Department of. The Department of Computer Science and the Big Data Institute, as you've just seen, we are recording this meeting.
So if you don't wish to be recorded, please keep your microphone off and your camera turned off.
And otherwise, I will hand over to Christoph to do the introductions. Thanks, Christine.
So it's a pleasure to introduce Professor, somebody from.
And Professor of Machine Learning and Public Health at the University of Copenhagen
in Denmark and Professor of Public Health and statistics at Imperial College.
So Sam is known to many in Oxford from his time.
From his time here, including the Big Data Institute, has done many contributions at the interface of statistics and public health first.
And he followed dynamics lots of important contributions to geospatial dynamics of infectious diseases, including on malaria and the global burden.
A very prolific and important contributions on the dynamics of COVID and always at the interface of applied and fundamental research.
And we'll be talking to us today based on the sort of fundamentals of learning based on the work Ontarians have achieved some.
Thank you. Thank you, Brazil. Thank you set for setting this up.
So I was I was a bit sort of at odds on what to what actually presented because we've been working
on a paper in my group and with my collaborators of months for a long time and we didn't know,
you know, I'm really proud of that. People are really happy of it and find its contribution very interesting.
But it can be a bit dry for those who are not used to or interested only in banking processes.
So I wanted to do a little bit of a history of infectious disease modelling
and the different approaches so that those in the audience were not used to.
The infectious disease modelling can sort of couch what our new contributions are,
and then I'll move on to some interesting facets that I've been working on.
So the first thing is, you know, what's our goal? Well, most of the time in infectious diseases, we're interested in some measure of the epidemic,
something like prevalence, the number of infected individuals at any time.
So I go somewhere, I find out how many people are infected with, say, SARS-CoV-2.
And that's my preference, right? Or you could look at something like incidence,
which is the number of newly infected individuals at time T and there are, of course, other measures in it.
But our goal essentially is to be able to model these two metrics.
Prevalence, you know,
when you can get from measure blood samples and seeing if people are positive for a virus or parasite or any disease of interest and incidence,
of course, you can just calculate using routine cases.
So the question is, you know, why don't we just use Gaussian process regression and call it a day?
Well, why don't we just use common filters or Roraima or recurrent neural networks?
I mean, these are all incredible and they're very good for prediction, and they may give you a really good metric of uncertainty.
And they're empirically motivated from the field of machine learning and deep learning for at least the econometrics.
But you know, why don't we just use this and then end the talk here, right?
Why? Why am I? Am I? You know what? The reason of trying to even look at things like process?
And the key thing is that we want to try for the best of both worlds and, you know,
said that I use this term a lot fairly mechanistic models and the idea is we want to encode some notion of dynamics in there.
Now in physics, you know where we're ultimately mathematics, chemical and physical law with remarkable accuracy,
with things like quantum electrodynamics and even just the swing of a pendulum chaotic or not.
What we really want to do is try to disentangle some mechanisms that have some basis in epidemiological reality.
And the reason we want to do this is that if we can understand the mechanism,
then it goes a long way to understanding the causality of the whole process, why things happen, why certain hypotheses occur.
And so the idea is that we don't have to throw away things like a remnant of industrial processes.
In fact, we used, you know, random walk, for example, we've got a process with a specific precision.
We use them in all our models, but we want to embed these in a mechanism.
And so the question then comes in, you know, if we take this flexibility plus the mechanisms,
what is the mechanism that we want to use for modelling effects?
And then we move on and we think about, well, what is it we actually care about, right?
So we have data on incidents of Providence, right again.
You know, the number of infected individuals, a number of newly infected individuals.
Well, what we really care about is the rate of transmission, which, you know, for all of us familiar with now, the COVID 19 pandemic,
it's it is generally reflected the reproduction number and if the rate of transmission and it is simply the average number of secondary cases.
So the average number of people I infect after I'm infected and historically this traces all the way back
to the earliest models in infectious disease modelling from from Ronald Ross and lockdown in malaria.
So we actually care about this metric, Archie.
But, you know, many people who are familiar with statistical literature could say, Well, why don't we just estimate growth?
And this is one of the big contributions from what we call lingo, Marc Lipsitch in terms of translating some degree of epidemiological mechanism,
using probability generating functions for moment, generating functions and trying to calculate the.
But however, we have to take care when trying to estimate growth rates or rates of transmission.
And because, as has been shown by our implosive and co, you have to be really careful when trying to estimate growth rates or reproduction.
So again, part of the benefit of having a mechanism in there is this knowledge that growth rates can be very
difficult to estimate because of the exponential growth and especially when looking at the tails.
And so by providing a mechanism, we can somehow control a little bit this statistical process, although not in time.
And so the question is, what is Olmec now? Once again, I do like the physical analogy, and I can't I can't stress it enough.
You know, the field of mathematical biology is driven by this idea of physicists coming into into biology from the.
What about me and all the other contributions? Roy Anderson, et cetera.
And bringing in this mechanism for trying to do to epidemiology and biology what have been done in physics,
and there's so many mechanisms that concern infectious diseases that have grown in epidemiology,
from looking at network structures to looking at underlying dynamics both time and space.
And I'll give me the first model that really set the stage was the seminal work of CalMac and.
I mean, it's pretty hard even if you've never done anything infectious disease modelling to to avoid some of the work that they did.
And of course, they have the classic scenario models where they look at susceptible,
eye infected or recovered, and they have very, very simple series of equations here.
