A STOCHASTIC MODEL FOR EXPECTED TIME TO SEROCONVERSION UNDER CORRELATED INTERCONTACT TIMES USING EXPONENTIAL-GEOMETRIC DISTRIBUTION
*R. Kannan and **K. Chandrasekar
*Professor and **Research Scholar
Department of Statistics
Annamalai University
Annamalai nagar – 608 002
Tamil Nadu
Email id: *
ABSTRACT
This paper focuses on the study on a stochastic model for predicting the seroconversion time of HIV transmission under correlated intercontact time. Every individual’s immune capacity as varies from person to person and also the antigenic diversity threshold is different from person to person. A stochastic model assuming that intercontact times between successive contacts as correlated random variables are proposed. Sathiyamoorthi5 has been studied the shock models with correlated intercontact time. Shock models with intercontact time have been obtained by Sathiyamoorthi and Kannan7 assuming the threshold distribution as exponential. In this paper, it is assumed that threshold follows exponential-geometric distribution. The result of Gurland3 has been used for developing this model. The mean time to seroconversion and its variance are derived and the numerical illustrations are provided.
Key words
Human Immunodeficiency Virus, Acquired Immuno Deficiency Syndrome, Antigenic Diversity Threshold, Intercontact times, Seroconversion
Introduction
In the study of HIV infection, transmission and the spread of AIDS is quite common in the use of stochastic models. The transmission of HIV is possible through homo or heterosexual contacts, transfusion of infected blood product, use of unsterile needles and mother to fetus. The per contact transmission probability is known as infectivity. The expression for hazard rate and prevalence function using the available data from the partner studies have been obtained by Shiboski and Jewell8. The antigenic variation would be on the increase when the more HIV gets transmitted from the infected person to uninfected person. The time to seroconversion from the point of infection depends upon what is known as antigenic diversity, which acts against the immune ability of an individual. Stilianakis et al9 and Nowak and May5 has been discussed by the antigenic diversity threshold model.
In the estimation of expected time to seroconversion, there is an important role for the inter-arrival times between successive contacts and it has a significant influence. Sathiyamoorthi and Kannan7 have obtained the expected time to seroconversion under the assumption that the inter-arrival times between the contacts are not independent but constantly correlated. Any person who has contacts with unknown partners is likely to have the fear of getting infected. However, the person does not avoid the contacts. But there is a possibility that the person can postpone the event namely the contact due to fear complex.
A stochastic model to estimate the expected time to seroconversion and variance of the seroconversion time derived under the assumption that the inter-arrival times between contacts may be treated as correlated random variables and the threshold distribution follows exponential-geometric distribution is discussed. For a detailed study of exponential-geometric distribution, one can refer to Adamidis and Loukas1. Shock model with correlated intercontact times has been studied by Sathiyamoorthi6. In developing this model the results of Gurland3 has been used. In this study the theoretical results are substantiated using numerical data simulated.
Assumptions of the model:
· Sexual contact is the only source of transmission
· When an uninfected individual has sexual contact with a HIV infected partner, a random variable of HIV gets transmitted.
· An individual is exposed to a damage process acting on the immune system and damage is assumed to be linear and cumulative.
· The intercontact timings between successive contacts are not independent but are correlated.
· The total damage caused exceeds the threshold level Y which is itself a random variable. The seroconversion occurs and the person is recognized as a seropositive.
· The process with generated contacts, the sequence of damages and threshold are mutually independent.
Notations:
Xi Increase in the antigenic diversity arising due to the HIV transmission during the ith contact. We assume that are continuous i.i.d random variables, with p.d.f g(.) and c.d.f G(.).
Yi A random variable representing antigenic diversity threshold follows exponential-geometric distribution with parameter and p with p.d.f h(.) and c.d.f H(.).
Ui a continuous random variable denoting the inter-arrival times between successive contacts with p.d.f f(.) and c.d.f F(.).
gk(.) denotes the p.d.f of the random variable,
is correlation coefficient between xi and xj, ij
Vk(t) Probability of exactly k contacts in [0,t]
T a continuous random variable denoting the time to seroconversion with p.d.f l(.) and c.d.f L(.).
l*(s) is the Laplace transform of l(t)
f*(s) is the Laplace transform of f(t)
Result
S(t) = P(T t)
The threshold variable Y has exponential-geometric distribution with parameter and p, so that
If we assume that the intercontact timings between contacts follow exponential distribution with parameter , then
and
S(t) =
L(t) = 1 – S(t) is called the prevalence function as mentioned in Shiboski and Jewel8.
...(1)
If the intercontact timings are independent, it is easy to obtain the joint distribution of their sum, but if they are correlated the determination of the distribution of is very complex in any general case but Gurland3 has obtained of
When Ui’s form a sequence of exchangeable constantly correlated random variables, each having exponential distribution with p.d.f
such that the correlation coefficient between any Xi and Xj, is
...(2)
where and
The Laplace transform of the density function of Zk is given by
... (3)
Taking the pdf of the prevalence function L(t), we get
l (t) =
Laplace Transformation of l(t) is given by
l*(s) =
where,
(on simplification)
and
(on simplification)
Taking first order differentiation
and
and second order differentiation
and
Hence
(on simplification)
and
(on simplification)
Therefore,
(on simplification)
If
... (4)
It may be observed that the expected time to seroconversion remains unaffected even if the intercontact timings are correlated but the variance is a function of If put in equation (4) the expression for the variance coincides with that of the result in uncorrelated case obtained by R. Kannan and K. Chandrasekar4.
