Research article · 2017 · Full text
International Journal of Statistics and Applied MathematicsA stochastic approach to determine seroconversion time of HIV infected under alertness using exponentiated modified weibull distribution
Volume 2(1) · Pages 14–24
Research areasMathematical and Theoretical Epidemiology and Ecology ModelsHIV/AIDS Research and InterventionsHIV Research and Treatment
Abstract
Author-writtenThis study is based on a stochastic model for estimating the expected time to seroconversion of HIV infected using preventive strategy. The use of preventive strategy gives rise to the concept of alertness on the part of the individuals who has sexual contact with unknown partner. The effectiveness of the adoption preventive strategy in the elongation of the expected time to seroconversion is studied by introducing the stochastic model under the assumption that the threshold level of antigenic diversity is a random variable which follows exponentiated modified Weibull distribution. Numerical illustration is provided using simulated data.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Seroconversion occurs immediately when the antigenic diversity of HIV crosses a specific threshold level, leading to the collapse of the human immune system. View evidence ↗
- Individuals engaging in sexual contact with unknown partners may use preventive strategies, such as condoms, which introduces the concept of 'alertness' to reduce infection risk. View evidence ↗
The question
- The study aims to evaluate how preventive strategies (alertness) affect the expected time to seroconversion using a stochastic model where the antigenic diversity threshold follows an exponentiated modified Weibull distribution. View evidence ↗
What they did
- The theoretical results of the stochastic model were validated using simulated numerical data. View evidence ↗
What they found
- Increasing the parameter 'c' (related to interarrival time) leads to shorter times between contacts, which decreases both the expected time to seroconversion and its variance. View evidence ↗
- An increase in the antigenic diversity threshold parameter 'μ' results in an increase in the expected time to seroconversion. View evidence ↗
- Increasing the parameter θ of the exponentiated modified Weibull distribution leads to a decrease in both the expected time to seroconversion and its variance. View evidence ↗
- Increasing the threshold parameter 'γ' results in a decrease in the expected time to seroconversion and its variance. View evidence ↗
- As the probability of inalertness (q) increases, the mean time to seroconversion decreases. View evidence ↗
- Increasing the transmission probability (β) decreases the expected duration to seroconversion, regardless of the value of q. View evidence ↗
What it means
- The authors conclude that the mean time to seroconversion is longer when alertness is present, and this effect is inversely proportional to the probability of inalertness (q). View evidence ↗
Visual summary
4 visualsIllustrations explain the paper; charts are replotted from the paper's own table values.
Antigenic Diversity Threshold Concept
AI-generated conceptual illustration (human blood cells and HIV virions). It explains the idea and does not show study data.
View source evidence ↗
Alertness vs. Inalertness Impact
AI-generated conceptual illustration (HIV virus particles and human T-cells). It explains the idea and does not show study data.
View source evidence ↗
1: = 0.5, = 0.2... vs V(T)
Replotted from Table 1 (page 8) of the original paper. Every bar is the paper's own table value.
View source evidence ↗
2: = 0.5, = 0.2... vs V(T)
Replotted from Table 1 (page 8) of the original paper. Every bar is the paper's own table value.
View source evidence ↗Figures & tables
From the original paper
Table 1
Table 2
Figure 1
Figure 2
Table 3
Table 4
Figure 3
Figure 4
Table 5
Figure 5
Figure 5Introduction
Human Immuno-deficiency Virus (HIV) / Acquired Immuno Deficiency Virus (AIDS) research include all medical research that attempt to prevent, treat are cure HIV/AIDS, as well as fundamental research about the nature of HIV has an infectious agent and AIDS has the disease cost of by HIV. The HIV virus is commonly transmitted via unprotected sexual activity, blood transfusions, hypodermic needles and from mother to child. The transmission of more and more HIV from infected persons to the uninfected; the antigenic variations would be increase. The antigenic diversity threshold which means the antigenic diversity crosses a particular level, then the human immunity system collapses and seroconversion takes place immediately. The antigenic diversity threshold and its estimation has been discussed by Stilianakis et al. (1994) [7] and Nowak and May (1991) [6] . An individual who resort to sexual contacts with unknown partners may go in for preventive strategy to avoid the possible contraction of HIV. A universally recommended strategy to avoid the possible infection is the use of condemns by any person. It gives rise to the concept of alertness on the part of the person under risk. In deriving this model it is assumed that a person with alert uses the prevention strategy and if the person is inalert then there is a risk of transmission. The effectiveness of the adoption preventive strategies in the elongation of the expected time to seroconversion is studied by introducing a stochastic model. A stochastic model for the HIV transmission under alertness, under the assumption that the threshold level of antigenic diversity is a random variable which follows a gamma distribution, mixed exponential distribution, exponentiated exponential distribution and exponentialgeometric distribution has been discussed by Kannan et al. (2007, 2011, 2012 and 2015) [2] . In this paper using the concept of alertness and preventive strategy the stochastic model for the estimation of expected time to seroconversion and its variance are derived under the assumption that the threshold level of antigenic diversity is a random variable which follows an exponentiated modified Weibull distribution. In developing such a stochastic model, the concept of shock model and cumulative damage process developed by Esary et al. (1973) [1] is used. In this study the theoretical results are widely substantiated using numerical data simulated.