And you know, part of the reason that we consider this thing, the reproduction number is not only from Ross and locked up,
but also from from this paper where I say r t that there's not actually a very that's a constant f r note,
but it's actually trying to trying to estimate the reproduction number from these equations.
And since the initial contribution of come back in the, you know, the architecture that have grown from susceptible, infected recovered models,
arguably some of the most widespread in all of infectious disease, you can cast these stochastic differential equations.
You can cast these as Markov chains. You can actually model using reaction kinetics, which we'll talk about in a little bit.
There are connexions in this to two fundamental integral equations, including the Volterra equation and the Fred Helme equation.
So there are a lot that could go into this, and there are deep links with this really great paper, right by David.
Shampooed on also showed that this has a link to the kind of renewal equations that I'll be talking about that arise from branching processes.
After that, the great man himself, Richard Bellman and and Theodore Harris started studying what are called age dependent branching processes.
And one of the things and this is actually what started this on this entire journey quite a long time back.
One of the things we realised in the really remarkably short paper published in 1948,
they have this line that says standard probabilistic arguments yield the same nonlinear integral equation.
And when you look at this, this looks remarkably like a renewal equation that I will get to in a bit.
But one of the most problematic things is the argument that Richard Harris uses that this can easily be found from standard probabilistic arguments.
I know from working with very competent mathematicians now that it's it really is not as trivial as that single line says.
And of course, Theodore Harris expounded this on this a lot more in his book on budget processes further down the line.
So now we've looked at, say, all models do the very popular in modelling infectious diseases writing processes,
which we're going to go and come back to because that's essentially what these talks about.
There are a couple of other really interesting facets that I want to go through just to set the stage for interesting approaches.
The other one is Hoekstra's, which is now what processes introduced by Helen Allen.
Hoax in 1971 are a type of self exacting process through point process where events occur and events can trigger other events to occur,
with some given right. And we have used these processes in modelling malaria elimination in the past, and octopuses are very nice,
and I think they are found to be used more and more infectious disease modelling. But they also have some problems.
And then, of course, there are natural processes now. One of the most, it's very interesting how these worlds tend to stand very separate.
One of the most influential papers in network theory was this paper by by David campaign on maximising the spread of influence,
and he introduced a structure called the independent cascade, and the independent cascade mill is remarkably like susceptible infected.
The beautiful thing about this paper was it it showed that using a greedy algorithm,
you can solve an empty heart problem in network theory with some guarantees and really, really influence a problem.
But this idea of using networks was also used by, well, even to this in their paper in trying to estimate who infected who.
Now we've also used these in our work to estimate reproduction numbers in especially very.
But this network view is it's an entirely different world, but that does have, you know, some some people who really like it in epidemiology.
But a huge amount of deep mathematics that does connect with the rest and the benefit
of network microprocessors is that they can really disentangle who infected,
who quite well in a way that other approaches can't.
And then finally, they are agent based models. And of course, for those will use this, it's hard to avoid.
Daniel Gillespie's paper on the exact simulation of reactions and that simply use often and often
in infectious disease modelling because lots of infectious disease epidemiologists know the
disease extremely well and know the mechanisms of the disease very well and just create a population
of individuals and allow them to evolve by setting rules as as shown here for tuberculosis.
And such models have, you know, some some extremely influential applications.
For example, some of Crystal and Neal's work, along with other colleagues on reporting on arguably one of the most influential papers on COVID
19 mortality rate at the start of the pandemic was built entirely from agent based models.
And of course, they are comes with using agent based models that they can be very difficult to fit and very difficult to
unpick the underlying dynamics of them because there can be a lot of redundancy of cooling parameters,
et cetera. And then finally, we get two renewal questions which were introduced at least of the world of infectious disease,
probably a long time ago, but most by Christof seminal work in PLoS One,
where he derived the renewal equation that we love, and then Ann Curry and colleagues used this in a framework to estimate 200 reproduction numbers.
Now, renewal equations, the allure of them is is pretty great because they really intuitive.
And if you read crystal paper, you'll see that it makes a lot of sense when they arise,
you know, from from a humanistic, from an epidemiological basis. And I'm just putting the renewed equation there for all to see.
You see I of T, the infected number of individuals can be distributed as they are.
Plus a random variable will just say the mean of it by the reproduction number,
which you touched upon the rate of transmission changing over time and all the
previous infected individuals multiplied by some variable g of you extra back there,
unfortunately. Now of you is known as the generation time and it it.
It's something that actually is quite difficult to have precise meaning in epidemiology
because it's defined in what you will find later that from the work that we've done,
we give you an extremely precise. Well, the generation time is generally the time between infection between and in fact, infected pair.
That's generally what it's called.
And so in world, the renewal equation can generally be thought of as previous infections cause current infections depending on the stage of infection.
It's very loose now.
Renewal equations are, in my view, extremely powerful, and that's why I think they've been used so ubiquitously and are increasingly being used.
So I wanted to go through two to explain all the old or the previous approaches to set the stage for one renewal of criticism for useful.
You know, I almost require solving a series of differential equations,
and solving differential equations can be quite problematic when we're doing things numerically,
and they never available until form filled with stochastic differential equations are even more complicated.
Network models are really nice, except you need to use a lot of assumptions about what's going on in the network.
Agent based models that I've said are exceedingly complicated to do inference on very, very difficult and looks.
