Introduction
I have been wondering for a while how to characterise the difference in approaches to Coronavirus modelling of cases and deaths, between “curve-fitting” equations and the SIR differential equations approach I have been using (originally developed in Alex de Visscher’s paper this year, which included code and data for other countries such as Italy and Iran) which I have adapted for the UK.
Part of my uncertainty has its roots in being a very much lapsed mathematician, and part is because although I have used modelling tools before, and worked in some difficult area of mathematical physics, such as General Relativity and Cosmology, epidemiology is a new application area for me, with a wealth of practitioners and research history behind it.
Curve-fitting charts such as the Sigmoid and Gompertz curves, all members of a family of curves known as logistics or Richards functions, to the Coronavirus cases or deaths numbers as practised, notably, by Prof. Michael Levitt and his Stanford University team has had success in predicting the situation in China, and is being applied in other localities too.
Michael’s team have now worked out an efficient way of reducing the predictive aspect of the Gompertz function and its curves to a straight line predictor of reported data based on a version of the Gompertz function, a much more efficient use of computer time than some other approaches.
The SIR model approach, setting up an series of related differential equations (something I am more used to in other settings) that describe postulated mechanisms and rates of virus transmission in the human population (hence called “mechanistic” modelling), looks beneath the surface presentation of the epidemic cases and deaths numbers and time series charts, to model the growth (or otherwise) of the epidemic based on postulated characteristics of viral transmission and behaviour.
Research literature
In researching the literature, I have become familiar with some names that crop up or frequently in this area over the years.
Focusing on some familiar and frequently recurring names, rather than more recent practitioners, might lead me to fall into “The Trouble with Physics” trap (the tendency, highlighted by Lee Smolin in his book of that name, exhibited by some University professors to recruit research staff (“in their own image”) who are working in the mainstream, rather than outliers whose work might be seen as off-the-wall, and less worthy in some sense.)
In this regard, Michael Levitt‘s new work in the curve-fitting approach to the Coronavirus problem might be seen by others who have been working in the field for a long time as on the periphery (despite his Nobel Prize in Computational Biology and Stanford University position as Professor of Structural Biology).
His results (broadly forecasting, very early on, using his curve-fitting methods (he has used Sigmoid curves before, prior to the current Gompertz curves), a much lower incidence of the virus going forward, successfully so in the case of China) are in direct contrast to that of some some teams working as advisers to Governments, who have, in some cases, all around the world, applied fairly severe lockdowns for a period of several months in most cases.
In particular the work of the Imperial College Covid response team, and also the London School of Hygiene and Tropical Medicine have been at the forefront of advice to the UK Government.
Some Governments have taken a different approach (Sweden stands out in Europe in this regard, for several reasons).
I am keen to understand the differences, or otherwise, in such approaches.
Twitter and publishing
Michael chooses to publish his work on Twitter (owing to a glitch (at least for a time) with his Stanford University laboratory‘s own publishing process. There are many useful links there to his work.
My own succession of blog posts (all more narrowly focused on the UK) have been automatically published to Twitter (a setting I use in WordPress) and also, more actively, shared by me on my FaceBook page.
But I stopped using Twitter routinely a long while ago (after 8000+ posts) because, in my view, it is a limited communication medium (despite its reach), not allowing much room for nuanced posts. It attracts extremism at worst, conspiracy theorists to some extent, and, as with a lot of published media, many people who choose on a “confirmation bias” approach to read only what they think they might agree with.
One has only to look at the thread of responses to Michael’s Twitter links to his forecasting results and opinions to see examples of all kinds of Twitter users: some genuinely academic and/or thoughtful; some criticising the lack of published forecasting methods, despite frequent posts, although they have now appeared as a preprint here; many advising to watch out (often in extreme terms) for “big brother” government when governments ask or require their populations to take precautions of various kinds; and others simply handclapping, because they think that the message is that this all might go away without much action on their part, some of them actively calling for resistance even to some of the most trivial precautionary requests.
Preamble
One of the recent papers I have found useful in marshalling my thoughts on methodologies is this 2016 one by Gustavo Chowell, and it finally led me to calibrate the differences in principle between the SIR differential equation approach I have been using (but a 7-compartment model, not just three) and the curve-fitting approach.
I had been thinking of analogies to illustrate the differences (which I will come to later), but this 2016 Chowell paper, in particular, encapsulated the technical differences for me, and I summarise that below. The Sergio Alonso paper also covers this ground.
Categorization of modelling approaches
Gerard Chowell’s 2016 paper summarises modelling approaches as follows.
Phenomenological models
A dictionary definition – “Phenomenology is the philosophical study of observed unusual people or events as they appear without any further study or explanation.”
Chowell states that phenomenological approaches for modelling disease spread are particularly suitable when significant uncertainty clouds the epidemiology of an infectious disease, including the potential contribution of multiple transmission pathways.
In these situations, phenomenological models provide a starting point for generating early estimates of the transmission potential and generating short-term forecasts of epidemic trajectory and predictions of the final epidemic size.
