EXPECTED TIME TO CROSS THE ANTIGENIC DIVERSITY THRESHOLD IN HIV INFECTION USING SHOCK MODEL APPROACH ABSTRACT

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1 EXPECTED TIME TO CROSS THE ANTIGENIC DIVERSITY THRESHOLD IN HIV INFECTION USING SHOCK MODEL APPROACH R.Elangovan and R.Ramajayam * Department of Statistics, Annamalai University, Annamalai Nagar ABSTRACT In the study of HIV infection the extent of contribution to antigenic diversity is influenced by several factors, such as the different sources of infection, the frequency of contacts etc.,. Several models are developed and analyzed in order to access how variability infectiousness during the incubation period of the disease is likely to influence the early stages and general pattern of the epidemic. Nowak et. al. (1991) has studied the impact of antigenic diversity threshold in the progression of AIDS. As and when the total antigenic diversity crosses the so called threshold level, the seroconversion takes place. The expected time to cross the threshold of the infected person is a vital event in seroconversion. In this paper the expected time to cross the threshold level and its variance are derived using three parameter Generalized Rayleigh Distribution. In doing so, the concept of Shock model and cumulative damage process suggested by Esary et. al. (1973), Thangaraj and Stanley (199) has been used. The analytical results are substantiated with suitable numerical illustrations. KeyWords: Threshold level, Seroconversion, Generalized Rayleigh distribution, Shock Models and Cumulative Damage Process. INTRODUCTION Much uncertainly still surrounds the processes governing the development of acquired immunodeficiency syndrome (AIDS), after an individual is infected with the human immunodeficiency viruses, for a detailed study refer to Nowak et.al. (1991). Since public recognition of the acquired immunodeficiency syndrome epidemic in 1981, there are nearly 7 million people in worldwide infected by HIV/AIDS. Now, there are approximately 57 people everyday infected this epidemic, which is equivalent to a new infections occur every 16 seconds. Hence the HIV/AIDS pandemic is the greatest public health disaster of modern times, refer to Jiang et.al. (21). In the study of HIV infection the extent of contribution to antigenic diversity is influenced by several factors, such as the different sources of infection, the frequency of contacts. Several models are developed and analyzed in order to access how variability infectiousness during the incubation period of the disease is likely to influence the early stages and general pattern of the epidemic. Nowak et. al. (1991) have studied the impact of antigenic diversity threshold in the progression of AIDS. As and when the total antigenic diversity crosses the so called threshold level, the seroconversion takes place. The incubation period after infection with HIV is known to be extremely long and is measured in years rather than days. The time interval between the points of Page 57 * This work was already presented in the National Conference on Mathematical Modelling, Department of Mathematics, Annamalai University, Annamalai Nagar February, 21 22, 214.

2 infection to the time at which the antibodies specific to HIV are detected in the blood is called the seronegative period. The onset of production of antibodies in the infected is referred to as seropositive period. The process of change over from seronegative state to seropositive state is called the seroconversion. Many biological factors such as incubation periods and social factors affecting HIV/AIDS spread are subjected to considerable random variation so that the spread of the AIDS virus is in essence a Stochastic process, refer to Waema and Olowofeso (25). The expected time to cross the antigenic diversity threshold of the human immune system has been estimated by using suitable Stochastic models. Several variations of these models have been discussed by many authors cited in the recent literature. In this context, estimation of expected time to seroconversion of HIV infected when inter contacts times are correlated has been discussed by Niyamathulla et.al. (213). The progression of the HIV infection is indicated by seroconversion. Hence the seroconversion of an individual with HIV infection is a vital event. The seroconversion occurs due to contribution to the antigenic diversity of the antigens due to successive sexual contacts. The seroconversion occurs as and when the total antigenic diversity crosses the antigenic diversity threshold. Using this property the expected time to seroconversion and its variance are derived by Niyamathulla et.al. (214). Some interesting result can also seen in Sathiyamoorthi (198), Isham (1988), Boer et.al. (1994), Nowak (1994), Stilianakis et. al. (1994), Bonhoeffer and Bittnar et.al. (1997), Sathiyamoorthi and Kannan (21), Elangovan and Ramajayam (212).. In this paper the expected time to cross the threshold level and its variance are derived using, three parameter generalized Rayleigh distribution. In doing so, the concept of Shock model and cumulative damage process suggested Esary et. al. (1973) has been used. The analytical results are substantiated with suitable numerical illustrations. Assumptions of the Model: (i) An uninfected partner has sexual contacts with an infected person, unsterile needles for drug abuse and Transfusion of infected blood products. (ii) On every occasion of the above three behaviours there is a random amount of transmission of HIV, which together contributes to the antigenic diversity. (iii) The damages due to the events namely sexual contacts, sharing of needles and blood Products are statistically independent. (iv) If the cumulative damage due to successive events crosses the antigenic diversity threshold level the seroconversion takes place. The inter-arrival times between contacts, sharing needles and blood products are statistically independent. Notations: X i : a continuous random variable denoting the amount of damage/depletion caused to the system due to the exit of persons on the i th occasion of policy announcement, i = 1,2,3, k and X i S are i.i.d and X i = X for all i. Y : a continuous random variable denoting the threshold level having three parameter generalized Rayleigh distribution. g. : The probability density functions (p.d.f) of X i g k. : The k- fold convolution of g(. ) i.e., p.d.f. of g. : Laplace transform of g(. ) g k. : Laplace transform of g k (. ) k i=1 X i. : The probability density function of random threshold level which has three parameter generalized Rayleigh distribution and H(. ) is the corresponding Probability generating functions. Page 58