Processes tend to be really, really nasty.
To simulate on with renewal equations tend to be very easy to embed within probabilistic programming language.
It's very easy to optimise and very easy to compute compute, right?
It's just a thumb there with with with quadratic complexity.
And so we have renewed equations to allow us to embed this idea of complexity within a mechanism.
The renewal equation gives us some mechanism and we feed into this mechanism complexity to help explain epidemiology.
Now, an example is some of the work that we've done on understanding the effectiveness of governmental
interventions first in our initial paper and then subsequently in more work where we try to understand,
you know, what interventions actually work to control the spread of SARS-CoV-2 and the disease COVID 19?
One of them, or probably the most important question for a large period of this pandemic.
And to do that, you know, there are many approaches that you can be taken, but by having a mechanism,
we can we can account for the lags between infection and cases and infections of death.
We can account for a mechanism that is plausible to explain the epidemiology of the renewal equation.
And then we can embed some complexity in there in terms of putting in stochastic processes,
doing our standard regression of terms with sensible priors.
And we can put all this within state of the art probabilistic programming languages such that we
have an entire framework from start to finish that can answer extremely important questions.
And that's what you see on the figure of the effort. And it's hard to to not overemphasise this because, you know,
the benefit of doing this kind of working in probabilistic programming languages
is not simply a model can be extremely interesting and extremely relevant,
but it can be extremely inconvenient to actually actually implement and and these models tend to be very useful.
If we knew equation for the opposite is very easy to implement and you can do lots of sensitivity analysis,
and that's not what you see in this blocks, you know, sensitivity analysis,
the previous one prediction to the future and really allow you to understand the problem,
allow you to summarise the problem and communicate this to decision makers in a way that they can.
They can really understand. The other benefit of showing when you want equations.
Being very powerful is that they can test epidemiological hypotheses by this.
I mean, some of the work as an example that we did in Brazil, you know,
there was a complex situation in the city of Manaus that had a huge wave that you'd seen in proxy the the yellow line that.
And then they have a second wave. Now the question was, why did this second wave occur?
Well, a new variant turned up, so we got some information from Apollo genetics about when that happened.
But we really needed some sort of model with plausible epidemiological characteristics that could account for things like waning of immunity,
you know, transmissibility, escape, which one of these hypotheses,
how did these hypotheses linked together to really understand the situation on the ground
and and renewed the questions allowed us to to have a fundamental mechanism to think we
could tweak and used to test epidemiological hypotheses without having a mechanism in
that arguably all you're doing is fitting to some data and predicting part of something.
You're not really understanding the underlying mechanism that. And finally, some work done done by my colleagues at Imperial.
You know, you can use renewal equations as well to embed other really interesting and
complicated characteristics such as mobility to understand contact matrix patterns.
And my colleague colleagues is doing a lot of work on this extended before.
So the reason to do is to show these is not simply to highlight some papers in the work that we've done, but we're happy with.
But but more to the point that the unifying factor behind all of these applications is the renewal rate.
And the reason we stuck with the renewal equation is that it really embeds quite a lot of epidemiology in terms of the infection process.
And so now that, you know, from all the approaches that I've gone in,
I've explained to you and you know, I'm going to show some exciting aspects in the next slide.
You know what, if when you do well, if you want to learn about renewal theory, really,
the first and probably best paper on the topic was the work by fellow in 1941,
when he looked at many aspects of the renewal and the renewal equation for
those in other fields is also known as the military question of the second of.
Basically, many integral equations with similar forms. And the question is how do we link this equation to infectious diseases?
Well, previous research on the renewal equation tends to study robotic behaviour.
That is how the renewal equation changes or what it averages to in the limit of time going to infinity.
Or they look at single parameters, pull the multiuser employment now setting the stage.
I've said that what we're really interested in is the rate of transmission, not just some growth parameter or some some estimate.
And in truth, and I really think this is a really fruitful area of research.
There are lots and lots of exciting links in renewing equations.
The hawks process that I mentioned is because it turns out to be the expectation of the renewal of fantastic paper written by by and Andrew Hawks.
Processes, in turn, are an auto regressive, infinite process on integers by Matthias Curtin in 2018.
And when you look at the renewal equation, you can see that in this discrete sense, it looks remarkably like an auto regressive one.
We did an auto move with a very specific set of coefficient determined by the.
The generation tops these 50, there's something to one, but then Hawks processes another, great people by result are awesome if they are models.
And like I said, the equation bye bye bye, baby champion that actually CIO models have a specific renewable equation for.
And recently,
I've been thinking about the Stochastic Volturi equation of being another big into this world to connect renewal equations to brunching process.
So I procrastinate better because it's just something I'm interested in recently.
So these are all kind of the same things,
and that's that's quite cool because they all they're all encompassing the same idea that we have where one thing affects another.
And at its core, when you reduce down and remove all the mathematics at its core,
this is actually what all these approaches are doing from an intuitive level.
They are reflecting some degree of self reference and some degree of one person infecting another.
So after that, a brief history or rather long history, what is the question and the question is the any the question that you see there,
which is the equation that Christopher and Corey have used in the continuous sense
that the script and there is there a mathematical principle source for this equation.
But that is not to say that their work was not mathematically principle. What I mean is, can we?
Well, I'll tell you what I mean by that. Can we connect the stochastic process to this one?
So the idea that we had was we first started with looking at at Belmont's paper and we derive this.
