Previously

Markov models: The future of the process is independent of its past, given its present state
- Everything boils down to the probabilities or rates of transitions between states
Inference for Markov models: Based on the likelihood, or on other ways of assessing fit to the data
Epidemic models: Every individual in the population is in one of a limited number of states of health; transitions between states happen at rates which depend on how many individuals are in each state

Compartment Models

Many sciences deal with populations, where each individual is in one of a limited number of states
- Epidemiology (as we saw)
- Ecology
- Sociology (think: occupations, classes, education, racial/ethnic groups, religion, …)
- Demography (age, sex, ethnicity, religion, …)
- Chemical kinetics (molecular “species”)
All these use compartment models
- Compartments = status for individuals
- Each individual is in 1 and only 1 compartment at any time
  - Compartments are often intersections, e.g., “Black males aged 61–65”
- Rates of transitions from one compartment to another, often depending on occupancy of compartments

Basic Definitions and Assumptions

Start with a “closed” population, constant size \(n\)
Population is divided into \(r\) compartments
State of population at time \(t\), \(X(t)\), is a vector of counts, \[\begin{equation} X(t) \equiv [X_1(t), X_2(t), \ldots X_r(t)] \end{equation}\]
- \(X_i(t)=\) how many individuals are in compartment \(i\) at time \(t\)
\(X(t)\) changes because individuals move from compartment to compartment, randomly
\(p_{ij}=\) probability that individual in compartment \(i\) will move to compartment \(j\) at the next time step, the \(i\rightarrow j\) transition rate
- If \(p_{ij} =0\) the \(i\rightarrow j\) transition is forbidden (or suppressed)
- \(p_{ii}=\) probability of staying in compartment \(i\)
Assumption: \(p_{ij}\) is a function of the current state \(X(t)\) and not the past or future states
- \(p_{ij}\) might be constant for some \(i, j\)
Assumption: All individuals in the population make independent transitions, given \(X(t)\)
- Transitions from compartment \(i\) are independent and identically distributed, given \(X(t)\)
These assumptions could be wrong!

Consequences of the Assumptions

\(Y_{ij}(t+1) =\) number of individuals transitioning from compartment \(i\) to compartment \(j\) at time \(t+1\), the flow or flux from \(i\) to \(j\)
By the assumptions, \[\begin{eqnarray} Y_{ij}(t+1) & \sim & \mathrm{Binom}(X_i(t), p_{ij}(X(t)))\\ \mathbb{E}\left[ Y_{ij}(t+1)|X(t) \right] & = & X_i(t) p_{ij}(X(t))\\ \mathrm{Var}\left[ Y_{ij}(t+1)|X(t) \right] & = & X_i(t) p_{ij}(X(t))(1-p_{ij}(X(t))) \end{eqnarray}\]
Stock-flow consistency: \[\begin{eqnarray} \label{eqn:stock-flow} X_i(t+1) & = & X_i(t) + \sum_{j\neq i}{Y_{ji}(t+1) - Y_{ij}(t+1)}\\ \text{(new level)} & = & \text{(old level)} + \sum{\text{(in flow)} - \text{(out flow)}} \end{eqnarray}\]
Implication: \(X(t)\) is a Markov process
- Because \(Y_{ij}\)s only depend on current \(X(t)\), not past
- The state space is finite, but very large

Large-Population (\(n\)) Asymptotics

We want to consider very large populations (\(n\rightarrow\infty\)) making lots of small steps over short time intervals \(h\) (\(h\rightarrow 0\)) (so we go from time \(t\) to \(t+h\), not \(t\) to \(t+1\))
Assume \(p_{ij}(X) = h \rho_{ij}(X/n)\)
- Meaning that the distribution over compartments matters, not the raw numbers in each compartment
Define \(x(t) = X(t)/n\)
As \(n\rightarrow\infty\) and \(h\rightarrow 0\), we go to deterministic differential equations, \[ \frac{dx_i}{dt} = \sum_{j \neq i}{\rho_{ji}(x(t)) - \rho_{ij}(x(t))} \]
There are some caveats — see the handout

