Codeço, Claudia Torres

Revista de Saúde Pública

English

Cadernos Espinosanos (E-Journal)

Comentário:The reversible catalytic modeland the mathematical theoryof epidemicsO modelo catalítico reversível e ateoria matemática de epidemiasPrograma de Computação Científica.Fundação Instituto Oswaldo Cruz. Rio deJaneiro, RJ, BrasilCorrespondência | Correspondence:Claudia Torres CodeçoAvenida Brasil, 4365Antiga Residência Oficial21040-900 Rio de Janeiro, RJ, BrasilE-mail: codeco@fiocruz.brIn the early XXth century, mathematical models wereintroduced into infectious disease epidemiology withthe hope that formal procedures so successfully ap-plied to physics and chemistry would be helpful inuntangling complex causal relationships in epidemicsystems. This hope is clearly stated by Kermack &McKendrick2 (1927) at the start of one of their paperson the mathematical theory of epidemics: “One ofthe most striking features in the study of epidemics isthe difficulty of finding a causal factor which ap-pears to be adequate to account for the magnitude ofthe frequent epidemics of diseases which visit almostevery population.”The starting point for the mathematical modeling ofinfectious diseases is the analogy between the conta-gious process that lead to new infections andthe process observed in autocatalytic reac-tions, where the conversion from substrate toproduct is driven by the contact between thesetwo substances. In both systems, the product(or infective individuals in the case of epi-demic systems) acts as a catalyst and the rateof reaction increases as the concentration ofthe product increases. The reaction slowsdown when the substrate (or the amount ofsusceptibles) reduces below a certain thresh-old. Based on this analogy, Muench3 pub-lished a book in 1959 where he proposed theuse of catalytic models to estimate the rateof conversion (that is, the infection or inci-dence rate) from susceptible to infective,based on the fitting of these models to preva-lence data.Since its publication, Muench’s book hasbeen source of inspiration for an ever grow-ing series of papers (Figure). Part of this pub-lications concerns the estimation of age-spe-cific infection rate from serological data using cata-lytic models or their derivations. Rodrigues da Silva(1969) is one of the first examples of this type ofapplication. He estimated the rate of Chagas diseaseinfection in communities in Salvador city, Brazil, andcompared them to a community in Venezuela. Hefound similar values in all three communities andconcluded that the contagious process involved inChagas transmission was similar in both regions.When presenting his results, however, he showed howconcerned he was about the oversimplifying assump-tions of the catalytic model. He listed them: the as-sumption that the dynamics of disease transmissionis at an endemic steady-state, that is, during the wholeperiod since the birth of the oldest member of thecohort until the last recorded data, the rate of infec-Rev Saúde Pública 2006;40(2):205-6Claudia Torres CodeçoFigure - Cumulated number of papers citing Muench’s book (1959) onthe application of catalytic models in epidemiology.(Silva, 1969)1960 1970 1980 1990 2000YearsCumulated number of citations050100150206 Rev Saúde Pública 2006;40(2):205-6The reversible catalytic modelCodeço CTtion is assumed constant; 2) that migration was neg-ligible; 3) that mortality due to the disease was neg-ligible; 4) the model did not consider other compo-nents of the transmission process, as the vector popu-lation or the other vertebrate hosts populations.Even if these simplif ications may have broughtdoubts about the conclusions of the work, themodeling exercise per se was successful in itself asway to make explicit arguments regarding the natureof the contagious process and allowing a mechanis-tic model-based approach to the estimation of theinfection rate. Since then, the emphasis in mechanis-tic mathematical modeling has turned to the devel-opment of more refined models that seek to relax thestrict assumptions present in formulations as thoseproposed by Muench and applied by Rodrigues daSilva. In the case of the Chagasic disease, for exam-ple, new mathematical models have been recentlyproposed to evaluate alternative control strategies,more specifically, the pros and cons of removing ani-mals (specially dogs) from the domestic environment.With this aim in mind, Cohen & Gurtler1 (2001) elabo-rated a mathematical model to represent the dynam-ics of infection within a household where humans,dogs, chickens and insect vectors coexist. In this en-vironment, the parasite circulates among humans,dogs and bugs and the model seek to capture thiscontact process. Chickens, which are not suscepti-ble, have two potential impacts on transmission. Theymay help to reduce human exposition by divertingpart of bug biting activity. On the other hand, in-creased number of available hosts may increase thebug population, and consequently increase transmis-sion of the parasite to humans. They find that havingor not chickens at home have only a marginal impacton transmission while removing dogs from sleepingrooms would have a large beneficial effect.Still, although models used today are much morecomplex than the reversible catalytic models usedby Rodrigues da Silva, the most basic assumptionof an autocatalytic process still holds in most mod-els of infectious diseases dynamics. Moreover, theproblem of estimating infectious diseases parametersfrom serological data is still an open issue and muchwork is expected in the future on the developmentof mathematical models for disease. An interestingidea would be to apply modern methods to Rodriguesda Silva’s data.REFERENCES1. Cohen JE, Gurtler RE. Modeling householdtransmission of american trypanosomiasis. Science.2001;293(5530):694-98.2. Kermack WO, McKendrick AG. A contribution to themathematical theory of epidemics. Proc R Soc LondA. 1927;115(772):700-21.3. Muench H. Catalytic models in epidemiology.Cambridge (MA): Harvard University Press; 1959.

The reversible catalytic model and the mathematical theory of epidemics

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O modelo catalítico reversível e a teoria matemática de epidemias

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