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This is a blog article in progress, written by Jacob Biamonte. See also the network theory series. To see discussions of the article as it was being written, visit the Azimuth Forum. For the final polished version, go to the Azimuth Blog.
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For those of you following the network theory series, you already know we’ve been talking a lot about how process change with time. For example, how a virus might spread, or how a population of fish might depend on time. In most, though not all, of our posts we have assumed that “everything interacts with everything else” all of the time. What do I mean by this? Say you have a large population of rabbits, and a predator (a wolf), we typically assume that every rabbit in this population has an equal ‘opportunity’ to encounter the predator and be eaten. In other words, the predator is interacting with all of the rabbits all of the time. We know in real life this is not the case. The predator will only interact with a small number of rabbits at any given time: this depends heavily on spatial location.
Of course, throwing away spatial degrees of freedom is just one of many approximations that can be made. And as you’d expect, there are clearly many situations when these spatial degrees of freedom are not very relevant to a problem at hand. For example, the classic thought example is if you stir the contents of a test tube fast enough so that all of the contents could be considered as interacting all of the time. Such a situation is typically referred to as a ‘mean-field’ approximation.
Today we don’t want to talk about the mean-field approximation, since we’ve done a lot of that already. What we’ll do today is add physical location to our models. So far, we have not done this when considering stochastic Petri nets (or in the words of a chemist, reaction networks). We have however considered stochastic walks on graphs, for example in Part 16. A graph of course could be interpreted as depicting a physical situation, in which a system changes state on a 2D plane.
Today what we are going to do is study epidemics. We’ve done this before, through the lens of a commonly used model, called the SIR model (see Part 3). I hope you’ll agree that epidemics are no laughing matter, but that does not stop me from telling bad jokes.
In contrast to what we’ve done before, today we assume that the epidemic will spread in 1D space. The way I like to think about this is all about a bad pet shop. I like to think of a line of rabbits in cages. On any given day, say that most of the rabbits start out healthy but susceptible to catching a cold from an infected rabbit. If none of the rabbits in any of the cages are sick, then things go on just fine. However, if one of the rabbits is sick, then there is a chance that this rabbit could infect a neighbouring rabbit. And the epidemic could spread further through the chain of cages. Today we are going to talk more about this situation, in more detail and talk about some recent work which has proposed some methods of exact solution for just these exact sort of scenarios.
Stated in another way, in this article, we’re going to explain how a member of a bunny chain like this
can get sick like this
and then how stochastic mechanics takes over and infects the whole batch like so:
If you’re worried, please don’t, since they can recover:
Recently Blake Stacy pointed out the following recent preprint.
Abstract. We present an exact analytical solution to a one-dimensional model of the Susceptible-Infected-Recovered (SIR) epidemic type, with infection rates dependent on nearest-neighbor occupations. We use a quantum mechanical approach, transforming the master equation via a quantum spin operator formulation. We calculate exactly the time-dependent density of infected, recovered and susceptible populations for random initial conditions, and compare our results with a low connectivity SIR model reported by Schuetz et al.. Our results compare well to those of previous work, validating the model as a useful tool for additional and extended studies in this important area. Our model also provides exact solutions for the n-point correlation functions, and can be extended to more complex epidemic type models.
The paper [WMM11] has clear relevance to the types of things that have been talked about in the network theory series. As mentioned, the SIR model was considered by John Baez in the petri net field theory framework in Network Theory Part 3.
The purpose of this page is to cast this recent work [WMM11] into our framework, and see what we can learn. We will label our sections precisely as was done in [WMM11] and conduct our analysis of the paper sequentially.
In the introduction, the authors motivate the problem and cite the following reference.
Abstract. The susceptible-infectious-recovered (SIR) model describes the evolution of three species of individuals which are subject to an infection and recovery mechanism. A susceptible S can become infectious with an infection rate β by an infectious I type provided that both are in contact. The I type may recover with a rate γ and from then on stay immune. Due to the coupling between the different individuals, the model is nonlinear and out of equilibrium. We adopt a stochastic individual-based description where individuals are represented by nodes of a graph and contact is defined by the links of the graph. Mapping the underlying master equation onto a quantum formulation in terms of spin operators, the hierarchy of evolution equations can be solved exactly for arbitrary initial conditions on a linear chain. In the case of uncorrelated random initial conditions, the exact time evolution for all three individuals of the SIR model is given analytically. Depending on the initial conditions and reaction rates β and γ, the I population may increase initially before decaying to zero. Due to fluctuations, isolated regions of susceptible individuals evolve, and unlike in the standard mean-field SIR model, one observes a finite stationary distribution of the S type even for large population size. The exact results for the ensemble-averaged population size are compared with simulations for single realizations of the process and also with standard mean-field theory, which is expected to be valid on large fully connected graphs.
There is a related paper on arXiv by these authors, that seems to be the same paper, with a slightly different title (this one is free to download, for those without subscriptions to APS).
This is the work we will review today. Before we do this, we will present a quick overview of related ideas we’ve considered to date.
The traditional SIR model consists of a fixed number of individuals $T$ split into three classes. (note, the authors use $N$ but we use this elsewhere in this series to represent the number operator).
SIR (classes: S - susceptible/ I - infected/ R - recovered), SIR epidemic model.
In Network Theory 3 John Baez explained the SIR model. Network Theory Part 3
The SIR model is said to have been invented by Kermack and McKendrick in 1927:
At any particular time, the average number of individuals in each class is given by
The time evolution of the three classes is governed by the following set of coupled non-linear differential equations.
The first of these equations describes the reduction of $S$ via infection.