Epidemic Models

SIR model from last time: 3 compartments, transition rates depend on counts in each compartment
- We’d now write \(X_S(t), X_I(t), X_R(t)\) instead of \(S(t), I(t), R(t)\)
- \(Y_{SI}(t+1) = C(t+1)\) from last time, \(Y_{IR}(t+1) = D(t+1)\)
- \(p_{SI} \propto X_I(t)\), \(p_{IR} =\) constant
- All other transitions are forbidden
SIS model: 2 compartments, all transitions both ways
SIRS model: 3 compartments, allow an \(R\rightarrow S\) transition but no \(R\rightarrow I\) or \(I \rightarrow S\)
- Often used for influenza

Sociological Application: Modeling the Spread of Novelties

Including: ideas, practices, technologies, use of products, beliefs, …
Some practices spread by personal contact
- E.g., to learn how to make pots in a certain way, you (probably) need to learn from someone who already knows how to make that kind of pot
- Others can be spread by artifacts (you see a new kind of pot and can guess how to make it), or get re-invented
Assume that everyone either has adopted the new practice (and can/will spread it), or that they haven’t
- \(\Rightarrow\) 2 compartments, call them \(A\) (adopters) and \(N\) (non-adopters)
Assume that the new practice spreads by personal contact
- So \(p_{NA}\) depends on \(X_A\), say \(\propto X_A\)
Assume that no one gives up the practice, so \(p_{AN} = 0\)

Comments on the Assumptions

Assuming that the practice has to spread by personal contact:
- Some practices can be spread by artifacts
  - Maybe you see a new-to-you type of pot and you can figure out how to make it
  - Ideas can be spread by books, videos, etc., as well as by people
- Some practices get re-invented independently
Assuming that every adopter is a potential spreader:
- Not necessarily true by any means
  - Maybe you just want to make pots and aren’t interested in teaching anyone
Assuming the rate of adoption \(\propto X_A\):
- Presumes some sort of random mixing within the population
- Presumes no independent adoptions (could add a constant term to the rate)
- Presumes no “diminishing returns” to encountering lots of adopters
- Presumes presumes there aren’t other influences on the decision to adopt or not
  - Maybe un-common practices are always less attractive, because it helps to coordinate with others
Assuming no one gives up the new practice, \(p_{AN} = 0\):
- Plenty of people try new technologies, products, beliefs, etc., and then give them up!

\(\Rightarrow\) Whether the assumptions are good matches to reality is something you need to investigate case by case, not presume!

Comment: Analogy to Epidemic Models; What to Call This

This is like an SIR model:
- Adopters \(A\) \(\Leftrightarrow\) Infectious \(I\)
- Non-adopters \(N\) \(\Leftrightarrow\) Susceptibles \(S\)
- But set \(p_{IR} = 0\) (i.e., in terms of last time, \(\gamma=0\))
- \(\therefore\) The infectious stay infectious forever
Some kind of analogy between the transmission of ideas/behavior and the transmission of disease is very old and very common
- E.g., Siegfried (n.d.) is a careful working-out of the analogy by a French geographer in 1960
The mathematical models, and the realization that it matches epidemiological models, comes from the 1950s
Talk about “diffusion of innovations” (Rogers 2003) and it sounds good
Talk about “viruses of the mind” (Dawkins 1993) and it sounds bad
It’s hard come up with a really neutral name
- The anthropologist Dan Sperber suggested “epidemiology of representations” (Sperber 1996), which I like but is a bit of a mouthful

Consequences of the Model Assumptions

\[\begin{eqnarray} Y_{NA}(t+1) & \sim & \mathrm{Binom}(X_N(t), \alpha X_A(t))\\ X_A(t+1) & = & X_A(t) + Y_{NA}(t+1)\\ X_N(t+1) & = & X_N(t) - Y_{NA}(t+1)\\ \mathbb{E}\left[ X_A(t+1)|X_A(t) \right] - X_A(t) & = & X_N(t)\alpha X_A(t)\\ & = & \alpha X_A(t)(n-X_A(t)) \end{eqnarray}\]

Notice: expected increment to \(X_A\) is quadratic in \(X_A\), small if \(X_A\) is small, small if \(X_A\) is big, maximized when \(X_A = n/2\)
Notice: if \(X_A \ll n\), then the increment is roughly \(\propto X_A\), so initially there’s exponential growth
“Slowly, and then all at once” but also “Nothing grows forever”

