Editing Compartmental models (epidemiology) (section)

{{short description|Type of mathematical model used for infectious diseases}}

'''Compartmental models''' are a mathematical framework used to simulate how populations move between different states or "compartments." While widely applied in various fields, they have become particularly fundamental to the [[mathematical modelling of infectious diseases]]. In these models, the population is divided into compartments labeled with shorthand notation – most commonly '''S''', '''I''', and '''R''', representing '''S'''usceptible, '''I'''nfectious, and '''R'''ecovered individuals. The sequence of letters typically indicates the flow patterns between compartments; for example, an SEIS model represents progression from susceptible to exposed to infectious and then back to susceptible again.

These models originated in the early 20th century through pioneering epidemiological work by several mathematicians. Key developments include Hamer's work in 1906,<ref>{{cite journal |last1=Hamer |first1=William |year=1906 |title=On Epidemic Disease in England -- The Evidence of Variability and of Persistency of Type, Lecture III |url=http://dx.doi.org/10.1016/S0140-6736(01)80187-2 |journal=The Lancet |volume=167 |issue=4305 |pages=569-574 |doi=10.1016/s0140-6736(01)80187-2|url-access=subscription }}</ref> [[Ronald Ross|Ross]]'s contributions in 1916,<ref>{{cite journal |vauthors=Ross R |date=1 February 1916 |title=An application of the theory of probabilities to the study of a priori pathometry.—Part I |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |volume=92 |issue=638 |pages=204–230 |bibcode=1916RSPSA..92..204R |doi=10.1098/rspa.1916.0007 |doi-access=free}}</ref> collaborative work by Ross and [[Hilda Phoebe Hudson|Hudson]] in 1917,<ref>{{cite journal |vauthors=Ross R, Hudson H |date=3 May 1917 |title=An application of the theory of probabilities to the study of a priori pathometry.—Part II |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |volume=93 |issue=650 |pages=212–225 |bibcode=1917RSPSA..93..212R |doi=10.1098/rspa.1917.0014 |doi-access=free}}</ref><ref>{{cite journal |vauthors=Ross R, Hudson H |date=1917 |title=An application of the theory of probabilities to the study of a priori pathometry.—Part III |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |volume=89 |issue=621 |pages=225–240 |bibcode=1917RSPSA..93..225R |doi=10.1098/rspa.1917.0015 |doi-access=free}}</ref> the seminal [[Kermack–McKendrick theory|Kermack and McKendrick]] model in 1927,<ref name="Kermack–McKendrick2">{{cite journal |vauthors=Kermack WO, McKendrick AG |date=1927 |title=A Contribution to the Mathematical Theory of Epidemics |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |volume=115 |issue=772 |pages=700–721 |bibcode=1927RSPSA.115..700K |doi=10.1098/rspa.1927.0118 |doi-access=free}}</ref> and [[David George Kendall|Kendall]]'s work in 1956.<ref name="Kendall2">{{cite book |title=Contributions to Biology and Problems of Health |vauthors=Kendall DG |date=1956 |publisher=University of California Press |volume=4 |pages=149–165 |chapter=Deterministic and Stochastic Epidemics in Closed Populations |doi=10.1525/9780520350717-011 |mr=0084936 |zbl=0070.15101 |chapter-url=http://projecteuclid.org/euclid.bsmsp/1200502553}}</ref> The historically significant [[Reed–Frost model]], though often overlooked, also substantially influenced modern epidemiological modeling approaches.<ref>{{Cite journal |last=Engelmann |first=Lukas |date=2021-08-30 |title=A box, a trough and marbles: How the Reed-Frost epidemic theory shaped epidemiological reasoning in the 20th century |journal=History and Philosophy of the Life Sciences |language=en |volume=43 |issue=3 |page=105 |doi=10.1007/s40656-021-00445-z |issn=1742-6316 |pmc=8404547 |pmid=34462807}}</ref>

Most implementations of compartmental models use [[Ordinary differential equation|ordinary differential equations]] (ODEs), providing deterministic results that are mathematically [[Tractable problem|tractable]]. However, they can also be formulated within [[Stochastic process|stochastic frameworks]] that incorporate randomness, offering more realistic representations of population dynamics at the cost of greater analytical complexity.

Epidemiologists and public health officials use these models for several critical purposes: analyzing disease transmission dynamics, projecting the total number of infections and recoveries over time, estimating key epidemiological parameters such as the [[basic reproduction number]] (R₀) or [[effective reproduction number]] (R<sub>t</sub>), evaluating potential impacts of different [[Public health intervention|public health interventions]] before implementation, and informing evidence-based policy decisions during disease outbreaks. Beyond infectious disease modeling, the approach has been adapted for applications in [[population ecology]], [[pharmacokinetics]], [[chemical kinetics]], and other fields requiring the study of transitions between defined states.