ORIGINAL ARTICLE TRACING THE THRESHOLD LEVEL OF THE …journalijmrr.com/wp-content/uploads/2014/03/Pandian-27-revised.pdf · ORIGINAL ARTICLE TRACING THE THRESHOLD LEVEL OF THE HIV

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


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

)


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)


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)


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)


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)


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)


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)


= 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)


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)


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

)


= 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)


= 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




increases, the expected time is found decreasing. This is









increases, the expected time is found increasing. This is









increases, the expected time is found increasing. This is




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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ORIGINAL ARTICLE TRACING THE THRESHOLD LEVEL OF THE …journalijmrr.com/wp-content/uploads/2014/03/Pandian-27-revised.pdf · ORIGINAL ARTICLE TRACING THE THRESHOLD LEVEL OF THE HIV