Model
A person may be alert in any contact with probability ' p ' and inalert with probability ' q ', so that 1 pq . It is a medically accepted fact that in any particular contact during which a person is inalert need not result in the transmission of HIV with unit probability. In other words, transmission at a contact with inalertness need not be a sure event. Hence, it is assumed that the probability of transmission or infection during a contact occurs with probability 1 .
Hence one can visualize the following possibilities. i)
A person is alert with probability ' p ' and during that contact invasion does not take place.
ii) A person is inalert with probability ' q ' and transmission occurs with probability .
iii) A person is inalert with probability ' q ' and transmission does not occurs and the probability of this event is
2 Assumption of the model Human Immunodeficiency Virus transmits only through sexual contact.
The transmission of HIV is a sure event during any contact in which a person is in-alert.
A person is alert in a single contact with probability p , and inalert with probability q , so that 1 pq .
The damage caused to the immune system due to the antigenic diversity is linear and additive.
The total damage caused when exceeds the threshold level Y which itself is a random variable, the seroconversion occurs and a person is recognized as infected.
Notation
i X a random variable denoting the increase in the antigenic diversity arising due to the HIV transmitted during the i th contact
Result
The probability density function of exponentiated modified Weibull distribution is,
Substituting equation ( 2) in (1), we get
is called the prevalence function as mentioned in Jewell and Shiboski (1990).
g * s 1 g * s g * s g * s () 1 g * s q p q q p q q q fs p q q
Substituting equation ( 6) and ( 8) in (7) we get
d s ds p q q q q p q q p q q q p q q p q q q p q q q q p q q p q q pq f
Substituting equation ( 5) in (10) we get
Substituting equation ( 8) and (11) in equation ( 12) we get
The variance time to seroconversion is
Special case
When there is no alertness, then 1, 1
Therefore ta > t and this implies that the mean time to seroconversion is larger in the case of alertness, which is inversely proportional to the probability of non alertness '' q which is an interesting result.
, there is no alertness and variance in the case is given by,
Which are the results obtained by Kannan et al. (2015) [5] . In Table 1 shows the value of expected time to seroconversion corresponding to the variation in c the parameter of the distribution of interarrival time when , , , q and are kept fixed. As c increases, the value of c 1 decreases that means the interarrival time between contacts become smaller and so there is a corresponding decrease in expected time to seroconversion and also its variance. The graph is also plotted in Fig. 1. 2. It is observed from the contribution to the antigenic diversity threshold parameter '' which increases then expected time to seroconversion increases. This implies the fact that g (.) is the distribution of X(i), the magnitude of contribution to antigenic diversity. Since , μ 1 X E as increase there is a decrease in the contribution of antigenic diversity. Hence, the expected time to seroconversion and also its variance for time to seroconversion increase. 3. From Table 3, as the value of θ is the parameter of the exponentiated modified Weibull distribution of the threshold increases then expected time to seroconversion and variance of seroconversion decreases. 4. From the fixed values of θ, μ and c when threshold parameter 'γ' is allowed to increase then expected time to seroconversion and variance of seroconversion decrease as indicated in Table 4 and Figure 4. 5. From table 5, we observe that the value of q which is the probability of inalertness increases than the mean time to seroconversion decreases is quite plausible. It also quite reasonable has regard the variance it could be seen the value of the q increases the variance is decreases. It can be easily observed that when the value of β increases, there is a decrease in the expected duration of seroconversion irrespective of variation in q values. It is also quite reasonable as regards the variance for a fixed value.
