Compartment Models

Last update: 11 Dec 2024 15:10
First version: 11 May 2020 (or earlier?)

---

A "compartment model" is a model for the evolution of a population. At any give time, each individual in the population is of one of a finite number of types. That is, the population is divided up among "compartments", one compartment per type. Individuals can "transition" from one compartment to another, i.e., change types. (In some versions, individuals can also be born and die.) The rates at which they make these transitions are allowed to be functions of the current distribution of the population across compartments; sometimes we allow for the history of the distribution to matter as well. There are deterministic and stochastic versions; I am particularly interested in stochastic versions which approach deterministic limits as the population size grows.

These get used a lot as models of contagious disease (where the compartments are status like "infectious", "exposed but not yet infectious", "recovered", etc.), and in demography (where the compartments are things like "Male 18--22" or "Female 90--94", perhaps further sub-divided by marital status, caste, etc.) They are also formally identical to models in chemical kinetics and physical chemistry, which was the inspiration for one of the pioneers here, A. J. Lotka.

See also:
• Branching Processes
• Convergence of Stochastic Processes
• Demography
• Epidemic Models
• Markov Models
• Mean-Field Games and Mean-Field Control

Recommended (very misc.):
• Stephen P. Ellner and John Guckenheimer, Dynamic Models in Biology
• Thomas G. Kurtz, Approximation of Population Processes
• James H. Matis and Thomas R. Kiffe, Stochastic Population Models: A Compartmental Perspective

Recommended, close-ups:
• Michael Assaf and Baruch Meerson, "WKB theory of large deviations in stochastic populations", Journal of Physics A: Mathematical and Theoretical 50 (2017): 263001, arxiv:1612.01470

Recommended, historical, primary sources:
• A. J. Lotka, Elements of Physical Biology = Elements of Mathematical Biology

Recommended, historical, secondary sources:
• Nicolas Bacaër, A Short History of Mathematical Population Dynamics
• Sharon Kingsland, "Alfred J. Lotka and the origins of theoretical population ecology", Proceedings of the National Academy of Sciences (USA) 112 (2015): 9493--9495

Modesty forbids me to recommend:
• My lecture notes on compartment models for "Data Over Space and Time" (lecture 21 in the latest version, at the time I write)

To read:
• John A. Jacquez
  • "The inverse problem for compartmental systems", Mathematics and Computers in Simulation 24 (1982): 452--459
  • Compartment Analysis in Biology and Medicine
• John A. Jacquez and Carl P. Simon, "Qualitative Theory of Compartmental Systems", SIAM Review 35 (1993): 43--79 [JSTOR]
• J. H. Matis, T. E. Wehrly, "Stochastic Models of Compartmental Systems", Biometrics 35 (1979): 199--220
• Ning Ning and Edward Ionides, "Systemic infinitesimal over-dispersion on graphical dynamic models", Statistics and Computing 34 (2024): 147

To write:
• "Large Deviations for Markovian Compartment Models, and Related Population Processes" (I know what the result is, and I have a non-rigorous proof, but I need to turn it into a rigorous result, which means I need to relearn a lot of stuff I used to teach...)