It was reported that the Dresdner barrier in maps initially started doing.
And the idea was, you know, how do we even just understand the government responses?
And then as we moved on from that, we started thinking, Well, ANA, if the building process is an independent budget process,
why don't we just pick the most general branching process we have and see what that means?
And so most general budget process, at least to my knowledge, is what's called the general branching process or the Trump mode, Yeager's process.
And I'm going to tell you what this process is, but basically, I'm just going to say it now.
Once a couple of gigas processes the branching process where we start with one individual after a random amount of time,
that individual can give rise to new individuals throughout their entire lifetime, and the number of individuals they give rise to is also random.
So it's sort of a very general process, very akin to an individual based model.
And that's why I particularly liked it. Because when using brunching process,
we're essentially connecting the world of governing equations like the plastic infected recovered equations to the world of of agent based models,
right by by defining the stochastic processes based on individuals.
And I think I'm going to shoot towards the end that this is actually really important. It's something that lots are missing.
And you know, you could always ask the question why a stochastic process, anyway?
And I think this was best encompassed by a quote by Peter Yeager's, a plea for stochastic dynamics.
I mean, I guess the whateley in there already showed how much he believed in this.
You know, it is argued that biological populations are finite and consisting of individuals with a varying lifespan and reproduction,
and they should be model as such. Now what he writes here that you know, this is what underlies biological processes, at least in infections.
Epidemics are comprised of individuals with varying infection durations and varying amounts of who they infect, and they should be model as such.
And the key is the modern probability theory allows for this.
And so that's what we wanted to do. And and it goes way beyond just this philosophy of science, of why use stochastic process to to to do you know,
why stop with the stochastic process and see where that leads us,
even though I've already given you the spoiler that that it leads you to the renewal equation of kristoffersson and lockdown, et cetera.
But it is, you know, the stochastic process tells you a lot of things that we'll see further down the line.
If you simulate from an independent budget process and Harris proxies, you see the simulations in the right plot as the black lines.
And we're interested in completing these the green line of the red doubling, which is the mean the multicolour expectation of this diagnostic process.
Now, the underlying stochastic process has so many interesting aspects that we tend to lose by if we just look at the mean,
if we just look at compartments, we look at the level of profound overexpression when simulating from the right.
You're right, it really lets you know that you know, when you compute communicating risk of infectious diseases,
just looking at the mean probably isn't going to cut it. And this is something that I will touch on at the end.
And so I'm going to breeze past this because I don't want to spend too much time on this and also the brand alone for too long on this lecture.
But I'm going to just go through a little bit about point processes and the more creative process just to give you a flavour of of,
you know, what we do and how these things are computable and derivable from a mathematical sense.
The first let me give you the intuition of the branching process. I'll give two examples.
First, let me go to Belman Harris Brunch, Billman, Harris going to process that one individual after a random amount of time,
that individual will give rise to a random number of new individuals.
So in this case, the orange individual will give rise to three blue individuals after some random part.
Now, the random time here is very well connected to the generation time, and it has a very precise meaning in this sense.
But you could also consider a more complicated model based on a homogeneous point process where you start with your orange individual.
This orange individual remains infectious for a certain amount of time.
You know, they get infected. They they recover from that unless, let's assume.
But they can infect people all the way to infinity.
They go over their duration of infection based on how infectious they are and all of this process of infection.
They will generate new infections from in homogeneous point process with a very rigid transmission.
Now, these branching processes give us a very simple way to to to to sort of model very plausible processes.
What I'm telling you now lays bare all the assumptions behind it.
You know, it's not like I've given you some simple model and you you're looking at the compartments and you wondered,
but I guess the compartments, but how the dynamics between them, you know, it's much needed.
This is why people like agent based models, because it's very easy to understand the assumptions, the precise assumptions.
And of course, we can derive two very complicated models funded.
So using our branching process, can we actually understand these two scenarios easily?
And so let's just go through and I'll go through this because I know it's really dry and I'll go through it reasonably quickly so we can move on.
We start with the crumpled Yeager's process, which the general budget does stuff with one individual at some time.
I said this before that individual state infectious for a certain amount of time.
That's that this l parameter, l l l variable, and then we have X, which is a stochastic process called a random characters.
This is beautiful and collaborative. Mika has done all the heavy lifting on this aspect as well.
When he first introduced this, I thought it was so beautiful and intuitive to make to make life easy.
Because when we first started doing this from the derivation of of of of Harris and Belman,
it's really messy and complicated, and we have to go into major theory.
And then suddenly, you know, when looking at point processes, you realise just how powerful point processes are in simplifying it.
And then we have end, which is the counting process, which is highly intuitive.
It's just keeping track of the number of new infections generated by the individual.
So we have the right amount of time, a counting process and a stochastic process called the random characteristic.
And so for each individual, we have these three three bits.
So for example, now we can write down the moment Harris process, which is this equation the first equation there, which is extremely simple.
We say that, you know, for some, the number of infected individuals at some time you is zero until sometime elsewhere.
They give rise to within you infected. And this is exactly what I'd said there.
In fact, individual new fatality and then generate new ones. The point, of course, is very simple.
We just have a unit, a unit unit problem process with some transmission rate roll from infectiousness K.
And then the integral of that, as you do generally within your own body, is pretty process.
So it suddenly becomes very easy to define these characteristics.
