SMABSC: Disease Propagation

The SIR model was introduced as a mathematical model with differential equations (Kermack & McKendrick, 1927). The basic states are Susceptible, Infected, and Recovered.

[latex]N_i = \frac{dS}{dt}+\frac{di}{dt}+\frac{dR}{dt}[/latex]

In the SIR model, the fundamental trajectory of disease propagation could be captured, immunity was acquired after disease and the population is homogeneous.

But the SIR model has short-comings:

  • populations are not infinite and increase/decrease over time,
  • populations are spatial objects and have (voluntary) spatial interactions,
  • populations are heterogeneous, including isolated sub-populations with irregular interactions as well as significant distances.
  • populations are driven by endogenous social factors and constrained by exogenous environmental circumscription.

Additional states were introduces such as Exposed (i.e. dormant infections that does not infect others yet) and Maternal immunity (i.e. individuals that cannot be infected). The order has been rearranged such as SIS, SEIS, SIRS.

These models still did not address explicit spatial reference to population loci & transportation networks, distribution of social & medical information (temporal and spatial), mechanisms to simulate voluntary & forced quarantines, treatment options and their delivery (temporal and spatial), and characteristics of pathogens and disease vector.

Questions that needs to be answered about a model is the form of circumscription (Carneiro, 1961, 1987, 1988) (e.g. social and environmental forces), instantiation topologies (i.e. abstract or logical relationships, social networks, or space), activation schemes (e.g. random, uniform random, Poisson), and encapsulation.


Carneiro, R. L. (1961). Slash-and-burn cultivation among the Kuikuru and it implications for cultural development in the Amazon Basin. In J. Wilbert (Ed.), The evolution of horticultural systems in native South America, causes and consequences. “Antropológica” (pp. 47–67). Caracas, Venezuela: Editorial Sucre.
Carneiro, R. L. (1987). Further reflections on resource concentration and its role in the rise of the state. In L. Manzanilla (Ed.), Studies in the Neolithic and Urban Revolutions (pp. 245–260). Oxford, UK: Archaeopress.
Carneiro, R. L. (1988). The Circumscription Theory: Challenge and response. The American Behavioral Scientist, 31(4), 497–511.
Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 115(772).