A Stochastic Model for the Estimation of Time to. Seroconversion of HIV Transmission Using. Geometric Distribution

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1 Applied Mathematical Sciences, Vol. 8, 014, no. 157, HIKARI Ltd, A Stochastic Model for the Estimation of Time to Seroconversion of HIV Transmission Using Geometric Distribution S. C. Premila and S. Srinivasa Raghavan* Department of Mathematics Mar Gregorious College of Arts & Science Chennai , India *Veltech DR.SR & R.R. Technical Uty avadi, Chennai, India Copyright 014 S. C. Premila and S. Srinivasa Raghavan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract This paper focuses on the study of a stochastic model for predicting the seroconversion time of HIV transmission. The breadown of the human immune system is very much based on the antigenic diversity of the antigen namely HIV acquire in successive sexual contacts. The antigenic diversity threshold is one at which the breadown of the immune system acquires leading to seroconversion. In this paper the stochastic model for the estimation of expected time to seroconversion is derived under the assumption that the threshold level of antigenic diversity is a random variable which follows Geometric distribution. The mean time to seroconversion and its variance are derived and the numerical illustrations are provided. Keywords: Human Immune deficiency Virus, Antigenic diversity threshold, Acquired Immune Deficiency Syndrome, Seroconversion Introduction The use of stochastic model in the study of HIV infection, transmission and spread of AIDS is quite common. The transmission of HIV from an infected

2 7804 S. C. Premila and S. Srinivasa Raghavan partner to an uninfected person taes place. It is really a matter of concern that the successive contacts with an infected person will hasten the process of seroconversion which means that a person moves from seronegative to seropositive state. It is interesting to note that the concept of antigenic diversity of the HIV plays an important role in this process of seroconversion. If the antigenic diversity crosses a particular level which is nown as antigenic diversity threshold, then there is a collapse of the immune system and seroconversion immediately taes place. The antigenic diversity threshold model has been discussed by Nowa and May (1991) and Stillianais et al (1994). A stochastic model based on the cumulative damage process is derived and using this model it is possible to obtain the expected time to seroconversion and its variance. Sathiyamoorthi and Kannan (001) used the shoc model and cumulative damage process due to Esary et al (1973) to estimate the expected time to seroconversion and its variances. In this paper, the stochastic model for the estimation of expected time to seroconversion and variance of the seroconversion time are derived under the assumption that the threshold level of antigenic diversity is a random variable which follows geometric distribution. In this study the theoretical result of substandiated using numerical data stimulated. Model Assumptions of the Model (i) (ii) (iii) (iv) (v) (vi) An uninfected individual has sexual contacts with a HIV infected partner and in every contact a random number of HIV are getting transmitted. Sexual contacts is the only source of transmission. An individual is exposed to a damage process acting on the immune system and damage process is linear and cumulative. Damages are caused by transmission of HIV at each contact, whose interarrival times are assumed to be i.i.d. random variables. If the total damage caused exceeds a threshold level Y which is itself a random variable, the seroconversion occurs and the person is recognized as a seropositive. The process which generates the contacts, the sequence of damages and threshold are mutually independent.

3 A stochastic model for the estimation of time to seroconversion of HIV 7805 Notations Xi A random variable denoting the amount of contribution to antigenic diversity due to the HIV transmitted in the i th contact, in other words the damage caused to the immune system in the i th 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 i th contact. Ui: A random variable denoting the interarrival time between successive contacts with p.d.f. f (.) and c.d.f F (.). 1 ( S) P S is the p.g.f of x. V (t) = Probability of exactly contacts in (0, t] 1 Results S (t) = P (T > t) = Pr {there are exactly contacts in (0, t]} 0 X Pr {the cumulative total of antigenic diversity < y} = V ( t) P ( 0 0 X y) Let P(y=1) = θ represent the probability that the threshold level is equal to one. P (y = ) = θ θ which implies that the probability the conversion taes place only when y =. Similarly, Pj = P (y = j) = θ (θ) j-1 ஃ The probability generation function φ is ( S) P S S