The second shows $I$ increasing via infection of the susceptibles and decreasing by spontaneous recovery
The third shows the increase of $R$ due to the spontaneous recovery.
These are called the “rate-equations”. They account for how an average number of things changes with time. The other option is to consider the master equation, which considers the complete history of a stochastic process. Most people suspect that you arrive at the rate-equation through a large number approximation of the master equation. However, very recently this view was smashed with a counter intuitive result by John actually. This correspondence is indeed replaced with a more subtle result which is physically relevant in several regimes.
TODO: explain the above better!
Ok, so back to these rate equations. The system can be solved numerically, yielding the time dependence of each class of individuals. Analytical solutions have several advantages, and are not known for this model in general. Special cases admitting exact solutions are interesting, and is what we will consider here.
The Petri net for the model is given as follows.
From this Petri net, we can arrive at the stochastic-Hamiltonian, using our all-too-familiar prescription.
Here we consider an initial state of a system. We will represent this as a formal power series in indeterminates $i, s, r$ as
We recall that the creation $(I^+, S^+, R^+)$ and destruction operators $(I^-, S^-)$ act as
and as
We’re reviewed all of these properties in much more detail throughout this series.
Looking at these equations, it’s useful to consider exactly what we mean by “spacial degrees of freedom”. Here, a term such as $i^n s^m r^k$ has no reference to spacial location for an individual.
In real life, we know very well that epidemics have a strong spacial dependence in most cases. (Examples where this is not true would include a scenario where you place all of your rabbits inside of the same cage and they all interact approximately equally).
To motivate how we will consider position in our model let’s consider now just two rabbits. These two rabbits are initially separate. One of them is infected and the other is susceptible. We could write their state as
When the rabbits are allowed to interact, they are exposed to a situation omitting the following transition.
Of course, the healthy rabbit could get lucky meaning that it would not become infected. Either of these two scenarios is possible. Assuming the healthy rabbit is infected after some time with probability $p$ the final state of the system would then be
In quantum mechanics, the interaction strength between two systems is typically determined by some sort of first principles law (e.g. dipole–dipole interaction etc.). Although we can use quantum techniques to model stochastic systems, such as epidemic modelling, it’s certainly not as clear how to determine what the interaction strength should be. For this one must turn to empirical data, leaving first principles account to be inaccurate. This is indeed much more difficult and must often by handled on a case-by-case basis when modelling epidemics.
In this subsection, we use
The chemical reaction network is given from the transition equations.
The set of species is
We will now calculate the deficiency of the resulting network.
The complexes. We let $v_1 := (1,1,0)^\top$, $v_2 := (0,2,0)^\top$, $v_3 := (0,1,0)^\top$ and $v_4 := (0,0,1)^\top$ denote the four complexes.
The stoichiometric subspace of the whole network is given by $\text{span}\{v_2 - v_1, v_4 - v_3\}$ which has dimension 2.
The total number of linkage classes is two. One for each connected component. These are (i) $v_1$ and $v_2$ connecting, (ii) $v_3$ and $v_4$ connecting.
The deficiency of the network is the number of complexes ($C=4$) subtract the dimension of the stoichiometric subspace (dim $= 2$), subtract the number of linkage classes ($l=2$), which is zero.
(Remark) The SIR model is not weakly reversible.
(Result) The SIR model has deficiency zero, but is not weakly reversible, so cannot admit a steady state. This follows from the deficiency zero theorem.
We apply a result from the following paper.
From Feinberg page 2252.
In contract to the fully general SIR model, in the paper [WMM11] the authors consider a stochastic one-dimensional adaptation of the SIR model. They show that this model admits exact solutions.
In other words, each node is in contact with only two other nodes, and the rate of infection of susceptible individuals increases when neighbours are infected. Not all transitions are allowed. The authors define the interaction of the considered model to be
The paper briefly describes mean field equations. We won’t consider this here at this stage.
They let $C$ represent a state $(S,I,R)$ of each of $N$ individuals. Here $N$ is adjacent to $1$ from periodic boundaries. The authors then state.
In quantum mechanics notation, a general state is then expressed as
where $P(C,t)$ is the probability that the system would be found in configuration $C$ at time $t$.
They then cast the master equation into a quantum form
and say that the pseudo-Hamiltonian $H$ has matrix elements.
with the known formal solution
In this formalism the expectation value of $M(C)$ is given as
We have the standard equation of motion for the time dependence of the expected value of $M$ as
Type S particles: $a_i^\dagger$ (creates); $a_i$ (annihilates)
Type I particles: $b_i^\dagger$ (creates); $b_i$ (annihilates)
A site with neither an S nor I is assumed R (recovered)
Operators at different sites commute.
Operators of the same species at the same site anti-commute.
The number operators $A_i := a_i^\dagger a_i$, $B_i := b_i^\dagger b_i$ take only values zero or one.
In this notation, the Hamiltonian is written as (Equation 8)
with
They use a method of $n$-point cluster functions.
(Symmetry) The Hamiltonian is translationally invariant and we consider periodic boundaries so we are dealing with the symmetry group $C_N$. If the initial state is translationally invariant, so are all future configurations.
Translational invariance make the quantities $K_r(n)$ and $G_r(n)$ independent of their initial starting point $r$.
The time dependence of the cluster functions are found from the equation of motion of the operators, which involves calculating the commutation relations between the Hamiltonian and cluster operator products.
The equations of motion for the particle density of susceptibles and infecteds are found in a similar fashion.
The paper was published as
H T Williams et al J. Stat. Mech. (2012) P01017 doi:10.1088/1742-5468/2012/01/P01017
It is likely best at this stage to define a basis, for the spin-1 particles.
We then have
and