Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
A STOCHASTIC APPROACH TO DETERMINE EXPECTED TIME TO CROSS
ANTIGENIC DIVERSITY THRESHOLD OF HIV INFECTED USING LARGEST
ORDER STATISTICS
S.C. PREMILA & S. SRINIVASA RAGHAVAN
Mar Gregorious College of Arts & Science, Chennai – 37
Vel-tech Dr. R.R. & Dr. S.R. Technical University, Avadi, Chennai.
ABSTRACT
This paper focuses on the stochastic approach to determine the expected time to cross antigenic diversity
threshold of HIV infected using order statistics. In the estimation of expected time to cross the antigenic diversity
threshold, there is an important role for the interarrival time between successive contacts and it has a significant
influence. We propose a stochastic model under the assumption that the interarrival time between contact form
order statistics. In developing such a stochastic model the concept of shock model and cumulative damage process
are used. Numerical illustration are also given using simulated data.
Keywords:
Human Immuno Deficiency virus. Acquired Immuno Deficiency Syndrome, Antigenic Diversity Threshold,
Order Statistics, Seroconversion.
Introduction:
In the study of HIV infection and AIDS, one of the important aspect of investigation is the estimation of
time to cross antigenic diversity threshold of HIV infected. The HIV can be transmitted through a variety of
contact mechanism that include homo or hetro sexual contacts, transfusion of infected blood product, needle
sharing among intravanion drug abuses and mother to fetus.
The most common means of spread of this infection is only by sexual contacts. If more and more of HIV
are getting transmitted from the infected person to the uninfected, the antigenic variation would be on the increase.
If the antigenic diversity crosses a particular level which is known as the antigenic diversity threshold, the immune
system collapses and seroconversion takes place, for a detailed study of antigenic diversity threshold and its
estimation one can refer to Nowak and May (1991) and Stilianaleis et al. (1994).
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
In the estimation of expected time to seroconversion there is an important role for the interarrival time
between successive contact and it has a significant influence Sathyamoorthi (2001) and Kannan et al (2007) have
obtained the expected time to seroconversion using the shock models and cumulative damage process as discussed
in Essary et al (1973). Using the results of Gurland (1953), they have obtained the expected time to seroconversion
assuming that the interarrival time between successive contacts are correlated random variables.
Ratchagar etal (2003) have derived a model for the estimation of expected time to seroconversion of HIV
infected using order statistics. In this paper it is assumed that threshold follows geometric distribution and
interarrival time form an order statistics and so they are not independent.
This is due to the fact that the smallest order statistics is taken, it implies that the interarrival are becoming
smaller. Hence frequent contact would be possible which will have impact on the time to seroconversion. If the
largest order statistics is taken, it implies that the interarrival time are becoming larger. Hence, frequent contact
would not be possible which will have it’s impact on the time to seroconversion. Numerical illustrations are
provided using simulated data.
Assumptions of the Model:
 The transmission of HIV is only through sexual contacts.
 An uninfected individual has sexual contacts with HIV infected partner, and a random number of HIV are
getting transmitted, at each contact.
 An individual is exposed to a damage process acting on the immune system and the damage is assumed to
be linear and cumulative.
 The interarrival times between successive contacts are taken to be identically and independently distributed
random variables.
 The sequence of successive contacts and threshold level are independent.
 From the collecton of large number of interarrival times between successive contacts of a person, a random
sample of ‘k’ observations are taken.
Notations
Xi – A random variable denoting the amount of contribution to antigenic diversity due to the HIV transmitted in
the ith contact, in other words the damage caused to the immune system in the ith contact with p.d.f.g (.) and c.d.f
G(.).
Y – A random variable representing antigenic diversity threshold which follows geometric distribution with
parameter θ.
T: A random variable denoting the time to seroconversion.
Pn = P (xi = n) the probability that ‘n’ particles or HIV are transmitted during the ith contact.
Ui : A random variable denoting the interarrival time between successive contacts with p.d.f. f(.) and c.d.f F(.).
Φ(S) =
k
∞
𝑘=0 PkS
is the p.g.f of x.
Vk (t) = Probability of exactly k contacts in (0, t).
θ=1–θ
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
U(n) – The largest order statistics with p.d.f. fu(m)(t)
f*(S) – Laplace stieltjes transform of f(t)
l*(S) – Laplace stieltjes transform of l(t).
Results:
S(t) = P (T > t)
=
∞
𝑘=0 Pr
{there are exactly k contacts in (0, t)}
X Pr {the cumulative total of antigenic diversity < y}
=
∞
𝑘=0 Vk(t)
∞
𝑘=0 Xk<
P
y
Let P (y = 1) = θ represent the probability that the threshold level is equal to one. P (y = 2) = θ,θ which implies that
the probability the conversion takes place only when y = 2.
