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Incorporating drug resistance

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, because this will generate an increase of the proportion of resistant strains to drugs (Rieder, 1993; Brenner and Pozsik, 1993, Manalo et al, 1990). To analyze the consequence at the population level of the incompleteness of treatment Castillo-Chavez and Feng (Castillo-Chavez and Feng, 1996) modified their simple model, incorporating the presence of another strain, which is favored by such behavior of the patients. They have added, to the previous four classes (S,E1,I ,T), the classes infected with the resistant strain (J) and latents with the resistant straing (E2) and made the following system of differential equations:

 

where
  is the rate of birth of the population,
  is the average proportion of susceptible individuals, infected by contact with an infectious individual with the strain 1 per unit time,
  is the average proportion of treated individuals, infected by contact with an infectious individual with strain 1 per unit time,
  is the average proportion of susceptible or treated individuals, infected by contact with an infectious individual with the drug-resistant strain per unit time,
k1 and k2 are the per capita rates of conversion from infected to ill, depending on wether they were infected with the strain 1 or with the drug-resistant strain,
  is the per capita contact rate,
  is the per capita natural mortality rate,
d1 and d2 are the per capita mortality rate due to the disease,
r1 and r2 are the per capita rates of beginning treatment,
p is the proportion of infected individuals that do not complete their treatment and do not develop resistance to drugs,
q is the proportion of treated infected individuals that do not complete their treatment and that develop resistance to drugs,
q * r2* I represents the rate at which the individuals develop resistance to tuberculosis because they do not complete the treatment against tuberculosis.
It is supposed that the treatment rate against tuberculosis is very small and, therefore, it is ignored.

The basic reproductive numbers for the two strains are given by:

R 1 and R 2 can be interpreted as the average number of infeccions caused, during the infectious period, by a typical infectious with strain 1 and the drug-resistant strain, respectively. Ro is th maximum of { R 1 , R 2 } and again gives the threshold condition for the maintenance or not of the disease in endemic state, if  the disease turns into endemic, if  the infection disappears.

Castillo-Chavez and Feng (Castillo-Chavez and Feng, 1996) have analyzed the equilibrium of the system, taking into account two possible situations: (a) when the resistance is not generated by the abandonment of the treatment, q = 0, and (b) when the resistance is generated by the incompleteness of treatment,  . In Fig. 1 that represents the space R 1 - R 2 the results are summarized. In the case (a) four situations of equilibrium can occur, depending on the relationship between the values of R 1 and R 2 . In the area I of the Figure 1a, the equilibrium is given at zero level of the disease. In the area II, the resistant strain is the one that maintains in endemic state and the other disappears. In the area III there is coexistance of the two strains, and in the area IV thenon-resistant strain remains while the other disappears. In the case (b), when the resistance is generated by the incompleteness of treatment, the area IV disappears (Figure 1b), there are 3 possible equilibria: 1) no disease, 2) endemic only in the resistant strain and 3) endemic in which both strains coexist.
The existance of resistant strains has been aknowledged in 1946, almost immediately after the introduction of streptomicine (Youmans et al, 1946). Obviously the resistance to drugs occurs by mutations and the drugs are only selection agents that allow the resistant strains to remain in the population. Since the resistance sites are not linked to the chromosome, the probability that spontaneous mutations occur to more than one drug is difficult (Kent, 1991). Patients with resistance to only one drug may be treated with two or more antibiotics. Nevertheless, the drugs used result inefficient for the treatment of multidrug strains. These strains are common in developing countries, but were not frequent in developed countries until end of the 80's (Gangadharm, 1993). Several microepidemics occurred in the U.S.A. and other developed countries were possibly favored by the presence of patients with HIV infection (see revisions Kent, 1991; Gangadharm, 1993). There is an important controversy regarding pathogenicity of resistant strains. In guinea pigs it was found that strains resistant to isoniazida presented a reduced virulence compared to the non-resistant strains, but they did not show that decrease in virulence in mice (Gangadharm, 1993; Middlebrook and Cohn, 1953). In humans Liebknecht (Liebknecht, 1964) showed that children exposed to strains resistant to isoniazida presented a rate of infection of 4.5% while the exposed to non-resistant strains presented a rate of infection of 11%. Epidemiological data support this observation, since the rate of infectivity of isoniazida-resistant strains was of 1.33 during 100 months to exposition of 57 contacts, against a rate of 5.94 during the same period of 77 contacts of patients with active tuberculosis that were infected with non-resistant strains (WHO, 1967). Nevertheless, the microepidemics that arose in the last years in the U.S.A. would be showing that these resistant strains are not less pathogenic (Kent, 1991; Gangadharm, 1993).

The results of the analysis made by Castillo-Chavez and Feng (1996) show that the coexistance of both strains is possible, although rare, without the presence of the antibiotics as agents of selection (model with q = 0) that would represent the epidemiology of the primary infection (persons that acquire resistant strains only by transmission from another patient with the same strain). The analysis of the model when incorporating the resistance to drugs acquired by the effect of incompleteness of treatment (  ) shows us the resistant strain coexisting with the natural strain. The drug-resistant strains are considered of low pathogenicity (Kent, 1991) and therefore are going to have low values for the per capita mortality rate due to the disease ( d ) and of the per capita rate of conversion from infected to ill ( k ), therefore their basic reproductive numbers could be high. Nevertheless, the absence of dominant resistant strains in the population suggests that it is more difficult for them to survive. It is important to note that the use of antibiotics makes that the natural strains are effectively not much virulent, i.e. that they cause relatively low mortality due to the accessibility to treatment. The multidrug resistant strains cause higher mortality rates than the ones caused by natural strains (taking into account the use of antibiotics), and in this sense they can be considered as strains with a high mortality rate ( d ) and, therefore, with a low basic reproductive number. The fact is that the mutations due to the use of antibiotics increases the possibility of coexistence between non-resistant strains y multidrug resistant strains .
 

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 183-206. Marcel Dekker Inc, New York.

Castillo-Chavez, C, Feng, Z. (1997) Treat or not to treat: The Case of Tuberculosis. J. Math. Biol. 35, 629-659.

Gangadharm, P. R. J. (1993) Drug resistance in tuberculosis. En: Reichman, L.B., Hershfield, E.S. (Eds.): Tuberculosis. A Comprehensive International Approach. Lung Biology in Heath and Disease 66, pp 293-328. Marcel Dekker Inc, New York.

Kent, J. H. (1991) The epidemiology of multi-drug resistant tuberculosis in the United States. Medical Clinics of North America 77: 1391-1409.

Liebknecht, W, (1964). comunicación personal al Dr. Meissner, citado por Meissner, G. en The bacteriology of tuberculosis. En: Barry, V.C.: Chemotherapy of Tuberculosis. Butterworths, London, p. 82.

Manalo, F., Tan, F., Sbarbaro, J. A., Iseman, M. D. (1990). Community-based short-course treatment of pulmonary tuberculosis in a developing nation. Am. Rev. Respir. Dis. 142: 1301-1305.

Middlebrook, G, Cohn, M. L.(1953) Some observations on the pathogenicity of isoniazid resistant variants of tubercle bacilli. Science 118: 297- 299.

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 167-182 Marcel Dekker Inc, New York .

WHO: World Health Organization Research Project Kenya, unpublished observations. (1967), citado en Gangadharm, P. R. J. (1993).

Youmans, G. P., Williston, E. H., Feldman, W. H, Hinshaw, C. H. (1946). Increase in resistance of tubercle bacilli to streptomycin. A preliminary report. Proc. Mayo Clin. 21: 16.

 

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