Numerical Illustrations:
Table 1
/ p=0.5, = 2, = 0.1E(T) / V(T)
= 0.3
K=2 / K=5
1 / 0.03304 / 0.20615 / 0.01982
2 / 0.01652 / 0.05154 / 0.00496
3 / 0.01101 / 0.02291 / 0.0022
4 / 0.00826 / 0.01289 / 0.00124
5 / 0.00661 / 0.00825 / 0.00079
6 / 0.00551 / 0.00573 / 0.00055
7 / 0.00472 / 0.00421 / 0.00041
8 / 0.00413 / 0.00322 / 0.00031
9 / 0.00367 / 0.00255 / 0.00025
Figure 1
Table 2
/ p=0.5, = 2, = 2E(T) / V(T)
= 0.3
K=2 / K=5
0.1 / 0.00826 / 0.05154 / 0.00496
0.2 / 0.01588 / 0.09911 / 0.00953
0.3 / 0.02295 / 0.14318 / 0.01377
0.4 / 0.02951 / 0.18416 / 0.01771
0.5 / 0.03564 / 0.22239 / 0.02138
0.6 / 0.04137 / 0.25816 / 0.02482
0.7 / 0.04675 / 0.29172 / 0.02805
0.8 / 0.05181 / 0.32327 / 0.03108
0.9 / 0.05657 / 0.35302 / 0.03394
Figure 2
Table 3
/ θ = 0.1, = 2, = 2E(T) / V(T)
= 0.3
K=2 / K=5
0.1 / 0.011303 / 0.07053 / 0.006782
0.2 / 0.010653 / 0.066478 / 0.006392
0.3 / 0.009944 / 0.062053 / 0.005967
0.4 / 0.009156 / 0.057135 / 0.005494
0.5 / 0.008259 / 0.051538 / 0.004956
0.6 / 0.007210 / 0.044993 / 0.004326
0.7 / 0.005950 / 0.03713 / 0.00357
0.8 / 0.004400 / 0.027456 / 0.00264
0.9 / 0.002459 / 0.015342 / 0.001475
Figure 3
Table 4
/ θ = 0.1, = 0.5, = 2E(T) / V(T)
= 0.3
K=2 / K=5
1 / 0.015882 / 0.099105 / 0.009529
2 / 0.008259 / 0.051538 / 0.004956
3 / 0.005581 / 0.034826 / 0.003349
4 / 0.004215 / 0.026299 / 0.002529
5 / 0.003386 / 0.021126 / 0.002031
6 / 0.002829 / 0.017654 / 0.001697
7 / 0.002430 / 0.015162 / 0.001458
8 / 0.002129 / 0.013286 / 0.001278
9 / 0.001895 / 0.011824 / 0.001137
Figure 4
Table 5
/ θ = 0.1, = 0.5, = 2, β = 2V(T)
K=2 / K=5
0.1 / 0.04757 / 0.0446
0.2 / 0.04823 / 0.04522
0.3 / 0.05154 / 0.05056
0.4 / 0.05749 / 0.05282
0.5 / 0.06608 / 0.07782
0.6 / 0.07731 / 0.11893
0.7 / 0.09118 / 0.20318
0.8 / 0.1077 / 0.31055
0.9 / 0.12686 / 0.44105
1.0 / 0.14867 / 0.59467
Figure 5
Conclusions:
Ø From the table 1, we observe that for fixed p, θ, β and ρ, when μ increases which means that the average inter-arrival time become smaller, so the mean time to seroconversion decreases and also variance time to seroconversion decreases. There is no impact of ρ on the mean time to seroconversion.
Ø From the table 2, we observe that for fixed μ, p, β and ρ, when θ which is the parameter of the random variable denoting the contribution to the antigenic diversity increases then it is seen that the mean time to seroconversion and variance time to seroconversion increases.
Ø From the table 3, we observe that for fixed μ, β, θ and ρ, when p which the parameter of the threshold distribution increases, the mean time to seroconversion as well as the variance time to seroconversion decreases.
Ø From the table 4, we observe that for fixed μ, p, θ and ρ, when β which is the parameter of the threshold distribution increases, the mean time to seroconversion as well as the variance time to seroconversion decreases.
Ø From the table 5, it is observe that for fixed μ, β, θ and p, when ρ which is the constant correlation between successive contacts increases then the variance time to seroconversion increases.
Acknowledgement
The authors are immensely thankful to Dr. R. Ramanarayanan, Professor of Mathematics, Chennai, Dr. R. Sathiyamoorthi, Professor of Statistics, Annamalai University and Dr. G. S. Harisekharan, WIPRO, Chennai, for their invaluable suggestions, guidance and moral support.
References:
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3. Gurland, J. (1955) ‘Distribution of the maximum of the arithmetic mean of correlated random variables’, Ann. Mathematical Statistics, 26: 294-300.
4. Kannan, R. and Chandrasekar, K. (2013) ‘A Stochastic Model for Estimation of Expected Time to Seroconversion of HIV Infected Using Exponential-Geometric Distribution’, Bioscience Research Bulletin, 29: 29-38.
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