Similarly, Pj = P (y = j) = θ (θ)j-1
The probability generating function ϕ is
Φ(S) =
Sθ
k
∞
𝑘=0 PkS
k-1
∞
𝑘=0(S 𝜃)
P(X < Y) =
k-1
∞
𝑘=1 θ θ
=
-
(4.1)
-
(4.2)
𝑆𝜃
= 1−𝑆𝜃
1
∞
𝑛 =1 Pn
(P (y > n))
∞
𝑛=1 P
Let P(y > n) =
Sk
(y=n+i)
P (y > n) = θn
P (X < Y) =
∞
𝑛 =1 P’n
(θ)n = Ψ (θ)
Which is the probability that an individual is not getting infected in a single contact. The probability that the
cumulative damage has not crossed the threshold level in k contacts is equal to Sk and
P (X1 + X2 + … + Xk < Y) = [Ψ(θ)]k
Now 1 – Sk = 1 – [Ψ(θ)]k is the prevalence function mentioned in Jewell and Shiboski (1990).
Let F(t) be the distribution function of a random variable Ui which is the interarrival time between contacts and
Ui’s are i.i.d random variables having exponential distribution with parameter C.
Vk(t) = P [U1 + U2 + ……. + Uk < t < U1 + U2 + Uk+1]
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
P (k contacts in (0, t]) = (Fk (t) – Fk+1(t)], where Fk(t) is the distribution function of
U1 + U2 + U3 …… + Uk
∞
𝑘=0 Vk(t)
S(t) =
=
[Ψ(θ)]k
∞
𝑘=0 Vk(t)
[Ψ(θ)]k
Let L(t) = 1 – S(t)
= [1 – (Ψ(θ))]
k-1
∞
Fk
𝑘=1[Ψ(θ)]
(t)
k-1
∞
f*(s)}
𝑘=1[Ψ(θ)]
The Laplace stieltjes transform of L(t) is L*(S) = {[1– (Ψ(θ))]
=
1-Ψ(θ) f*(s)
1-Ψ(θ)f*(s)
The inter-arrival times U1, U2, U3, … UN are i.i.d random variables and U(1) < U(2), … < U(N)form k order
statistics which are random variables that are not independent.
The largest order statistics is fu(k)(t) = k(F(t)]k-1 f(t) -
(4.4)
The Laplace transform of fu(k)(t) is
f*u(k) (t) =
∞ -st
𝑒
0
k[F(t)]k-1 f(t)dt
Assuming that f(t) follows exp (θ). It can be shown that
f* u(k)(t) =
K!θk
(θ+s)(2θ+s)(3θ+s)……(kθ+s)
-
(4.5)
When k = 1, 2, …………
𝜃
f* U (1)(S) = 𝜃+𝑆
f* U(2)(S) =
2!θ2
(θ+S) (2θ+S)
Therefore in general
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
f* U(k)(S) =
K!θk
(θ+s)(2θ+s)……(kθ+s)
-
(4.5)
Substituting (4.5) in (4.3) and it can be shown that
(1-ψ(θ))
(k1θkk))
(k!θ
(θ+S)(2θ+S)……(kθ+S)
(θ+s)(2θ+s)……(kθ+s)
- (4.6)
l* (S) =
1-ψ(θ)
k
k!θ
(θ+s)(2θ+s)……(kθ+s)
From (4.1) we have, if Pn = P (x=n) = (1/2)n
ψ(θ)
=
1 1
∞
1 Pn θ
=
n
∞
𝑛=1(t)
=
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1− 𝜃
θn
- (4.7)
1+𝜃
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
Substituting (4.7) in (4.6)
1-θ
1+θ
1-
(k!θk)
(θ+s)(2θ+s)……(kθ+s)
l* (S) =
1-θ
1+θ
1-
k!θk
(θ+s)(2θ+s)……(kθ+s)
2θk! θk
(1+θ)(θ+s)(2θ+s)……(kθ+s)- (1-θ) k!θk
=
-dl*(S)
ds
ds
E(T) =
On simplification
S=0
d
k!θk
ds ((1+θ)(θ+s)(2θ+s)……(kθ+s)-(1-θ) k!θk)
=-2θ
2θk!θk
(1+θ)
(2θ.3θ…kθ)+(θ…3θ-4θ…kθ)
+…+(θ…2θ-3θ…(k-1)θ
=
(θ+s)(2θ+s) ……(kθ+s)-(1-θ) k!θk
=
1+θ
2θ
𝑘
𝑖=1 1 /𝑖
2
-
(On simplification)
-
4.8
Mean time to seroconversion is
E(T) =
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1+θ
2θ
𝑘
𝑖=1 1 /𝑖
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
It implies that E(T) becomes larger as k increases since
𝑖/1 increases with k. Hence it may be concluded that the
maximum value U(k) increases with increase in k and also the interarrival time U(k) becomes larger.
