International Journal of Current Medical Sciences- Vol. 5, Issue, 1, pp. 1-7, January, 2015

Transcription

1 Available onle at INTERNATIONAL JOURNAL OF CURRENT MEDICAL SCIENCES RESEARCH ARTICLE ISSN: EXPECTED TIME TO SEROCONVERSION OF HIV INFECTED WHEN BOTH ANTIGENIC DIVERSITY THRESHOLD AND VIRULENCE THRESHOLD SATISFY SCBZ PROPERTY R.Ramajayam, R.,,3Department AR TIC L E Elangovan 3R.Sathiyamoorthi of Statistics, Annamalai University, Annamalaagar- 68, Tamilnadu, India I NF O Article History: th Received, December, 4 Received revised form st, December, 4 Accepted 9th, January, Published onle 8th, January, Key words: ABS TR AC T The Human Immune Deficiency Virus entraps the body of the human begs causes the so called HIV fection. The antigenic diversity as well as virulence is generated by the antigens. As when the total antigenic diversity generated or the virulence crease by the antigens side the human body the seroconversion takes place. In this paper the expected time to seroconversion is derived under that assumption that if the contribution to antigenic diversity or virulence exceeds a particular level called the threshold level both aspects a seroconversion takes place. Numerical examples are also provided. HIV, AIDS, Seroconversion, Antigenic Diversity Threshold, Virulence Threshold. INTRODUCTION The cidence of Human Immune Deficiency Virus HIV all over the world is a really a matter of great concern due to the fact that there is no cure available for this fection till today. Also the spread of this fection is really alarmg. The progression of HIV to Acquired Immune Deficiency Syndrome AIDS is a matter of concern due to the fact that the affected person suffersboth physical mental torture. The governments admistration suffer a great burden both fancially social. Many authors have used Mathematical Stochastica model to depict the progression of this fection among the affected. Nowak May 99 have identified the antigenic diversity as the ma cause for the progression of the fection. They also describe a particular level of antigenic diversity as antigenic diversity threshold. It is observed that not only the antigenic diversity of an vadg antigen plays a vital role the progression of the fection but also the virulence of the antigens. May Anderson 983 have given an terpretation of virulence its impact. The concept of virulence threshold AIDS has been discussed by Boer et.al.994. Bull 994 has discussed about the virulence of the vadg antigens its perspective. In the present paper the expected time to seroconversion is determed by usg the concept of shock models cumulative damage process due to Essary, Prochan Marshall 973. It may be observed that the cidence of the seroconversion occurs whenever the cumulative antigenic diversity of the vadg antigens crosses the so called antigenic diversity threshold. Similarly if the cumulative level of virulence crosses its threshold level it Copy Right, IJCLS, 4, Academic Journals. All rights reserved. leads to seroconversion of the fected. In this paper the expected time to seroconversion of the fected is derived under the assumption that both the antigenic diversity threshold virulence threshold satisfy the so called Settg the Back to Zero Property SCBZ due to Raja Rao Talwalkar 99. Assumptions. A person is exposed to sexual contacts with an fected partner on each occasion of contact the transmission of HIV takes place.. The mode of transmission of HIV on successive occasions results the contribution to the antigenic diversity of the vadg antigens. Also there is crease the virulence of the vadg antigens. 3. As when the total antigenic diversity crosses a particular level called the antigenic diversity threshold, then the seroconversion takes place. Similarly if the total virulence of the vadg antigens crosses the virulence threshold, then the seroconversion will occur. 4. The crossg of both antigenic diversity threshold virulence threshold simultaneously is considered to be an impossible event.. The two thresholds are rom variables are mutually dependent. 6. Both the thresholds satisfy the SCBZ property. Notations : a rom variable denotg the contribution to antigenic diversity on the th contact,,3,.., with probability density function g. with cumulative distribution function G. *Correspondg author: R. Ramajayam Department of Statistics, Annamalai University, Annamalaagar- 68, Tamilnadu, India

2 the crease the virulence due to the th contacts, So it is seen that,,3,.., with probability density function q. cumulative distribution function Q. Now, sce the virulence threshold is also satisfyg the :a rom variable denotg antigenic threshold SCBZ property we have ~. as p.d.f.. is the has probability density c.d.f. function h. cumulative distribution function H. a rom variable denotg the virulence threshold Sce satisfies the SCBZ property with probability density function m. cumulative We have if distribution function M. > T time to seroconversion *denotes Laplace transform where ~exp a rom variable denotg the terarrival times As above it can be shown that between contact,,3,.., with probability density function of f. cumulative distribution. function F. where RESULTS The survivor function is given by > The antigenic diversity as well as the virulence due to k successive contacts do not cross the respective thresholds Hence, ℎ is given by if > if where is it-self followsexp. a rom variable which ℎ Also it can be seen that Where denotes the probability that there where are exactly contacts, as per renewal theory. Hence it is seen that It can be proved that Now, the rom variable denotg the antigenic diversity threshold virulence threshold respectively both undergo parametric changes their respective probability distributions, sce both the thresholds satisfy the SCBZ property. Now, the p.d.f. of {that there are exactly contacts, the antigenic diversity, virulence developed do not cross the respective threshold levels} There are contacts, t the total antigenic diversity as well as total virulence do not cross the respective thresholds Now, Page

3 Now, Now Expected time to seroconversion where Similarly on simplification on, simplification. ~exp,. ~exp, q. ~exp Now, it is seen that Now given Let us assume that Now, sce takg the Laplace transform we get, 3 Page

4 Let. 7 Substitutg 6, 7, 8, 9 we get, on simplification. 6.. Numerical Examples The followg numerical example provides an idea of due to changes the values of different parameters.,.4,.,.,.,.4, 4 Page

