CastilloChavez and Feng's basic model for tuberculosis.
In the basic model of tuberculosis transmission proposed by CastilloChavez and Feng (CastilloChavez and Feng, 1996) the population is divided in four classes: susceptibles (S), latent (E, infected but not infectious), infectious (I) and treated (T).
where:
is the rate of birth of the population,
is the average proportion of susceptibles individuals infected by contact with an infectious individual per unit time,
is the average proportion of treated individual infected by contact with and infectious individual per unit time,
is the per capita rate of conversion from infected to ill,
is the per capita rate of contact,
is the per capita rate of natural mortality,
is the per capita rate of mortality due to the disease,
are the per capita rates of beginning treatment.
CastilloChavez and Feng have demonstrated that the basic reproductive number is given by:
where represents the average number of susceptibles infected by an infectious during his period of effective transmission, and represents the fraction of the population of infected that survives the latent period. Therefore Ro results being the number of new cases of infection produced by each infectious individual during his period of transmission, when a typical infectious is introduced in a population of susceptibles.
The population system has two possible equilibria: a) when , presenting an equilibrium with a number of infected different from zero, i.e. an endemic is generated, and b) when , the equilibrium occurs when the number of infected, latents and treated is zero, i.e. an equilibrium free of disease. Ro depends inversely of the rates of treatment r_{1} and r_{2} This shows the importance of prevention treatment of latent and infectious to make the endemic levels disappear. For this reason, the follow up of contacts and the active search of cases are widely used strategies in the developed countries (see revision Rieder, 1993). Formulae like the one of Ro allow the relative quantification and planning of diverse control strategies, based on the reduction of epidemiological and social factors that determine the value of Ro. For example, theoretically it is possible to predict the levels of necessary intervention that leads to the extinction of TB in long term with the previous model. Of course that these predictions are of first order, i.e. they give the possibility of approximately estimating the magnitude of the necessary investment required to control TB. Elements not included in the previous model and that may result fundamental in the elaboration of effective intervention policies include epidemiological, sociological, socioeconomical and demographic factors. For example, delays in the diagnosis favor the maintenance of the disease, because they produce a dicrease of the rate of beginning treatment, increasing the value of Ro . Macintyre et al. (Macintyre et al., 1995) have found that a delayed diagnosis of an office worker permitted him to transmit the disease to 24% of his office mates. Examples of transmission in the comunity like churches (Dutt et al., 1995) and pubs (Kline et al., 1995) have also been related to delayed diagnoses.Nevertheless, even though the search of cases can be improved, this would not be effective if an increase in the completeness of the treatment is not achieved by the patients (Brenner y Pozsik, 1993).
References
Brenner, E, Pozsik, C. (1993). Case Holding. En: Reichman, L.B., Hershfield, E.S. (Eds.): Tuberculosis. A Comprehensive International Approach. Lung Biology in Heath and Disease 66. pp 183206. Marcel Dekker Inc, New York
CastilloChavez, C, Feng, Z. (1996) Mathematical Models for the Disease Dynamics of Tuberculosis. Submited to Proceedings of the Fourth International Conference on Mathematical Population Dynamics
Dutt, A. K., Mehta, J. B., Whitaker, B. J., Westmoreland., H.(1995) Outbreak of tuberculosis in a church. Chest 107 : 447452
Kline, S. E. , Hedemark, L. L., Davies, S. F. (1995) Outbreak of tuberculosis among regular patrons of a neighborhood bar. N. Engl. J. Med. 333: 222227.
Macintyre, C. R., Plant, A. J., Hulls, J., Streeton, J.A., Graham, N.M.H., Rouch, G.J. (1995) High rate of transmission of tuberculosis in an office: Impact of delayed diagnosis. Clin. Infect. Dis. 21: 11701174.
Rieder, H. L.(1993): Case Finding. En: Reichman, L. B., Hershfield, E. S. (Eds.): Tuberculosis. A Comprehensive International Approach. Lung Biology in Heath and Disease 66, pp 167182 Marcel Dekker Inc, New York .
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