3 U : a continuous random variable denoting the inter-arrival times between decision epochs. f(. ) : p.d.f. of random variable U with corresponding Probability Generating function. V k t F k t F k+1 t F k t : Probability that there are exactly k policies decisions in (,t] S. : The survivor function i.e. P T > t 1 S t = L(t) Model Description and Results The two-parameter generalized Rayleigh distribution, introduced by Surles and Padgett (21) has the following cumulative distribution function for x > F x; α, λ = 1 e λx 2 α ; x > ; and the corresponding probability density function becomes f x; α, λ = 2αλxe λx 2 1 e λx 2 α 1 ; x > ; Here α > and λ > are the shape and scale parameters, respectively. From now on, a two parameter generalized Rayleigh distribution with shape parameter α and scale parameter λ will be denoted by GRD (α, λ). A three-parameter generalized Rayleigh distribution can be obtained from a two-parameter generalized Rayleigh distribution by introducing the location or threshold parameter. Therefore, for α >, λ > and < <, a three-parameter generalized Rayleigh distribution has the cumulative distribution function is F x; α, λ, = 1 e λ(x )2 α ; x >, α, λ > Here α > and λ > are the shape and scale parameters, respectively. And the corresponding probability density function probability density function is f x; α, λ, = 2αλ x e λ x 2 1 e λ x 2 α 1 ; It will be denoted by GRD (α, λ, ). The corresponding survival function is λ(x )2 H X = 1 1 e λ(x )2 = e Assume that shocks occur randomly in time in accordance with a three parameter generalized Rayleigh distribution. Taking the shape parameter as α = 1 P x i < y = g k (x)h X dx = g k (x) e λ(x )2 dx = g λ(1 ) 2 k The survival function which gives the probability that the cumulative threshold will fail only after time t. S t = P(T > t) = Probability that the total damage survives beyond t (1) Page 59

4 = P tere are exactly k contacts, t k= P{ te total cumulative tresold (, t)} It is also known from renewal process that P (exactly k policy decisions in (, t)) =F k t F k+1 t with F t = 1 P T > t = k= V k t P x i < y = F k t F k+1 t k= g λ(1 ) 2 k (2) Now, the life time is given by P T < t = L t = The distribution of life time (T) L t = 1 S t = 1 F k t F k+1 t k= Taking Laplace transformation L (t) we get g λ(1 ) 2 k l s = 1 g λ + λ 2 2λ f (s) 1 g λ + λ 2 2λ f (s) Let the random variable U denoting inter arrival time which follows exponential with parameter. Now f (s) = c, substituting in the above equation (3) we get c+s = c 1 g λ + λ 2 2λ c + s g λ + λ 2 2λ c E T = d ds l s Given S = 1 E T = c 1 g λ + λ 2 2λ g ~ exp, g λ =. + λ, g 2 λ = + 2λ, g (λ 2 ) = + λ 2 On simplification we get E T = c 1 +λ + 1 +λ 2 +2λ (3) E T 2 = d2 ds 2 l s Given S = E(T 2 ) = 2 c 2 1 g +λ + On simplification we get +λ 2 +2λ E T = λ λ λ 3 + 2λ λ 4 c 2 + 2λ λ E(T 2 ) = 2 λ λ λ 3 + 2λ λ 4 2 c λ λ (4) Page 6

5 Expected Time (T) V T = E T 2 E T 2 = 2 λ λ λ 3 + 2λ λ 4 2 c λ λ On simplification we get λ λ λ 3 + 2λ λ 4 c 2 + 2λ λ = λ λ λ 3 + 2λ λ 4 2 c λ λ (5) Numerical Illustration On the basis of the expressions derived for the expected time and variance, the behaviour of the same due to the chance in different parameter is shown in Figure.1a. to Figure.2b. Table 1: Variation in E T and V T for changes in the Inter Arrival Times between successive Contacts, keeping λ =. 5 fixed. C µ = 1.2 µ = 1.4 µ = 1.6 µ = 1.8 E(T) V(T) E(T) V(T) E(T) V(T) E(T) V(T) Inter Arrival Time c µ = 1.2 µ = 1.4 µ = 1.6 µ = 1.8 Fig. 1a. Variation in E T for Changes, c Page 61

