ of what we reported). Surprisingly though, they went on to say

“The reduced significance is entirely consistent with noise, and so we conclude that the analysis of Abedi et al. does not provide any observational evidence for the existence of Planck-scale structure at black hole horizons.” !!!

Needless, to say that this didn’t make too much sense to us, a point that I made publicly on arXiv, and on a (let’s say less diplomatic) exchange on facebook, with Thomas Dent.

What they really are saying is that, since General Relativity has been such a successful theory for the past 100 years, we don’t really think echoes are there, and we need really strong evidence (e.g., p-value of 10-6) to be convinced otherwise. Fair enough, but that is a very subjective statement. Someone like me may argue that we have known for nearly fifty years that if you consider quantum mechanics, something funny is happening at the black hole horizons. Why is the entropy proportional to horizon area? How could information get out of the horizon of evaporating black holes? We can also explain the scale of dark energy (the infamous 10-120), by assuming a “quantum equilibrium” at the horizons of astrophysical black holes.  So for me the bar is probably lower. Therefore, it is important to separate the objective statistical statements (e.g. p-value) which only depend on data, from subjective “priors” that varies from theorist to theorist.

For more, have a look at Sabine’s update on the state of echoes and controversy in Quanta Magazine.

Untitled

BlackHoleEchoes_2880x1620-2880x1620
James O’Brien for Quanta Magazine

Black Hole Echology

Out of the “tentative evidence” for echoes and the resulting controversy, emerged the need for a clearer understanding of what echoes should really look like. In a recent arXiv preprint: “Black Hole Echology: The Observer’s Manual”, Qingwen Wang and I provided the most comprehensive study of echo templates, and their model dependence (and independence), for a spinning black hole.

Untitled
Nominal echoes from GW150914 event, for different wall distance to the horizon

Qingwen Wang

 

 

 

 

 

 

 

 

We made some surprising discoveries, e.g., that the echoes decay as a power-law ∝ time-4/3, not exponentially, as we had originally assumed, or that the signal below the superradiance frequency is insensitive to initial conditions. Echoes_Western_Mar_2018.001

These findings set the stage for the new observational evidence for echoes that was about to emerge.

Echoes Strike Back!

There is a quote that I have often heard in Physics and Astronomy gatherings, but I don’t know who it can be attributed to:

“If you have to argue about statistics, it means that you need more data.”

and indeed that will be the way to unambiguously settle the argument about the significance of black hole echoes.

First came the surprising results by Conklin, Holdom, and Ren from University of Toronto, who developed a “model-agnostic” search for echoes by cross-correlating data from the two detectors and looking for periodic signals. This was very complementary to our original search, as it assumed very little about the exact template, but looked for repeating echoes that lasted much longer.  Indeed, they think they see echoes in 5 of the LIGO/Virgo events that we did not find (or look for) echoes in, with p-values 0.2%-0.8% (roughly 3σ evidence).

Echoes_Western_Mar_2018.001

The grand finale came about, after I gave a talk about echoes at Yukawa Institute in Kyoto, during the CosPa 2017 meeting.

cospa1

During the meeting, both Shinji Mukohyama and Lam Hui asked me whether I expected to see echoes from the binary neutron star merger GW170817, which had made headlines a couple of months earlier. I first dismissed the idea as it was a very different frequency regime from what we had for binary black holes, and given the lack of any detectable post-merger signal by LIGO/Virgo, it wasn’t even clear when a black hole remnant would form, if at all.

 

 

 

However, it then occurred to me that we might have an opportunity to probe a very different regime, consisting of the first few harmonics of the echo chamber. This is at too low a frequency for binary black hole mergers, but is squarely within the LIGO sensitivity band. Indeed, a simplified version what the Toronto group did, with proper inverse noise weighting, gave a huge and surprising signal for echoes at 4.2σ (or p-value of 10-5) exactly where you expected for the mass and spin of a black hole remnant of GW170817 binary neutron star merger.

spectrogram_color
cross-spectrum of the two LIGO detectors (after Wiener filtering and summing over integer multiples of the frequency). The peak at 72 Hz and 1 second is the most significant in the diagram, at 4.2σ.

peaks_in_frequency_05_sec
Same cross-power spectrum as above, but plotted at 1 second after the merger. Blue region shows the expected range for echo frequency.

For more details, you should read our paper, watch my talk, and/or see my presentation slides.

So, is this just a statistical or systematic fluke? Or, could it be the beginning of the end for the black hole information paradox, as well as our first crack at the quantum gravity nut? Only time will tell.