Consequences of the assumptions

alpha <- 1e-6
n <- 1e5
xa <- vector(length=200)
xa[1] <- 1
for (t in 2:length(xa)) {
    xa[t] <- xa[t-1] + alpha*xa[t-1]*(n-xa[t-1])
}
plot(1:length(xa), xa, xlab="t", ylab=expression(X[A](t)), type="l")

This looks like the logistic function we use in logistic regression, \(\frac{e^t}{1+e^t}\)
This is, in fact, called a “logistic growth curve”
- Actually, the name “logistic regression” comes from this sort of logistic growth curve, not the other way around
- “Logistic” here from the sense of logistics, of supplying something (Bacaër 2011)

Consequences of the assumptions

sim.diffusion <- function(n, alpha, xa.1=1, T) {
    xa <- vector(length=T)
    xa[1] <- xa.1
    for (t in 2:length(xa)) {
        xa[t] <- xa[t-1] + rbinom(n=1, size=n-xa[t-1], prob=alpha*xa[t-1])
    }
    return(xa)
}

“Conversion to Islam in the Medieval Period”

An interesting, important, and analytically-clean example from Bulliet (1979)
Islam is a “missionary” religion; its adherents (often) go out to seek converts
- Some other religions aren’t missionary ones and conversion is rare or even impossible (Judaism, Shinto, etc.)
Conversion requires personal interaction; you are converted by someone
- In some religions you might get converted just by reading (and believing) a book, but in all religions that’s rare
During the initial expansion, say +630 to 900, lots of people converted to Islam
- Previously they were polytheists, Christians, Jews, Zoroastrians, Buddhists, Hindus, Manicheans, etc.
- There were many reasons for conversion — there were certainly practical advantages if living under Islamic rule — but part of it was definitely sincere religious conversions
  - Hodgson (1974), Bulliet (1994), Ansary (2009) all try to present why people might’ve wanted to convert to Islam
Bulliet (1979) used medieval biographical dictionaries to determine when many different conversions happened, in different countries
- The biographies give the subject’s dates, and their formal name, in the style “A, son of B, son of C, son of D, … son of Z”
- Converts typically assumed a Muslim name
- Tracking back to the latest non-Muslim name in a genealogy gives the number of generations to conversion
  - Especially clear for Persian-speaking countries (modern Iran, Afghanistan, Uzbekistan, etc.) where the pre-Islamic names were linguistically very different from Arabic names, and also there weren’t many Jews and Christians with Biblical names (which are also common in Islam)
- Assuming so many years per generation says (roughly) when the conversion happened
- Bulliet grouped by 25-year intervals

“Conversion to Islam in the Medieval Period”

How could we fit the model to this data?

Regress the increments \(X_A(t+1) - X_A(t)\) on \(X_A(t)(n-X_A(t))\)
Fit the curve \(X_A(t)\) to the deterministic curve implied by the model
Use maximum likelihood

“Conversion to Islam in the Medieval Period”

Summary

Divide the population into mutually exclusive and jointly exhaustive compartments
Assume both that transition rates between compartments only depend on the present state of the population, and that individuals transition (conditionally) independently
Result: Markov model for the state of the population, with stable large-\(n\) behavior
Applications: epidemiology, spread of new ideas/practices, lots of other social processes, demography, …

References

Ansary, Tamim. 2009. Destiny Disrupted: A History of the World Through Islamic Eyes. New York: Public Affairs.

Bacaër, Nicolas. 2011. A Short History of Mathematical Population Dynamics. London: Springer-Verlag.

Bulliet, Richard W. 1979. Conversion to Islam in the Medieval Period: An Essay in Quantitative History. Cambridge, Massachusetts: Harvard University Press.

———. 1994. Islam: The View from the Edge. New York: Columbia University Press.

Dawkins, Richard. 1993. “Viruses of the Mind.” In Dennett and His Critics: Demystifying Mind, edited by Bo Dahlbom. Oxford: Blackwell.

Hodgson, Marshall G. S. 1974. The Venture of Islam: Conscience and History in a World Civilization. Chicago: University of Chicago Press.

Rogers, Everett M. 2003. Diffusion of Innovations. Fifth. New York: Free Press.

Siegfried, André. n.d. Germs and Ideas: Routes of Epidemics and Ideologies. Edinburgh: Oliver; Boyd.

Sperber, Dan. 1996. Explaining Culture: A Naturalistic Approach. Oxford: Basil Blackwell.