Such methods include curve fitting, as used by Michael Levitt, where an equation (represented by a curve on a time-incidence graph (say) for the virus outbreak), with sufficient degrees of freedom, is used to replicate the shape of the observed data with the chosen equation and its parameters. Sigmoid and Gompertz functions (types of “logistics” or Richards functions) have been used for such fitting – they produce the familiar “S”-shaped curves we see for epidemics. The starting growth rate, the intermediate phase (with its inflection point) and the slowing down of the epidemic, all represented by that S-curve, can be fitted with the equation’s parametric choices (usually three or four).
A feature that some epidemic outbreaks share is that growth of the epidemic is not fully exponential, but is “sub-exponential” for a variety of reasons, and Chowell states that:
“Previous work has shown that sub-exponential growth dynamics was a common phenomenon across a range of pathogens, as illustrated by empirical data on the first 3-5 generations of epidemics of influenza, Ebola, foot-and-mouth disease, HIV/AIDS, plague, measles and smallpox.”
Choices of appropriate parameters for the fitting function can allow such sub-exponential behaviour to be reflected in the chosen function’s fit to the reported data, and it turns out that the Gompertz function is more suitable for this than the Sigmoid function, as Michael Levitt states in his recent paper.
Once a curve-fit to reported data to date is achieved, the curve can be used to make forecasts about future case numbers.
Mechanistic and statistical models
Chowell states that “several mechanisms have been put forward to explain the sub-exponential epidemic growth patterns evidenced from infectious disease outbreak data. These include spatially constrained contact structures shaped by the epidemiological characteristics of the disease (i.e., airborne vs. close contact transmission model), the rapid onset of population behavior changes, and the potential role of individual heterogeneity in susceptibility and infectivity.“
He goes on to say that “although attractive to provide a quantitative description of growth profiles, the generalized growth model (described earlier) is a phenomenological approach, and hence cannot be used to evaluate which of the proposed mechanisms might be responsible for the empirical patterns.
Explicit mechanisms can be incorporated into mathematical models for infectious disease transmission, however, and tested in a formal way. Identification and analysis of the impacts of these factors can lead ultimately to the development of more effective and targeted control strategies. Thus, although the phenomenological approaches above can tell us a lot about the nature of epidemic patterns early in an outbreak, when used in conjunction with well-posed mechanistic models, researchers can learn not only what the patterns are, but why they might be occurring.“
On the Imperial College team’s planning website, they state that their forecasting models (they have several for different purposes, for just these reasons I guess) fall variously into the “Mechanistic” and “Statistical” categories, as follows.
“Mechanistic model: Explicitly accounts for the underlying mechanisms of diseases transmission and attempt to identify the drivers of transmissibility. Rely on more assumptions about the disease dynamics.
“Statistical model: Do not explicitly model the mechanism of transmission. Infer trends in either transmissibility or deaths from patterns in the data. Rely on fewer assumptions about the disease dynamics.
“Mechanistic models can provide nuanced insights into severity and transmission but require specification of parameters – all of which have underlying uncertainty. Statistical models typically have fewer parameters. Uncertainty is therefore easier to propagate in these models. However, they cannot then inform questions about underlying mechanisms of spread and severity.“
So Imperial College’s “statistical” description matches more to Chowell’s description of a phenomenological approach, although may not involve curve-fitting per se.
The SIR modelling framework, employing differential equations to represent postulated relationships and transitions between Susceptible, Infected and Recovered parts of the population (at its most simple) falls into this Mechanistic model category.
Chowell makes the following useful remarks about SIR style models.
“The SIR model and derivatives is the framework of choice to capture population-level processes. The basic SIR model, like many other epidemiological models, begins with an assumption that individuals form a single large population and that they all mix randomly with one another. This assumption leads to early exponential growth dynamics in the absence of control interventions and susceptible depletion and greatly simplifies mathematical analysis (note, though, that other assumptions and models can also result in exponential growth).
The SIR model is often not a realistic representation of the human behavior driving an epidemic, however. Even in very large populations, individuals do not mix randomly with one another—they have more interactions with family members, friends, and coworkers than with people they do not know.
This issue becomes especially important when considering the spread of infectious diseases across a geographic space, because geographic separation inherently results in nonrandom interactions, with more frequent contact between individuals who are located near each other than between those who are further apart.
It is important to realize, however, that there are many other dimensions besides geographic space that lead to nonrandom interactions among individuals. For example, populations can be structured into age, ethnic, religious, kin, or risk groups. These dimensions are, however, aspects of some sort of space (e.g., behavioral, demographic, or social space), and they can almost always be modeled in similar fashion to geographic space“.
Here we begin to see the difference I was trying to identify between the curve-fitting approach and my forecasting method. At one level, one could argue that curve-fitting and SIR-type modelling amount to the same thing – choosing parameters that make the theorised data model fit the reported data.
But, whether it produces better or worse results, or with more work rather than less, SIR modelling seeks to understand and represent the underlying virus incubation period, infectivity, transmissibility, duration and related characteristics such as recovery and immunity (for how long, or not at all) – the why and how, not just the what.