5 .4,,.6,,..4, ET VT Expected Time Variance Table Variation E T V T for Changes ET VT Expected Time Variance ET VT,.,.,.4,.8,,..8,,.. ET ,.4,.6,,. Table Variation E T V T for Changes VT Expected Time Variance.,.4,.4,.6,.4,,. VT ET VT ET Figure Variation E T V T for Changes. Figure 4 Variation E T V T for Changes,., ET VT. Figure Variation E T V T for Changes.,.4,.8,,.. ET ,.4,.6,,. Table 3 Variation E T V T for Changes θ VT Expected Time Variance.8,, ET ,.4,.4,.6,.4,,. Table Variation E T V T for Changes,.4,. Table 4 Variation E T V T for Changes.,.4, Figure 3 Variation E T V T for Changes θ Expected Time Variance VT ET VT. Figure Variation E T V T for Changes Page

6 .,.4,.4,.4, 4.4,.,.6,,. Table 6 Variation E T V T for Changes ET VT Expected Time Variance,., ET VT Expected Time Variance. Figure 8 Variation E T V T for Changes ET VT,.,.,.4, ET ,,.6 Table 9 Variation E T V T for Changes.4,.4,.,,. VT Figure 6 Variation E T V T for Changes.4,.,.4,.6,.4,,. Table 7 Variation E T V T for Changes ET VT ET 4 VT Expected Time Variance 9 Expected Time Variance,.,.,.4,. Figure 9 Variation E T V T for Changes CONCLUSION On the basis of the numerical examples worked out for this model the followg conclusions can be drawn. ET VT. Figure 7 Variation E T V T for Changes,.,.,.4,.4,.,.4,,.4,,. Table 8 Variation E T V T for Changes.. ET VT If which is the parameter of the density function of. rom variable which is the contribution to antigenic diversity shows an crease, then the mean contribution per contact is due to fact that. has exponential distribution. Therefore decreases as creases. So creases also the variance. It is seen from table.. is the parameter of the distribution of a rom variable which denotes the antigenic diversity threshold. It satisfies the SCBZ property the parameter is prior to the truncation pot. It follows exponential distribution also. As creases also the shows a very small decrease. It is seen from table. 3. If which is the parameter of the distribution of the rom variable after the truncation pot creases a small decrease is noted. But the shows an crease. It is observed table 3. 6 Page

7 4. Elangovan, R. R. Ramajayam 4. Expected 4. When the values of which is the parameter of the time to cross the antigenic diversity threshold HIV rom variable which is truncation pot follows fection usg shock model approach. Asia Pacific exponential distribution with parameter, then an Journal of Research. Vol., Issue XI, pp creasg results a small decrease both. Elangovan, R., Jaisankar, R., Sathiyamoorthi, R. as observed from table 4.. On the time to cross the Antigenic Diversity. The rom variable denotes the terarrival time Threshold HIV fection a Stochastic Approach. between successive contacts. It follows exponential Ultra Scientist. Vol., No., pp distribution with parameter. So the mean 6. Elangovan, R., Jaisankar, R., Sathiyamoorthi, R. terarrival time is. It implies that as. On the time to cross the Antigenic Diversity creases decreases so the terarrival time Threshold HIV fection A Stochastic Approach. between contacts are smaller. Hence there will be Ultra Scientist. Vol., No., pp more contribution to antigenic diversity 7. May, R.M., R.M.Anderson 983. virulence. Therefore as creases, both Epidemiology Genetics the Coevolution of decrease as observed from table. Parasites Hosts, Philosophical Transactions of 6. The virulence threshold satisfies the SCBZ property the Royal Society B, Biological Sciences, Vol. follows exponential distribution. The parameter 96, pp is prior to the truncation pot it is after 8. Niyamathulla, U., Elangovan, R Sathiyamoorthi, the truncation pot. As creases both R.. A Stochastic Model to Determe the shows a small decrease. It is observed table Expected Time to Seroconversion of HIV Infected 6. due to Antigenic Diversity or Virulence. Similarly as creases decreases also International Journal of Current Medical Sciences, as observed table 7. Vol., Issue, 3, pp Niyamathulla, U., Elangovan, R Sathiyamoorthi, 7. If which is the parameter of the truncation pot of R. 4. Estimation of Expected Time to the virulence threshold which satisfies the SCBZ Seroconversion of HIV Infected When the Antigenic property, if this rom variable has exponential Diversity Thresholds has SCBZ Property. distribution with parameter then International Journal of Ultra Scientist of Physical shows a small decreases. It is observed table 8. Sciences, Vol. 6A, pp If which is the parameter of the distribution of the. Nowak, M. A., R.M. May 99. Mathematical virulence threshold after the truncation pot has an Biology of HIV Infections: Antigenic Variation crease both decrease as observed Diversity Threshold. Mathematical Biosciences, table 9. Vol.6, pp. -.. Rao,B.R., S. Talwalker 99. Settg the Clock References Back to Zero Property of a Class of Life. Boer, R.J. M.C. Boerlist 994. Diversity Distributions. The Journal of Statistical Planng Virulence Thresholds AIDS. Applied Inference, Vol.4, pp Mathematics, Vol.94, pp Sathiyamoorthi, R., Kannan, R On the. Bull, J.J Perspective: Virulence, Evolution. time to Seroconversion of HIV patients under Vol.48, Issue, pp Correlated ter contact times. Pure Applied 3. Easary, J.D., A.W. Marshall F. Proschan 973. mathematical Science, Vol. XLVIII, No.-, pp.7shock Models Wear Processes. Annals of 87. Probability, pp ******* 7 Page