Similarly, the random characteristics, which you know, it becomes very easy to define incidence and prevalence rate as incidence,
that cumulative incidence and prevalence, of course,
is just the number of individuals between two time points, and that's exactly what it's showing there.
It's been very easy to include the measures that we're interested in.
So I'm just going to go in some derivation and I'm going to zoom past this because I don't bore everyone.
You know, we define some, some across our random characteristics. We single out the index case and this is, of course,
the most important and really the most beautiful aspect of Belmont came about and how it came about,
which is the first generation separating out the index case from all other cases.
And the intuition behind it is that doing this, the first generation we can create some degree of self-protection,
some self similarity in terms of what the fifth generation thought. And so we then split our our we can look at the second part of first generation
and we can we can split that into each one of the first generations there.
And that's what we have. Kate, as part of this set, I have Kate.
And then, depending on the statistic of choice, we can use a tower property of expectations to create an expected value.
Remember, we start with this stochastic process, right, which is generating realisations,
and what we're interested in is the average and as we'll see later, the second higher or the moment I'm almost done, we make things look easier.
Change that sum into into an interval.
And really as simple as that, because just algebraic operations, even though they look, they may look complicated.
Now, you know, they're actually not. They're very straightforward.
There's nothing there's nothing really, really complicated about this other than having to drive it in first place.
But you know, this would make a huge difference for. And we get to renew the question.
And this is this is the first point where, you know, we realise this with the bellman Harris courses with a complicated process.
It's wonderful because after all of this, we have a branching process that tells us and we can imbue dynamics of individuals and we calculate the
average of that budget process for a measure that we're interested in for a specific counting process.
And what we get is a renewed equation, a very general equation.
Now this equation looks nothing like the equation of Christoph and course, it's far, far more general.
But. You know, to get this sorry for all the months,
but to get to the point what this equation does is it allows us for the first
time to actually have a time varying reproduction number for general brunching,
just so we can now use this framework, which I think, you know, although the MAX looks complicated, I think, you know,
if you if you if an individual wants to put time now, they can go and they can say, well, give my individual branching process.
I want to, you know, tweak it in certain ways to have it behave in certain dynamics.
Can I then derive an unexpected formula from that? Or can I use that renewal equation to do that?
That's that's exactly what what we did in our processes.
And so we should really question for the first time that you can do many new and interesting aspects.
Individuals can infect an individual after some random time, and the number can vary.
They can it at random times, a person process not leaving, but in principle, you can look at compound based on processes.
And that would be interesting. And the number they can infect changes over time.
So this is the really new contribution. At least, you know, it seems very esoteric, but from from our viewpoint,
it's connecting the world of branching processes to the world of renewal equations, which already have a huge amount of epidemiological basis.
But doing it in such a way that it's stuff becoming extremely customisable to to to change it.
And so an example of that customisable image, we can actually create a renewable equation for the Belmont harvest process or the homogeneous process.
And what is really interesting is that these two epidemiologically are almost the same.
So we found it really confusing at first when you know the equation that everyone uses where, you know, previous infections affect future infections.
And that equation explicitly abutment Harris equation model.
We found this assumption that all the infections have to happen at one time.
This idea of an instantaneous are we thought that that was a very poor approach to modelling infectious diseases.
And so why do the renewal equations?
Well, it turns out that the, you know, genius point processes is, for all intents and purposes, practically the same epidemiologically.
And so you can interpret the renewal equation either through a bellman harris view or through an homogeneous point of view.
And this is well to me, mind blowing, but maybe, you know, new to us.
And in fact, you know, after much struggling, I really struggled with this because it took so long made me and my postdoc at the time, Thomas Mallon.
We proved that via induction, at least in the discrete case,
that the common renewed equation that Christoph Anon and others introduced is actually a very special case of our.
But what we do is we provide renewal equations for cumulative incidence incidents and most importantly, prejudice.
To my knowledge. Up to now, no one has a renewed equation for presidents.
And everyone uses this what's called a back calculation approach where you could evolve incidents with the generation interval to get pregnant.
This requires you to have latent functions and processes. It's not elegant, to say the least, but also not practically very nice.
What we do is we unify prevalence and incidence under the back calculation approach now.
This is one aspect which I truly understand. And also you'll have to ask me about the specifics of this proof.
But essentially, we prove that and I say proven elusive the system.
There's still some more bits on this that that need to be done in terms of full rigour.
But we prove that our relationship with prevalence and incidence in this new found renewal
equations are they are exactly they conform exactly to what we know from epidemiology.
And this is a really, you know,
it might be a beautiful thing that you derive all this complicated maths and the complicated maths reflects what people already know in epidemiology.
Be sort of doing informal mathematics. What is known heuristic it to be true in the field of epidemiology.
So we show that not only can we give you, can I give you a renewed equation for prevalence of incidence,
but I can show you that these two equations are consistent under the common definition that unify prevalence in this century,
we provide the first framework that can that can unify Palestinians.
Now, starting from the basis of Christof or Brown's equation, you can't immediately go into writing because this is we have tried this in the past.
This is really difficult, and it's because you need to do this from the underlying stochastic process, and it's much easier to do it that way.
Starting from very simple principles and building up, then starting from the end and trying to do that.
Now, all of this is pointless. If it's not easy to code and it's trivial to go, this is solve the entire renewal equation for prevalent in this block.
This block requires nothing but element wise multiplications, and Rosso's now in modern statistical computers on GPUs.