The (nonlinear) differential equations are then solved numerically (rather than analytically with exact functions) and there does have to be some fitting to the initial known data for the outbreak (i.e. the history up to the point the forecast is being done) to calibrate the model with relevant infection rates, disease duration and recovery timescales (and death rates).
This makes it look similar in some ways to choosing appropriate parameters for any function (Sigmoid, Gompertz or General Logistics function (often three or four parameters)).
But the curve-fitting approach is reproducing an observed growth pattern (one might say top-down, or focused on outputs), whereas the SIR approach is setting virological and other behavioural parameters to seek to explain the way the epidemic behaves (bottom-up, or focused on inputs).
Metapopulation spatial models
Chowell makes reference to population-level models, formulations that are used for the vast majority of population based models that consider the spatial spread of human infectious diseases and that address important public health concerns rather than theoretical model behaviour. These are beyond my scope, but could potentially address concerns about indirect impacts of the Covid-19 pandemic.
a) Cross-coupled metapopulation models
These models, which have been used since the 1940s, do not model the process that brings individuals from different groups into contact with one another; rather, they incorporate a contact matrix that represents the strength or sum total of those contacts between groups only. This contact matrix is sometimes referred to as the WAIFW, or “who acquires infection from whom” matrix.
In the simplest cross-coupled models, the elements of this matrix represent both the influence of interactions between any two sub-populations and the risk of transmission as a consequence of those interactions; often, however, the transmission parameter is considered separately. An SIR style set of differential equations is used to model the nature, extent and rates of the interactions between sub-populations.
b) Mobility metapopulation models
These models incorporate into their structure a matrix to represent the interaction between different groups, but they are mechanistically oriented and do this by considering the actual process by which such interactions occur. Transmission of the pathogen occurs within sub-populations, but the composition of those sub-populations explicitly includes not only residents of the sub-population, but visitors from other groups.
One type of model uses a “gravity” approach for inter-population interactions, where contact rates are proportional to group size and inversely proportional to the distance between them.
Another type described by Chowell uses a “radiation” approach, which uses population data relating to home locations, and to job locations and characteristics, to theorise “travel to work” patterns, calculated using attractors that such job locations offer, influencing workers’ choices and resulting travel and contact patterns.
Transportation and mobile phone data can be used to populate such spatially oriented models. Again SIR-style differential equations are used to represent the assumptions in the model about between whom, and how the pandemic spreads.
Summary of model types
We see that there is a range of modelling methods, successively requiring more detailed data, but which seek increasingly to represent the mechanisms (hence “mechanistic” modelling) by which the virus might spread.
We can see the key difference between curve-fitting (what I called a surface level technique earlier) and the successively more complex models that seek to work from assumed underlying causations of infection spread.
An analogy (picking up on the word “surface” I have used here) might refer to explaining how waves in the sea behave. We are all aware that out at sea, wave behaviour is perceived more as a “swell”, somewhat long wavelength waves, sometimes of great height, compared with shorter, choppier wave behaviour closer to shore.
I’m not here talking about breaking waves – a whole separate theory is needed for those – René Thom‘s Catastrophe Theory – but continuous waves.
A curve fitting approach might well find a very good fit using trigonometric sine waves to represent the wavelength and height of the surface waves, even recognising that they can be encoded by depth of the ocean, but it would need an understanding of hydrodynamics, as described, for example, by Bernoulli’s Equation, to represent how and why the wavelength and wave height (and speed*) changes depending on the depth of the water (and some other characteristics).
(*PS remember that the water moves, pretty much, up and down, in an elliptical path for any fluid “particle”, not in the direction of travel of the observed (largely transverse) wave. The horizontal motion and speed of the wave is, in a sense, an illusion.)
Concluding comments
There is a range of modelling methods, successively requiring more detailed data, from phenomenological (statistical and curve-fitting) methods, to those which seek increasingly to represent the mechanisms (hence “mechanistic”) by which the virus might spread.
We see the difference between curve-fitting and the successively more complex models that build a model from assumed underlying interactions, and causations of infection spread between parts of the population.
I do intend to cover the mathematics of curve fitting, but wanted first to be sure that the context is clear, and how it relates to what I have done already.
Models requiring detailed data about travel patterns are beyond my scope, but it is as well to set into context what IS feasible.
Setting an understanding of curve-fitting into the context of my own modelling was a necessary first step. More will follow.
References
I have found several papers very helpful on comparing modelling methods, embracing the Gompertz (and other) curve-fitting approaches, including Michaels Levitt’s own recent June 30th one, which explains his methods quite clearly.
Gerard Chowell’s 2016 paper on Mathematical model types September 2016
The Coronavirus Chronologies – Michael Levitt, 13th March 2020
COVID-19 Virus Epidemiological Model Alex de Visscher, Concordia University, Quebec, 22nd March 2020
Empiric model for short-time prediction of Covid-19 spreading , Sergio Alonso et al, Spain, 19th May 2020
Universality in Covid-19 spread in view of the Gompertz function Akira Ohnishi et al, Kyoto University) 22nd June 2020
Predicting the trajectory of any Covid-19 epidemic from the best straight line – Michael Levitt et al 30th June 2020