6 Variance (T) µ = 1.2 µ = 1.4 µ = 1.6 µ = Inter Arrival Time c Fig. 1b. Variation in V T for Changes, c It is observed that when λ is kept fixed, the inter arrival time c follows exponential distribution is an increasing case by the process of renewal theory. Therefore, the value of the expected time E(T) to cross the threshold of seroconversion is found to be decreasing, in all the cases of the parameter value = 1.2, 1.4, 1.6, 1.8. When the value of the parameter increases, the expected time is found decreasing, which is shown in Figure.1a. The same case in found in V (T) which is observed in Figure.1b. Table 2: Variation in E T and V T for changes in the Inter Arrival Time between Successive Contacts, keeping = 1. 5 fixed. C λ =.2 λ =.4 λ =.6 λ =.8 E (T) V (T) E (T) V (T) E (T) V (T) E (T) V (T) Page 62

7 Variance (T) Expected Time (T) Inter Arrival Time c λ=.2 λ=.4 λ=.6 λ=.8 Fig. 2a. Variation in E T for Changes λ, c Inter Arrival Time c λ=.2 λ=.4 λ=.6 λ=.8 Fig. 2b. Variation in V T for Changes λ, c It is observed that, when is kept fixed and the inter-arrival time c increase, the value of the expected time E(T) to cross the threshold of seroconversion is found to be decreasing,in all the cases of the parameter value λ =.2,.4,.6,.8. When the value of the parameter λ increase, the expected time is found decreasing, this is indicated in figure.2a. The same case is observed in the threshold of seroconversion of V (T) which is observed in figure.2b. Conclusion The model discussed in this paper leads to clearly testable by assuming suitable hypotheses, which survive comparison with the admittedly limited data that are available from longitudinal studies of patients over the incubation period of AIDS. Data collected from longitudinal studies of infected people are used to generate a hypothesis of how infectiousness varies through time from the point of infection. To analyze AIDS epidemiological data, many parametric distribution have been assumed for the HIV infection and seroconversion without due regard to the dynamics of the HIV epidemic and the biological and clinical features of the HIV. It would be worthwhile to Page 63

8 collect HIV/AIDS epidemic data with regard to time to seroconversion, time cross the antigenic diversity threshold etc., and suitable probability distribution can be assumed, and test for the goodness of fit to be carried out in order to validate the model. References 1. Bittner, B., S. Bonhoeffer, and M.A. Nowak (1997). Virus Load and Antigenic Diversity. Bulletin of Mathematical Biological Sciences, Vol.59, No.5, pp Bonhoeffer, S., and M.A. Nowak (1994). Mutation and the Evaluation of Virulence. Biological Sciences, Vol, 258, No.1352, pp De Boer, R.J., and M.C. Boerlust (1994). Diversity and Virulence Threshold in AIDS. Journal of Antigenic Diversity Threshold, Vol.94.pp Easary, J.D., A.W. Marshall, and F. Proschon (1973). Shock Models and Wear Processes. Annals of Probability, pp Elangovan, R., and R. Ramajayam (212). A Stochastic Model in the study of HIV/AIDS Epidemic and its Progression. International Transactions in Applied Sciences, Vol. 5, No.3, pp Isham, V. (1988). Mathematical modeling of the Transmission Dynamics of HIV Infection and AIDS. A review. Journal of Royal Statistical. Society. A, A. 157(1), pp Kundu, D., and M.Z. Raqab (213). Estimation of R=P[Y<X] for three parameter Generalized Rayleigh Distribution. Journal of Statistical Computation and Simulation Niyamathulla, U., R. Elangovan, and R. Sathiyamoorthi (213). Estimation of Expected Time to Seroconversion of HIV Infected when Inter Contact Times are Correlated. Asia Pacific Journal of Research, Vol.1, No. XI, pp Niyamathulla, U., R. Elangovan, and R. Sathiyamoorthi (214). Seroconversion of HIV Infected when Virulence Threshold undergoes Changes. International Journal of Current Advanced Research, Vol.3, No.1, pp Nowak, M.A., R.M. Anderson, A.R. Mclean, T.F.W. Woifs, J. Goudsmit, and R.M. May (1991). Antigenic diversity thresholds and the Development of AIDS. American Association for the Advancement of Science, New series, Vol.254, No.534, pp Sathiyamoorthi R. (198). Cumulative Damage Model with Correlated Inter Arrival Time of Shocks. IEEE Transactions on Reliability, R-29, No Sathiyamoorthi, R., and R. Kannan (21). A Stochastic Model for time to seroconversion of HIV transmission. Journal of Kerala Statistical Association, No.12, pp Stilianakis, N., D. Schenzle, and K. Dietz (1994). On the Antigenic Diversity Threshold Model for AIDS. Mathematical Biosciences, Vol.121, pp Surles, J.G., and W.J. Padgett (21). Inference for Reliability and Stress-Strength for a Scaled Burr Type X distribution. Lifetime Data Analysis, Vol.7, pp Thangaraj, V and A.D.J. Stanley (199). General Shock Models with Random Threshold. Optimization, Vol.21, No.4, pp Waema, R., and O.E Olowofeso (25). Mathematical modelling for Human Immunodeficiency Virus (HIV) Transmission using Generating Function Approach. Kragujevac Journal of Science, Vol.27, pp Page 64