These two computations element wise multiplication and growth are exceedingly fast,
so fast such that the bottleneck is actually starting from the underlying posterior distribution.
We're optimising it across some non-contact space that actually takes more time than actually solving the equations themselves.
And so although our equations are slightly more complicated than the real equations
probably use because they are far more general and you can do a lot more with them. They actually don't take that much time to solve.
And in fact, I've been playing around a lot with Julia recently because I really like Julia now, and you can just you can write this even simpler.
Julia, this is a series of moves. So who that was a lot of content we arrive.
What have we learnt? We've arrived at the commonly used equation in a principle that was outgoing.
Is it groundbreaking in terms of changing the face of epidemiology? No, obviously not.
But, you know, we show where this equation arrived for the principal mathematical way from osteopathic priests.
And I think the most important thing is we connect the world of agent based models to govern immigration.
So that people can build on, we put these two equations equivalent,
i.e. the renewal equation currently being used in the special case of a more general emergency.
We unified pretty privileged nations and we provide an efficient computational scheme.
So applications, you know, the most obvious one is go, you know, analyse the grid.
And this is a great thing that I'm going to release all the data in, too, so that everyone can analyse it.
We can go and model. It was using these equations in a full Bayesian framework really easily.
And we get exactly what you'd expect using stochastic processes. The benefit is that we don't have to use any arbitrary assumptions behind,
you know, the functional form of our team here when you try to move processes. But you could do the governing process.
You could use piece planes, which are basically random walks, which are basically consequences you can use.
You can use any functional form you want in there to estimate it.
So our renewal equations can do what the previous equations did.
And you know, that's not new, but I'm just telling you that you haven't lost anything here, but you can do new things.
And this is a recent example that I did before presidents. So this is the U.K. owner's infection survey.
And in the plot on the on the top right, you see the prevalence,
the population prevalence over time and you can see the access that this is
just the percentage of individuals that are testing positive for SARS-CoV-2,
the number that have COVID 19 at any given point.
And we want to estimate the reproduction number. If you want to do this, you have to use some form of that calculation, which is very difficult,
and I haven't been able to find how the, you know, the owners do this.
I need to speak to Thomas Health about this at some point.
But I guarantee it's not as simple as what I'm doing right here, where I have a renewal equation that I can fit,
for instance, that is the same with you with the question is this one? I just added an extra term on that and solve it in the exact same way.
I again even have to modify the code,
and I could get a very good estimate of the case reproduction number to validate this using the same renewal equation without doing any extra fitting.
I get incidents straight away and I and that's what the bottom left plot to show you with the incident and the blue bars and the actual cases,
and the red is the fitted incidence.
I look at that and I see a line of about seven days, which is completely in line with what you'd expect when I get infected,
it takes about seven days until that to manifest of reported case. You look at them from that, you can calculate the ascertainment ratio,
which is around 2.5 and is remarkably stable, apart from weekly fluctuations and from different places.
You know, if we could run the tape of the pandemic again, I have no doubt that such a framework would be extremely useful.
Maybe I'm being good to highlighting. I would do much,
but I think would be extremely useful for something like react to audio in this setting because we can now have a renewal
question for a president that links back to a budget process that has epidemiological mechanisms that we can then build on.
As me and my colleagues have done on several different applications in the past, and I think this is really powerful.
But in the last few bits, I'm just going to talk about an application for the variance.
Now this is this is really one of the cool things you can get with these renewal equations are generating funds with generating functions.
You can get all the higher order moments for not just the mean, but the very. And there is a really important question of superspreading, right?
You know, superspreading has been there all the time in terms of, you know,
not the people in that superspreading, this big thing, one person, in fact. How does superspreading actually arise from it?
You know, closet and co. have also set the scale free networks of rice.
Where does this scale free of heavy tailed behaviour in budget processes actually come from, right?
Is it only from the secondary distribution,
i.e. the reason that these these branching processes in epidemics have really heavy tails is because I might go to a festival and in fact,
100 people is not the only dynamic at play. And no, actually.
And I thought of this a lot in the view of the central limit theorem that that superspreading is really not a big deal,
but I was using the new kind of central libertarian, which does not account for weak dependence.
And I realised after that that I was entirely wrong because there is actually no central limit for these budget processes because
the dependence on the time and you can see the effect of superspreading when you simulate from these branching processes,
look at how profound the over dispersion is there. And let's take a simple experiment to look at this in the top left corner.
We have the reproduction number at the rate of transmission oscillating between growth, you know, reduction growth reduction growth reduction.
In in the top right, you have the rather intuitive mean presidents that have a large first wave, a smaller second wave, smaller, smaller and smaller.
What do you see in the bottom left is really fascinating.
Is the variance now the variance in the second wave, despite the mean being smaller, is much, much bigger.
And this is, in my view,
a little bit unintuitive because we're used to thinking of profound likelihoods or likelihoods where the variance is some function of the mean.
And in truth, the variance is huge and you know, each subsequent wave, the variance grows in uncertainty.
You know, even the you in the third wave is quite quite small, but the variance is still one of the biggest at the variance in the first wave.
What is going on here, right? This is a dynamic that I haven't seen, personally discussed or highlighted much in the literature,
but you can see it in the simulations where superspreading is emerging and you can see other dynamics from this,
which all of us in infectious diseases know about extinction. Yet we rarely integrate these into our modelling framework, right?
