Bancada deformada

6.1

In trod u ction

The two models presented in C hapters 4 and 5 have accounted for the most im portant determ inants of the spread of tuberculosis w ithin a population and they provide a means of understanding the very basic intrinsic dynamics of the infection. Nevertheless there are some further interesting features of TB th a t, for simphcity, were not taken into account (for instance the possibility of reinfection of individuals w ith an old latent infection) which we are going to investigate in this chapter.

Also, our ultim ate goal is to study the effects of the various control measures currently available (vaccination, chemoprophylaxis, and chemotherapy). W ith this goal in m ind the model presented in C hapter 5 will now be modified to account for some fea­ tures of TB th a t previously were either om itted or simplified. For instance in C hapter 5 we assumed th a t the population is divided into three classes: susceptibles, infectives, and inactives. Chemoprophylaxis is recommended only for those w ith a latent infection and close contacts of infectious cases. Since homogeneous mixing is assumed we will assume th a t chemoprophylaxis is used only for latents and therefore these individuals m ust be in a separate class in the model from the other inactive cases.

An interesting question from a public-health point of view is w ith respect to the effect of treating non-infectious cases (which constitute about half of the TB cases - see C hapter 2). In order to answer this question the num ber of non-infectious cases must

p q i ^ X Y (/i + y.\)Y

ÔW

IàX

U fiU

V I

(y,

Z ) = Q2I3Z+ q ^ ^ Y Z V2( y , Z) = ( l - q2)P Z + ( 1 - q3) f Y Z

Figure 6.1: A detailed model for TB: Model Zeus

be known.

Therefore now we consider a population subject to homogeneous mixing and divided into five classes:

(a) uninfected: individuals who have never been infected w ith TB

(b) latents: individuals who have been infected (at least once) b ut the infection has rem ained latent (they are non-infectious and healthy)

(c) infectious TB cases: individuals who have developed chnical disease and are infectious

(smear-positive TB cases)

(d) non-infectious cases: individuals who have developed chnical disease but they are

not infectious (this class includes all the smear-negative TB cases: patients with non- pulm onary TB and those w ith smear-negative pulm onary TB)

(e) recovered: individuals who have developed chnical TB and recovered spontaneously

w ithout treatm ent

The sizes of these classes a t tim e t will be denoted by X (t), Z { t ) , Y(t), W(t), U{t), respectively, and the size of the population by N {t) = A’(t) + y {t) -\-Z { t)-h W {t) + U{t) (for sim phcity these classes will be sometimes referred to as the X class, the Z class, and so on). The initial sizes of the five classes are %(0) = xq,

y(0)

= yo, Z(0) = zq, W (0) = wq, 17(0) = uo, and N{0) = xq yo + zq wq uq = n, where 1 < xq < n — 1, yo > 0, %o ^ 0, Wo > 0, uo > 0, and n > 2 . Occasionally we will use the notation X (t) and X for the vectors { X { t),Y { t), Z ( t) ,W { t ) ,U { t) ) and ( x , y , z , w , u ) , respectively.

lation then the probability of one new infection occurring in th e interval [t, t dt] is a X { t ) Y { t ) d t / n + o{dt) where a is the efiective contact rate (as was explained in Sec­ tion 4.1). Among those who get infected a proportion p develop clinical TB w ithin a year after infection (prim ary TB) and the rem aining proportion, 1 — p, become latents; those who develop TB are infectious or non-infectious w ith probabilities q\ and 1 — gi, respectively. T he difference between prim ary and secondary TB (i.e. w hether an in­ fected develops TB w ithin a year or later) could be modelled w ith a time-delay model, b u t in this chapter we will not investigate this possibility and we keep th e stru cture of a basic Markov process.

