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Effect of control policies on Tuberculosis dynamics

The purpose of the proposed model of tuberculosis infection is to project the trends of the disease with the incorporation of control systems, consisting of vaccination campaigns, prophylaxis and treatment of the cases detected. It consists of 9 categories, groups or classes that describe the different stages an individual can go through when infected by Mycobacterium tuberculosis (including the transference rates for the passage from one category to another). It describes the further behaviour of the system in a specific time period, using 9 differential equations.

There are four different stages of the disease : susceptible, inactive cases, active cases and recovered inactive cases. The susceptible (S) are individuals that have never been infected with the bacteria, but that could be if sufficiently exposed. The inactive cases (E) are individuals that have been infected, but that are incapable of transmitting the disease and have not yet developed pulmonary damage. The active cases (I) are infectious individuals that have developed pulmonary damage, and can transmit the disease to the susceptible. The recovered inactive cases (T) are those active cases who have naturally recovered from the disease, with the possibility of returning to the infectious category.

The infectious process is represented in the passage of individuals from the S category to the E category. In all groups, we consider the same mortality rate, giving the same value to all cases, except for I, where we add the mortality rate due to the disease. We consider that all newly born individuals are susceptible.

We describe 3 basic ways of TB control :

1. Chemoprophylaxis in infected individuals (that are not yet either infectious or active cases), treated with the same drugs applied to active cases.
2. Chemotherapeutic cure of active cases using isoniazide and supporting drugs (isoniazide can be administered alone or combined with other drugs). The administration of the three drugs has 100% effectiveness, whereas the administration of two drugs (such as isoniazide and PAS -paraminosalicilic acid) has a maximum effectiveness of 90%. Only isoniazide may be prescribed, with results of up to 70% effectiveness.
3. Vaccination of susceptible individuals with BCG. Vaccination does not prevent infections, but delays the development of active cases from non active cases. The estimations of the effectiveness of vaccination are variable.

These methods of control are applied in a particular stage of the disease, each of these policies having a characteristic cost (Re Velle, 1967).

The prophylaxis applied to the individuals in the E category creates a new category in the model, Ep. According to the author, the advantage of the application of prophylaxis can be measured by the value of the rate of progression of the individuals in class Ep to class I, which is half the rate of progression from E to I. With the application of chemotherapy to individuals in category I, their cure can be achieved, resulting necessary to group them in a different category than the recovered non active cases (T). In this case, the relapse rate (i.e. the flow of individuals from class T to I) is less than the rate for those recovered naturally.

We consider that vaccination with BCG creates in the model a new system, parallel to the described previously. The individuals who enter the system and are vaccinated with BCG are transferred from category S to a new category, Sbcg, that groups the vaccinated susceptible. In the studies carried out, Re Velle found that when resisting infection, the vaccinated susceptible do not differ greatly from the susceptible without BCG. But once infected, the risk of an individual passing to the active category differs notably. The infection of a vaccinated susceptible makes the individual to transfer from the Sbcg to the Ebcg category. According to the data taken from the British Medical Journal (1963), the rate of progression from Ebcg to I has been estimated to represent the fifth part of the rate of progression from E to I. In case of applying prophylaxis to the individuals in class Sbcg, they transfer to the category Sbcgp, where the risk of contracting the disease is even lower.

The model proposed can be described by a system of 9 differential equations. The period of time considered is divided in intervals of 1 year, and the equations were written considering a range of change during i years. (Go to equations)

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