You can model using these renewal equations in our framework. The precise expected extinction probability and you know,
you can change these from having a possibly secondary distribution of negative binomial secondary dispersion.
And so in plot eight, you can see the index of dispersion, which is huge, right?
It's huge. You know, it's massive. And then you can see the extinction probabilities which again conform to what we're seeing as time goes by.
If there's no new importation event, some epidemics are just going to burn out, especially when the Arctic is less than zero.
And that's what that extinction probability is very short.
And so finally, in the final slide, what I'm really interested in looking at now is what is the appropriate likelihood, right?
And let's just take our example,
if we simulated the planting process in and we looked at the underlying distribution at those five points that, well, the histogram,
we see that the blue blueberries, what you would get with a profound likelihood and the possible likelihood is telling you,
well, my main problem at that point two is about 60. So let me put a bar at 60 and have some discussion on that.
But when you look at the underlying distribution that simulated from an individual based model,
the branching process, the dynamics are very different. Long story short, what the dynamics tell you is that.
Actually, what happens is many epidemics go extinct and some blow up really large towns.
And this is true even when your secondary distribution is forced on you.
Very finite, it doesn't have a habitat. This idea of heavy handedness emerges is an emergent property of branching processes, right?
Getting superspreading, you don't need individuals to be super spreaders, the brunching process itself.
Has high variance in high teens. And so I am now conjecturing and start to wonder whether using a negative binomial likelihood or
unlikelihood alone is actually extremely inappropriate for capturing the dynamic that we know to be true.
I actually don't know if it'll make a difference. I suspect it won't. Unfortunately, isn't, as in the case of many of these things.
You know, sometimes an approximation is often good enough,
at least for epidemiological purposes doesn't actually change the underlying estimation, but I think it's extremely interesting.
And so the questions for the audience of those who have health is, you know, I have these underlying distribution here.
How do I get a parametric choice for maximum entropy seems good, but I have other couple of techniques up my sleeve.
But you know, what's the best solution to the moment?
Probability generating distribution from a BGF or any approximation forms can result the renewal equation even more
efficiently than just using some to where you transformed in all that last time from stuff doesn't make things much faster.
So where do we go next from here? I want to do more unification.
I want to unify all these approaches.
I think they are all essentially the same thing could benefit from more and more linkage together to understand the underlying roots behind.
I think that what we've described here is a perfect fit for genetics as opposed to using a general model.
Why not use an age dependent Bacik process with a fixed splitting of two and then estimating a time varying generation time,
which would be a replacement for Scotland blocks? Simple to simulate from.
Simple do inference from and you can then link it straight to epidemic's using the same equations.
More efficient ways of competition exist.
I'd like to embed these processes on graphs so we can measure the worth of networks to these well, and the question of immigration is huge.
Bringing new infections from outside is very important and requires proper treatment.
That's challenging, so I want to say thanks.
And first and foremost, amigo who really did the heavy lifting of the max on this election has been interacting with me throughout all of this.
He's really lovely to work with and brilliant, actually brilliant.
And to Thomas Thomas Mellon and senior moderator who helped with all aspects of this in three or four draughts.
Transdermal drug the work in the intermediate paper Charlie helped me later on in subcommittee meetings came up with this idea in the first place,
long, long time ago. There is no project with amazing collaborators,
and I hope I hope it's useful for the epidemiological community rather than just a mathematical curiosity.
Thanks. All right.
We have 10 minutes for questions, please write them in the chat or raise your hand.
And Chris, you're. Thanks.
Great talk. You met you said there was a difficulty with relating the standard generic equation of epidemiology to prevalence.
If we're talking numerically and not just not analytically,
is it not just a question of involving the past incidence with the probability of contributing to the prevalence at a given moment,
whether by being PCR positive or seropositive or whatever? It is exactly that, except you,
you need to first create a latent function for incidents and then involve that and then put that into a likelihood for governments right now.
It probably has the same impact in terms of posterior computation.
It's just that when you type one thing into another thing it sometimes can be can cause lots of issues with some people from a difficult question.
Whereas here it's just the equivalent of having a renewed, aggressive Bridgerton's. You don't need a latent function first and then controlled.
I mean, that's already in there anyway. It would be interesting to test and see how bad the posterior teeth are.
I mean, you know, nothing we've put in here hasn't already had a solution for the epidemiology and lot of political epidemiology.
The question is, you know, is it better? In some ways, I think it's more intuitive and more succinct.
Is it better? I think it's better from a computational perspective, but that remains to be.
The fact that we tested it probably changes on on different and I don't know how the noise propagate, propagating them.
There's a lot of things here.
If I want to fit to react to the cases, I can trivially have to likelihood that that really cost me very little, much more,
whereas it can sometimes get a little bit more difficult when you have to do one consulting to the other in principle.
Same thing, you. Krista, yeah, thanks.
Great talk. I have two questions really about Stochastic City when we submit a stochastic process sort of to to to tune the renewal equation
estimates as we tend to switch between Belmont Harris and then homogeneous crescent representation more or less at random.
You mentioned that you've shown that they behave the same and then the men feel, do they behave the same stochastic play?
So that's the first question. And then once we got over dispersion, we tend to do that in Belmont Harris because it's easier now.
But we've never found a good way to estimate the amount of over dispersion from the time source.
So we always end up looking for other data sources. Do you think that's that's even possible?
So as we enter in the degree of statistical?