Latents may develop chnical disease a t some point as a result of endogenous reactivation of an old infection or exogenous reinfection (acquiring a new infection). There have been doubts about w hether exogenous reinfection is possible for TB (see Section 2.3) b u t in the recent hteratu re it has become more certain th a t reinfection is possible and should be taken into account, especially in areas w ith high risk of infection (see, e.g., Vynnycky 1996, Dye et al. 1998). Therefore, the possibility of exogenous reinfection for the latents is included in this model. On the other hand, we assume th a t reinfection is possible only for the latents and not for the non-infectious and recovered cases, since the relapse rates from these two classes (to the infectious class) are very high, so th a t th e effect of reinfection is negligible for these cases.

The reactivation rate is denoted by ^ so th a t the probability of an endogenous reactivation occurring in [t, t+ d t] is (3Z{t)dt-\-o{dt). After reactivation the individual has infectious or non-infectious TB w ith probability Q2 and 1—92, respectively. For exogenous

reactivation we assume th a t the effective contact rate between latents and infectious cases is P r«, where 0 < Pr ^ 1- If Pr = 0 then reinfection is not possible; if Pr = 1 this means th a t past infections confer no im m unity a t all, so th a t latents are equally likely to get infected as the uninfected. This is not the case w ith TB, since infection does provide im munity (at least partial an d /o r tem poral) and therefore pr must be strictly less th an 1. A more realistic approach would be to assume th a t pr is an increasing function of the tim e since infection since most results (see, e.g., Styblo 1991, Dye et al. 1998) suggest th a t im m unity conferred by an old infection wanes in time. This approach though would increase substantially the complexity of the model (since th a t implies keeping track of the tim e since infection for each infected individual) and therefore for simplicity we assume

th a t Pr is constant. After reinfection an individual develops clinical disease w ithin a year (prim ary TB) or remains latent w ith probabilities ps, 1 — ps, respectively; those who develop TB are infectious or non-infectious w ith probabihties 1 — Ç3, respectively. For simplicity we will denote Q2 = P3P r« so th a t the probability of a reinfection leading to prim ary TB occurring in [t,t-\-dt] is a2Z { t) Y { t) d t/ n + o(dt). We assume th a t additional

infections do not change the reactivation rate /3 or the effective contact rate Q2.

Non-infectious TB cases become infectious a t a rate SW . Infectious and non- infectious cases recover spontaneously a t rates 70 and So per capita, respectively, and those who have recovered may relapse later and become infectious or non-infectious cases a t rates eiU and €2U, respectively.

Finally, there is im m igration of susceptibles at a constant rate A, norm al death at rate p per capita, and excess death due to TB at rates p i and p2 (per capita) for the infectious and non-infectious cases, respectively. Individuals w ith latent infection and those who have recovered are healthy and hence there is no excess death for these two classes. A t some points the special case A = p n will be investigated.

T he definitions of the variables and param eters used for this model are sum ­ marised in Table 6.1. The possible transitions and their rates are illustrated in Fig­ ure 6.1.

It has to be noted th a t this formulation assumes th a t the values of çi, 92, Qs may be different in general. This means th a t when an individual develops clinical disease the probability th a t the form of disease will be infectious or non-infectious depends on (a) w hether the individual had an old infection or not (%, Qs for the former, qi for the latter)

(b) for an individual who had an old infection, whether the current incidence of disease is a result of the old infection (endogenous reactivation) or of a new additional infection (exogenous reinfection); the probabilities are Q2 and %, respectively.

In the literature there is not enough evidence to prove either th a t the çi, Ç3 are equal or th a t they are not, and therefore modellers have taken either line (for instance. Dye et al. (1998) assumed th a t qi = Q2 = %; Blower et al. (1995) assumed th a t

qi and % are not equal and % = 0 ) . In this chapter we have used the three different param eters çi, since th a t allows for b o th approaches to be adapted, and in some cases we investigate the situation çi = 92 = 93 as a special case.

X (t)

Y{t)

In document Diseño de un plan de mantenimiento preventivo de motores diesel basado en análisis de aceite (página 174-178)