1.3 UMTS: SISTEMA UNIVERSAL DE TELECOMUNICACIONES MÓVILES
1.3.2 ARQUITECTURA DE RED UMTS
1.3.2.1 Principales Componentes de la arquitectura UMTS R’99:
0 .3 06 7,... ,0.5994 77o < 1, 77i > 1 01 stable 02 feasible, stable 0.5 995,... ,1 77g > 1, 77i > 1 01 unstable 02 feasible, stable
Table 6.5: Feasibility and stability of the deterministic equilibria in a subspace of the parameter space: grid for qi
{/3) Close to the point where 7^i — 1 changes sign or changes between being positive and pure complex (and as long as IZq < 1) the determ inistic system tends to 0% w ith both initial conditions xJ, Xg. For instance th a t happens w ith 7g = 0.3244 (7^i > 1 ) and
7g = 0.3245 (771 non-real) and also for p = 0.07. Table 6.6 shows th a t the value p = 0.07 is close to the point where 77i changes from being non-real to real and greater th a n one.
It seems therefore th a t close to the point where 77i — 1 changes behaviour (and as long as 77o < 1) the system is attracted to 01; in most of the numerical solutions this happened a t a slower rate th an when the corresponding param eter value (th at was varied) was far away from th a t point. O n the other hand, close to the point where 77q — 1 changes sign (and as long as 77i remains greater than 1) the system is attracted to 02- In some cases it is more accurate to say th a t the system is ultim ately attracted to 02; Figure 6.3 shows the value of z over a period of 6000 years ((a) shows the first 1000 years and (b) the whole period) for a system th a t begins from xJ = (n — 1,0,1,0,0) w ith the
V alue o f p S ta b ility 0.0001,... ,0.0672 7^0 < 1, 97.1 not real 01 stable 02 infeasible 0.0673,... ,0.1057 TZq < 1, 7^1 > 1 01 stable 02 feasible, stable 0.1058,... ,1 TZq > 1, 7Zi > 1 01 unstable 02 feasible, stable
Table 6.6: Feasibility and stability of the deterministic equilibria in a subspace of the parameter space: grid for p
(a) (b)
T im e t
T im e t
Figure 6.3: The value of z(t) as obtained from numerical solution of the system (6.1). (a) shows the first 1000 years and (b) the first 6000 years. The parameter values are as shown in (6.16) and 7o = 0.1588. The initial conditions are xo = n - 1 and zq = 1.
param eter values shown in (6.16) and 70 = 0.1588. In this case TZq < 1 and 7Zi > 1 and both 6i and 02 are stable. Initially the value of z decreases and reaches its minimum at z(226) = 0.2023. Then it starts increasing slowly and after 3500 years it jum ps to a peak of z = 33.2 and then drops to the endemic value = 27.38. A similar behaviour was observed for y, and u. It has to be stressed th a t from these results it appears that the final equilibrium may not be so significant for any practical purposes, since it is reached after a very long time and for such a long timescale it is unreasonable to assume that the values of the parameters remain the same.
Summarising the results for th e equilibrium of th e determ inistic system, we have found the following:
• T he determ inistic system has three equilibrium points: the disease-free equilibrium e i, and two endemic equilibria 62 and 63.
• e i is locally stable when 7 ^ < 1 and unstable when TIq> 1.
• 62 is feasible when 7^i > 1 and infeasible otherwise. T he numerical results presented in this section prove th a t 62 is stable in some subsets of th e param eter space where 7^i > 1
(and w ith either %o > 1 or %o < 1) and suggest th a t maybe this is the case throughout the region > 1 (even when 72,o < 1).
• 63 is unstable in th e space where it is feasible.
Finally it has to be noted th a t the fact th a t the endemic equilibrium can be stable even when 7 ^ < 1 has serious implications for the control of the disease. If public health policies aim a t reducing 'Rq in order to control th e disease, then for TB this is not enough: reducing 72.q to a value less th a n one makes the disease-free equilibrium stable, b u t still the disease may not tend to extinction (depending on the initial conditions) if
7^1 is still greater th an one. Therefore R \ has to be reduced (to a value less th a n 1) as well, in order to "guarantee" the extinction of th e disease. In any case, though, the time until extinction (if extinction is achieved) can be very long, as the results in the following sections will show.
6.3
T h e stoch astic m odel
6 .3 .1 T h e tr a n s ie n t p h a se
Let Px(^) = p(a;, y, z, w, u; t) be the probability th a t there are x uninfected individuals, y infectious cases, z latents, w non-infectious cases, and u recovered cases in the population at tim e / > 0:
Px{t) = p { x , y , z , w , u ; t ) = P[X{t) = x , Y { t ) = y ,Z { t ) = z , W { t ) = tu, U{t) = u], (6.17)
for t > 0, X e 5 = Z^., and Px(t) = 0 otherwise. T he initial conditions are Pxo(O) =
1 and Px(0) = 0 for any x xq where xq = (xo,yo,zo,wo,UQ) 6 «So as defined in (6.3). T he corresponding Kolmogorov forward equations for Px(^) are given in th e Ap pendix, equation (A.7). The joint probability generating function P (0 i, 02,0 3 ,04,05; t) =
satisfies the equation d V dV dV
+ [(/i + /^i)(l — ^2) +70(^5 — ^2)]^^
dV + [m(1 ~ ^3) + Q2 ^ ( ^ 2 — ^3) 4- (1 — g2)/?(^4 — ^3) ] ^ ^ dV + [(/^ + /U2) ( l — O4) + 0 ( 6 2 — Oa) + #o(#5 — ^4) ) ^ ^ (6.18) dV + [/^(l “ ^5) + fl(#2 — ^5) + (2(^4 — ^3) ] ^ ^ 4— 02[~&i 4- pg i^2 4- (1 — p)^3 4 -p (l — 91)^4] n d9i d0 2 4— - ^2[93^2 4- (1 — 93)^4 — ^3] n 0 6 2 8 6 3 ’w ith the initial condition V (6 1,6 2,6 3,6 4,6 3] 0) = 6^^6 2^6^°6^ °6^ .
From equation (6.18) a system of diflFerential equation for the first and second moments of X , Y , Z^W, and Uis deduced; the equations for th e means are the following