Yeah, I mean, the same is the same in the mean, but they are not they're not the same in the individual which processes.
But, you know, at the end of the day, I think it only changes the degree the variance, you know, changes.
But in terms of the mean, they're the same. But in terms of the simulating, yeah, they are different.
I think it depends what you what you're going to summarise from them after that.
That makes it in terms of the over this question, I'm convinced that the over dispassionate as you'd expect, is.
The benefit of using generating functions is actually I can provide a renewed equation for
the DeFi of the of the negative binomial that requires very little additional computation.
So I can provide you in close for what the variance should be from theory rather than you just estimating one fine for all of your data, right?
You get them incorrect by using linear equation you put in a fire.
And the problem is that that fire is not inflationary, but it should be not stationary, given what we know from individual simulations.
I could provide you with that equation, of course, for.
It's still not perfect because we have an accounting for the zero inflation, and I can provide you an increasingly platform for zero inflation,
but then fiddling with a zero inflated negative binomial and solving equations
algebraically I've been struggling with to to get it without using non-linear solving.
But essentially, I think we can provide you with with the higher order moments trivially.
And do you think it's identifiable? We've always struggled. No, I don't think it's I think it's because we only have one academic.
The truth is, it's completely not identifiable. You have to trust that the that that the assumptions of your stochastic process represent reality
reasonably well and then say that although I can identify what the variance is from observational data,
I think from theory it should be this. And so but I think it'd be completely identifiable.
I mean, you know, you'd need we need to rerun the tape of time on academics many times over actually get an estimate of the variance.
Otherwise, we have confounding and all these other problems.
But I think you know, what I want to do is and I'm going to do is fit the prevalence data from the over this,
you think, a more appropriate likelihood in this one.
And what I think we'll find is that the variance as the second wave goes up, sometimes you see artists spike, right?
If you randomly Archie goes up to six or seven, just because the small sample statistics, because we have got the variance correct.
I don't think the mean will need to do that and it'll keep Archie a little bit more stable.
This could be very useful for policy.
I think, you know, otherwise in the news you get, artist is six, but there's only 10 cases or something, you know?
And then it course. A big part of the reason that this exactly happened is because of this.
And. Question from Leonid in the chat.
Sam, you wanted to read this question earlier on this question.
I'm wondering if there would be a straightforward way to represent the secondary attack rate in the framework you presented.
Yeah, I mean, there is it is in a very it's baked in that right.
We have our and that's always the mean that we get in.
But you can always assume in these equations of distribution for the secondary, the secondary infections, if I can live to regret,
for example, I assume the secondary infection to be, you know, plus negative binomial discrete power law.
And based on all of these, as long as I can calculate second moment, I can get the variance until four.
Until I can, just from what I was telling Bristol,
I can I can actually provide a cure for immigration for the very it's not that's based on me assuming I've got the underlying facts,
the policies correct. But it's definitely better than just having a fixed variant for process.
We know it's more efficient. So neither is perfect. But I think if I did, did you?
A secondary attack rate, it means the percentage of people who are in some setting that are getting infected from the from the index case.
Yeah. Well, Leonid, you want to clarify.
Yeah, that's exactly what what's said. So but the very to you mean, what is the prevalence from an index case?
There's a time component as well. So if it's within, within, within a generation interval, I guess, I mean, you know, yeah,
in principle, we can we can't do this by just choosing the random characteristics of matches.
But obviously, you'd have to write down the marks for it. OK.
Let me go to someone who hasn't gotten a chance to ask a question, so Francesco writes a question.
Yeah. Francisco goes home at variability. Do you get when you look at the total package or at the end of the way, it seems quite high.
Well, I mean, at the end of the day in this block over here, you know, the population prevalence is impacted by the attack rate.
You mean the number of total infected or the number affected each time it's clarified.
Yeah, the total number of infected. Thank you. Yeah, I mean, that's that's very good.
We have cumulative incidents. You can generate that very trivially from this exact equation.
So I can fit the president's get up incidents here and actually get total cumulative incidence as well and then divide that by the population.
It requires no extra compute or computation. So how much variability do I get?
I mean, it's sort of a difficult question to answer because all these equations are measuring the exact same process.
The amount of variability you get in terms of incidents will be the same as you get in previous things.
But they're all essentially dependent on a single RTU estimate,
a single gene from a single acting and single gene you can to individual incidence and prevalence.
Christophe, you still have your hand up, but maybe that's all, I hope that answered your question.
I just got the pizza, you know, jump in, if not. And you're still on mute first off, OK, you've taken it down.
So let's do one more follow up. What about the risks? Yeah, Chris asked about exact results for the offspring distribution.
Yeah, you can. Well, I mean. When you calculate the higher order moments for the winning equation, you need to assume an offspring.
This is this is something you have to do.
And yeah, as long as an offspring distribution has a second moment, then you can compete if it doesn't have a second moment.
Well, the variance can be essentially unbounded. OK, so we are officially at the end, so let's take a moment to once again thank Samir.
So if people want to unmute and clap, there will be a way to do it if they want to use emojis, but also fine.
And then I will just volunteer Sam to stick around and informally answer my questions with some more feedback.
But this officially closes it. Well, you know, thanks for inviting me, Dominic.
Please reach out if you want to call this a programme that's in our Python, Juliet.
I'm happy and I've written up our notebook to to to make it simple with some examples.
How do you share? Just want to?