| q | = 0.5, = 0.2, µ=0.2, c=2 | — | — | — | — | — |
|---|---|---|---|---|---|---|
| β=0.2 | β=0.5 | β=0.8 | ||||
| E(T) | V(T) | E(T) | V(T) | E(T) | V(T) | |
| 0.1 | 35.71429 | 1479.592 | 14.28571 | 236.7347 | 8.928571 | 92.47449 |
| 0.2 | 17.85714 | 369.8980 | 7.142857 | 59.18367 | 4.464286 | 23.11862 |
| 0.3 | 11.90476 | 164.3991 | 4.761905 | 26.30385 | 2.976190 | 10.27494 |
| 0.4 | 8.928571 | 92.47449 | 3.571429 | 14.79592 | 2.232143 | 5.779656 |
| 0.5 | 7.142857 | 59.18367 | 2.857143 | 9.469388 | 1.785714 | 3.698980 |
| 0.6 | 5.952381 | 41.09977 | 2.380952 | 6.575964 | 1.488095 | 2.568736 |
| 0.7 | 5.102041 | 30.19575 | 2.040816 | 4.831320 | 1.275510 | 1.887234 |
| 0.8 | 4.464286 | 23.11862 | 1.785714 | 3.698980 | 1.116071 | 1.444914 |
| 0.9 | 3.968254 | 18.26657 | 1.587302 | 2.922651 | 0.992063 | 1.141660 |
| 1 | 3.571429 | 14.79592 | 1.428571 | 2.367347 | 0.892857 | 0.924745 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
View original table image
| 𝜸 | θ= 0.5, c = 2, µ=0.2, q = 0.5 =0.4 | — |
|---|---|---|
| E(T) | V(T) | |
| 0.1 | 5.000000 | 25.00000 |
| 0.2 | 4.375000 | 20.70313 |
| 0.3 | 4.000000 | 18.00000 |
| 0.4 | 3.750000 | 16.14583 |
| 0.5 | 3.571429 | 14.79592 |
| 0.6 | 3.437500 | 13.76953 |
| 0.7 | 3.333333 | 12.96296 |
| 0.8 | 3.250000 | 12.31250 |
| 0.9 | 3.181818 | 11.77686 |
| 1 | 3.125000 | 11.32813 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
View original table image
| 𝜃 | = 0.5, c = 2, µ=0.2, q = 0.5 =0.4 | — |
|---|---|---|
| E(T) | V(T) | |
| 0.1 | 3.750000 | 15.10417 |
| 0.2 | 3.571429 | 14.79592 |
| 0.3 | 3.437500 | 14.35547 |
| 0.4 | 3.333333 | 13.88889 |
| 0.5 | 3.250000 | 13.43750 |
| 0.6 | 3.181818 | 13.01653 |
| 0.7 | 3.125000 | 12.63021 |
| 0.8 | 3.076923 | 12.27811 |
| 0.9 | 3.035714 | 11.95791 |
| 1 | 3.000000 | 11.66667 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
View original table image
| c | = 0.5, = 0.2, µ=0.2, q = 0.5 =0.4 | — |
|---|---|---|
| E(T) | V(T) | |
| 1 | 7.142857 | 59.18367 |
| 2 | 3.571429 | 14.79592 |
| 3 | 2.380952 | 6.575964 |
| 4 | 1.785714 | 3.698980 |
| 5 | 1.428571 | 2.367347 |
| 6 | 1.190476 | 1.643991 |
| 7 | 1.020408 | 1.207830 |
| 8 | 0.892857 | 0.924745 |
| 9 | 0.793651 | 0.730663 |
| 10 | 0.714286 | 0.591837 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
View original table image
Numerical Illustrations
Numerical Illustrations
Acknowledgement
The authors are immensely thankful to Dr. R. Ramanarayanan, Professor of Mathematics, Chennai and Dr. R. Sathiyamoorthi Retired Professor and Head, Department of Statistics, Annamalai University , Annamalai Nagar for their invaluable suggestions, guidance and moral support.
Figures and tables from the paper
| μ | = 0.5, = 0.2, c=2, q = 0.5 =0.4 | — |
|---|---|---|
| E(T) | V(T) | |
| 0.1 | 3.035714 | 11.63903 |
| 0.2 | 3.571429 | 14.79592 |
| 0.3 | 4.107143 | 18.27168 |
| 0.4 | 4.642857 | 22.06633 |
| 0.5 | 5.178571 | 26.17985 |
| 0.6 | 5.714286 | 30.61224 |
| 0.7 | 6.250000 | 35.36352 |
| 0.8 | 6.785714 | 40.43367 |
| 0.9 | 7.321429 | 45.82270 |
| 1 | 7.857143 | 51.53061 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
View original table image
References
7 in the original paper- Esary JD, Marshall AW, Proschan F. Shock model and Wear Processes. Ann. Probability. 1973; 1(4):627-649.
- Kannan R, Venkatachalam KA, Sathiyamoorthi R, Malarvizhi G. A Stochastic Model for HIV transmission under alertness. Bioscience Research Bulletin. 2007; 23(1):25-38.
- Kannan R, Thirumurugan A, Sathiyamoorthi R, Malarvizhi G. A Stochastic Model for the Estimation of time to seroconversion using Preventive Strategy, Journal of Indian Acad. Maths. 2011; 33(2):553-569.
- Kannan R, Kavitha S, Sathiyamoorthi R. A stochastic approach to determine time to cross antigenic diversity threshold of HIV transmission under alertness. Ultra scientist. 2012; 24(2):327-336.
- Kannan R, Chandrasekar K. A stochastic model for estimation of expected time to seroconversion of HIV infected using alertness. International Journal of Mathematical Archiv. 2015; 6(4):16-22.
- Nowak MA, May RM. Mathematical Biology of HIV Infection: Antigenic variation and Diversity Threshold. Mathematical Biosciences. 1991; 106:1-21.
- Stilianakis N, Schenzle D, Dietz K. On the antigenic diversity threshold model for AIDS. Mathematical Biosciences. 1994; 121:235-247.