original article tracing the threshold level of the...
TRANSCRIPT
ORIGINAL ARTICLE
TRACING THE THRESHOLD LEVEL OF THE HIV INFECTED PATIENTS THROUGH
STOCHASTIC MODEL
1P.Pandiyan,
2V.S.Bhuvana,
3K.Kannadasan and
4R.Vinoth
1,3Department of Statistics, Annamalai University, Annamalai Nagar.
2Department of Statistics, Manonmaniam Sundaranar University, Tirunelveli. 4Department of Community Medicine, Shri Sathya Sai Medical College and Research Institute, Kanchipuram.
Article History: Received 2 th Jan,2014, Accepted 5 thFeb,2014,Published 25th Feb,2014
ABSTRACT
HIV/AIDS has also been, until recently, the leading cause of death among young adults. Statistical tools were derived from the
firmly established theory of epidemic modeling, although some adjustments became necessary, because of specific
characteristics of HIV infection. In this paper Shock model approach to estimate the threshold level is been derived by Three
parameter generalized exponential distribution. The numerical illustrations are also used to support the model development.
Keywords: Estimation, HIV/AIDS, Three parameter generalized exponential distribution, Epidemic Model.
1.INTRODUCTION
There is now considerable variation in the timing and intensity
of the HIV epidemic in different regions of the world. Intuition
and exploratory work in statistical disease models suggest that
sexual partnerships will amplify the spread of an infectious
agent such as HIV. From the viewpoint of the virus, there is less
time lost after transmission occurs in waiting for the current
partnership to dissolve, or between the end of one partnership
and the beginning of another.
This new family of distribution functions is always positively
skewed, and the skewness decreases as both the shape
parameters increase to infinity. Interestingly, the new three-
parameter generalized exponential distribution has increasing;
decreasing, Uni-modal and bathtub shaped Hazard Functions.
One can see for more detail in Sathiyamoorthy (1980), Pandiyan
et al., (2010), Pandiyan et al., (2012) about the expected time to
cross the threshold level of the seroconversion.
ASSUMPTIONS
A person is exposed to HIV infection. At every epoch
of contact with an infected there is some contribution
to the antigenic diversity.
Anti Retroviral Therapy is administed to the infected.
There is a particular level of antigenic diversity of the
invading, and it is called the antigenic diversity
threshold. If antigenic diversity crosses this threshold
the seroconversion takes place.
The interarrival times between the successive contacts
are random variables which are identically
independently distributed.
ATIONS
A continuous random variable denoting the amount of
contribution to the 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 (.).
A continuous random variable denoting the threshold for
two components which follows three parameter
generalized exponential distribution
The probability density functions of Xi
Laplace transform of g (.)
The k- fold convolution of g (.) i.e., p.d.f. of
Laplace transform of
A random variable denoting the inter-arrival times
between contact with c.d.f. ,
p.d.f. of random variable denoting between successive
contact with the corresponding c.d.f. F (.)
The k-fold convolution functions of F (.)
The survivor function, i.e.
1 - S (t)
Probability that there are exactly k contacts.
Volume 2, Issue 2, pp 107-111 February,2014
107
DESCRIPTION OF STOCHASTIC MODEL
It can also be proved that
Probability that exactly k decision epochs in (0,t] and the combined threshold level is not crossed
Therefore on simplification it can be shown that
Using convolution theorem for Laplace transforms, and on simplification, it can shown that,
Taking Laplace transformation we get
Then
Similarly
By taking Laplace-Stieltjes transform, it can be shown that
Let the random variable denoting inter arrival time which follows exponential with parameter c. Now ,
substituting in the above equation we get
P.Pandiyan et al., 2014
108
MODELING THE PERFORMANCE MESURES
We know that
Now, ,
on simplification we get,
On simplification
Volume 2, Issue 2, pp 107-111 February,2014
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Vari
an
ce V
(T)
Inter arrival time-c
= 0.5
= 1
= 1.5
= 2
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Exp
ecte
d t
ime
E(T
)
Inter arrival time-c
1 = 0.5
1 = 1
1 = 1.5
1 = 2
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Var
ian
ce V
(T)
Inter arrival time-c
1 = 0.5
1 = 1
1 = 1.5
1 = 2
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Expecte
d t
ime E
(T)
Inter arrival time-c
2 = 0.5
2 = 1
2 = 1.5
2 = 2
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Expecte
d t
ime E
(T)
Inter arrival time-c
1 = 0.5
1 = 1
1 = 1.5
1 = 2
0 2 4 6 8 10
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Vari
an
ce V
(T)
Inter arrival time-c
1 = 0.5
1 = 1
1 = 1.5
1 = 2
0 2 4 6 8 10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Expecte
d t
ime E
(T)
Inter arrival time-c
2 = 0.5
2 = 1
2 = 1.5
2 = 2
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Expecte
d t
ime E
(T)
Inter arrival time-c
= 0.5
= 1
= 1.5
= 2
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Vari
an
ce V
(T)
Inter arrival time-c
2 = 0.5
2 = 1
2 = 1.5
2 = 2
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Vari
an
ce V
(T)
Inter arrival time-c
2 = 0.5
2 = 1
2 = 1.5
2 = 2
Fig. 1a
Fig. 2a Fig. 2b
Fig. 3a Fig. 3b
Fig. 4a Fig. 4b
Fig. 5a
P. Pandiyan et al., 2014
Fig. 1b
Fig. 5b
110
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Exp
ecte
d t
ime
E(T
)
Inter arrival time-c
= 0.5
= 1
= 1.5
= 2
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Vari
an
ce V
(T)
Inter arrival time-c
= 0.5
= 1
= 1.5
= 2
2.NUMERICAL ILLUSTRATIONS In seen the following tables and figures these performance
measures are calculated by varying the parameters one at a
time and keeping the parameters and fixed.