4 7806 S. C. Premila and S. Srinivasa Raghavan S 0 ( S ) 1 S 1 S P ( X Y) n 1 P ( P( y n)) ' n Let P ( y n) P( y n) n 1 n P( y n i) P ( X Y) ' P n n 1 n ( ) ( ) 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 contacts is equal to S and P(X1+X+ +X < Y) = Now 1 - S = 1 - Shibosi (1990). is the prevalence function mentioned in Jewell and 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. V (t) = P [U1 + U + + U < t < U1 + U + + U+1] P ( contacts in (0, t]) = [F (t) F+1(t)], where F (t) is the distribution function of U1 + U + U3 + U S ( t ) 0 V ( t ) ( )

5 A stochastic model for the estimation of time to seroconversion of HIV 7807 =0 = V (t) [ψ (θ)] Let L (t) = 1 S (t) F (t) The Laplace stieltjes transform of L (t) is L*(S) = { 1 1 f * f ( s) * ( s) f*(s)} On simplification Assuming that f*(.) follows exp (c) Then f*(s) = C C+S L*(s) = [1 ψ(θ)][c/c+s] [1 ψ(θ)(c/c+s)] µt = - dl (S) ds S = 0 µt = E (T) = 1 C[1 [ψ(θ)] ' * d L ( s) d s = c s 0 [1 ] V (T) = C 1 1 Special case If Pn = P(x = n) = 1 n

6 7808 S. C. Premila and S. Srinivasa Raghavan ' 1 Pn 1 n n n ( t) E (T) = 1 C (on simplification) 1 V ( T) (on simplification) (C ) Table 1 C Mean Variance θ= 0.4 θ= 0.6 θ= 0.4 θ=

7 E(T)&V(T) A stochastic model for the estimation of time to seroconversion of HIV 7809 Figure 1 1,8 1,6 1,4 1, 1 0,8 0,6 0,4 0, C θ= 0.4 θ= 0.6 θ= 0.4 θ= 0.6 Table θ E(T) V(T) C = 4 C = 6 C = 4 C =

8 E(T) & V(T) 7810 S. C. Premila and S. Srinivasa Raghavan 1,8 1,6 1,4 1, 1 0,8 0,6 0,4 0, 0 0 0, 0,4 0,6 0,8 1 θ C = 4 C = 6 C = 4 C = 6 It is observed from the table 1 and also the graph as the value of the parameter c which is namely the parameter of the distribution of the interarrival time between contact shows an increase it means that the average interval time which is given by E(U) = 1/C. Since U ~ exp (C). The interarrival time between contacts become similar. Then there would be more number of contacts. Hence the expected time to seroconversion decreases and the variance of the seroconversion decreases. It is observed from the table and also the graph as the value of θ which is the parameter of the geometric distribution of the threshold increases the mean time to seroconversion decreases and the variance of the seroconversion time are also decreases. References 1. Esary, J. D., Marshall, A. W. and Proschan, F. (1973). Shoc models and Wear processes, Ann. Probability, 1 (4):

9 A stochastic model for the estimation of time to seroconversion of HIV Jewel, N. P. and Shiboshi, S. (1990) Statistical analysis of HIV infectivity based on partner studies. Biometers. 46, Nowa, M. A. and May, R. M. (1991). Mathematical Biology of HIV Infections Antigenic Variation and Diversity Threshold, Mathematical Biosciences, Vol. 106: Sathiyamoorthi, R. and Kannan, R. (001). A Stochastic Model for Time to Seroconversion of HIV Transmission, Journal of Kerala Statistical Association, Vol.1: Stilianais, N., Schenzle, D. and Dietz, K. (1994). On the Antigenic Diversity Threshold Model for AIDS, Mathematical Biosciences, Vol. 11: Received: September 15, 014; Published: November 6, 014