2
E(T )=
d
ds
2θk!θk (2θ+s)(3θ+s)…(kθ+s)+(θ+s)(2θ+s)
…(kθ+s)+…+(θ+s)(2θ+s)…(k-1)θ+s]
((1+θ)(θ+s)(2θ+s)……(kθ+s)-(1-θ) k!θk)2
=
2
𝑘
𝑖=1 1 /i
2(1+θ)2
4θ2
1+θ
- 4θ
𝑘
𝑖=1
1 /i
2
𝑘
𝑖=1
1+θ
+
4θ
1 /i
2
2 24θ
4θ2 𝑘𝑖=1 1 /i2 2 (1+θ)
I (1+θ)
i
2
2
2
4θ
i 2θ 4θ
2θ
2
2
4θ
4θ
I 2 (1+θ)
4θ 2
i
𝑘
- (3.9)
E(T2) = 2
𝑖=1 1 /i
2
4θ
i
4θ
2θ
4θ2 4θ2
I
4θ
i
2 of seroconversion time is V(T) = E(T2) – (E(T))2
Then the variance
4θ
i
4θ
2
4θ& (4.9) we obtain
By using (4.8)
=
V(T) = E(T2) – (E(T))2
2
𝑘
𝑖=1 1 /i
4θ2
4θ2
4θ2
=
=
𝑘
(1+θ) 2 𝑖=1 1 /i
2θ2
I 2 4θ 𝑘 4θ2 2 i
(1+θ)
(1+θ)
1/𝑖]
2θ2
i4θ2 4θ [ 𝑖=1
4θ2
4θ2
(1+θ)2
4θ2
=
[
(1+θ) 2
2θ2
2
(1+θ)
2θ2
I
i
2
4θ
4θ
i
2
𝑘
𝑖=1 1/𝑖]
[
2
𝑘
𝑖=1 1/𝑖]
>0
- (4.10)
This implies that for fixed θ the variance of t increases as k increases
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
Numerical Illustrations
Table 4.1
Values of E(T) & V(T) w.r. to k
Θ = 0.5
K
E(T)
V(T)
2
2.2500
5.0625
3
2.7000
7.2900
4
3.0750
9.4556
5
3.3750
11.3906
6
3.6250
13.1769
7
3.8444
14.7456
8
4.0275
16.2208
9
4.1942
17.5896
10
4.3440
18.8703
11
4.4804
20.1152
25.0000
20.0000
15.0000
Θ = 0.5
10.0000
E(T)
V(T)
5.0000
0.0000
2
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3
4
5
6
7
8
9
10
11
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
Table 4.2
Values of E(T) & V(T) w.r. to θ
K=2
θ
E(T)
V(T)
0.5
2.2500
5.0625
1.5
1.2495
1.5613
2.5
1.0500
1.1025
3.5
0.9600
0.9216
4.5
0.9150
0.8372
5.5
0.8850
0.7844
6.5
0.8700
0.7582
7.5
0.8500
0.7310
8.5
0.8400
0.7056
9.5
0.8382
0.6806
6.0000
5.0000
4.0000
k=2
3.0000
E(T)
V(T)
2.0000
1.0000
0.0000
0.5
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1.5
2.5
3.5
4.5
5.5
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Asia Pacific Journal of Research
Vol: I. Issue XXXVI, February 2016
ISSN: 2320-5504, E-ISSN-2347-4793
Conclusion
In the case of the interarrial times distributed as largest order statistics, the following observations can be
made. In table 1 the value of ‘k’ which is the parameter of the threshold distribution increases E(T) also increases
so also V(T).
In table 2 the value of θ which is the parameter of distribution of random variable y namely H(y) denoting
the interarrival times. It shows that if the value of a increases then the value of E(T) and V(T) decrease. This is due
to the fact that the average of the interarrival times becomes smaller so E(T) decreases and also V(T).
References:
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1(4), 627-649.
 Gurland, J. (1955). Distribution of the maximum of the arithmetic mean of correlated random variables.
Ann. Math. Statist., 26, 294-300.
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 R. Kannan, A. Ganesan and A. Thirumurugan (2009) stochastic model for estimation of expected time to
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