3.CONCLUSIONS When is kept fixed with other parameters and
the inter-arrival time which follows Exponential
distribution, is an increasing parameter. Therefore, the value
of the expected time to cross the threshold of
seroconversion is decreasing, for all cases of the parameter
value when the value of the parameter
increases, the expected time is also found increasing, this is
observed in Figure 1a and the same case is found in variance
which is observed in Figure 1b.
When is kept fixed with other parameters and
the inter-arrival time increases, the value of the
expected time to cross the threshold of seroconversion
is found to be decreasing, in all the cases of the parameter
value When the value of the parameter
increases, the expected time is found decreasing. This is
indicated in Figure 2a and the same case is observed in the
threshold of seroconversion of variance which is
observed in Figure 2b.
When is kept fixed with other parameters
and the inter-arrival time increases, the value of the
expected time to cross the threshold of seroconversion
is found to be decreasing, in all the cases of the parameter
value When the value of the parameter
increases, the expected time is found decreasing. This is
indicated in Figure 3a and the same case is observed in the
threshold of seroconversion of variance which is
observed in Figure 3b.
When is kept fixed with other parameters
and the inter-arrival time increases, the value of the
expected time to cross the threshold of seroconversion
is found to be decreasing, in all the cases of the parameter
value When the value of the parameter
increases, the expected time is found increasing. This is
indicated in Figure 4a and the same case is observed in the
threshold of seroconversion of variance which is
observed in Figure 4b.
When is kept fixed with other parameters
and the inter-arrival time increases, the value of the
expected time to cross the threshold of seroconversion
is found to be decreasing, in all the cases of the parameter
value When the value of the parameter
increases, the expected time is found increasing. This is
indicated in Figure 5a and the same case is observed in the
threshold of seroconversion of variance which is
observed in Figure 5b.
4.REFERENCES
Kundu, D. and Gupta, R. D. 2005. Estimation of P(Y < X)
for Generalized Exponential Distribution, Metrika,
61(3), 291-308.
Esary, J.D., Marshall, A.W. and Proschan, F. 1973. Shock
models and wear processes. Ann. Probability, 1(4),
pp.627-649.
Gupta, R.D. and Kundu, D. 1999. Generalized Exponential
Distribution, Austral. N. Z. Statist. 41(2), pp.173-188.
Pandiyan, P., Agasthiya, R., Palanivel, R.M., Kannadasan,
K. and Vinoth, R. 2010. “Expected time to attain the
threshold level using Multisource of HIV
Transmission-Shock Model Approach”, Journal of
Pham tech research, 3(2):1088-1096.
Pandiyan, P., Bhuvana, V.S. and Vinoth, R. 2012. “Model
for calculated transmission of HIV threshold using
particular distribution”, International journal of
advanced scientific research and technology, 3:161-
167.
Gupta, R. D. and Kundu, D. 2001b. Generalized Exponential
Distributions: Different Methods of Estimation, J.
Statist. Comput. Simulations, 69(4), 315-338.
Sathiyamoorthi, R. 1980. Cumulative Damage model with
Correlated Inter arrival Time of Shocks. IEEE
Transactions on Reliability, R-29, No.3.
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