Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 452 https://internationalpubls.com A Stochastic Model for Hepatitis B Patients Using Two Sources of Transmission Dr. P. Pandiyan1, G. Sathyamurthy2 V.Sudha3, 1 Professor& HOD, Department of Statistics, Annamalai university, Annamalai Nagar, Cuddlaore Dist, Tamilnadu. India. (E-mail: pandiyanau@gmail.com) 2Research Scholar, Department of Statistics, Annamalai university, Annamalai Nagar, Cuddlaore Dist, Tamilnadu. India. (E-mail: statsathya@gmail.com). 3. Associate Professor, Dept of Community Medicine, Aarupadai Veedu Medical College and Hospital,.Vinayaka mission’s research foundation (DU), Puducherry Article History: Received: 12-10-2024 Revised: 15-11-2024 Accepted: 01-12-2024 Abstract The Hepatitis B (HBV) infection has led to a global pandemic. Addressing this issue requires the collaborative efforts of medical professionals, social workers, mathematicians, and statisticians to analyse various facets of the infection and its transmission. A particularly intriguing aspect of this research is estimating when an infected individual becomes seropositive. This brings us to the concept of the antigenic diversity threshold. This threshold refers to a specific level of antigenic diversity in the invading pathogen, beyond which the human immune system fails, resulting in seropositivity. In this paper, we derive the expected time to seroconversion by considering the antigenic diversity threshold comprising two elements: the natural threshold level of the human immune system and the threshold associated with antiretroviral therapy (ART). Additionally, numerical examples are provided.Keywords: Skin Lesion, Deep Learning, Ceroscopy, Classification, Neural Network, Melanoma. Keywords: Hepatitis B virus (HBV), Antigenic Diversity Threshold, Anti-Retroviral Therapy (ART), Seroconversion, Expected Time to Seroconversion. Introduction Hepatitis B virus (HBV) significantly contributes to chronic liver diseases, hepatocellular carcinoma, and liver failure. Despite the availability of vaccines and antiviral therapies, HBV continues to spread in inadequate and/or unsanitary healthcare environments. Its transmission routes are like those of HIV, including unprotected sexual contact (both homosexual and heterosexual), blood exposure, use of unsterilized needles, and perinatal transmission. In the U.S., about 5 to 15% of individuals diagnosed with HIV also report HBV infection (1-3). The progression to chronic cirrhosis happens more easily for individuals coinfected with both HIV and HBV than for those with HBV alone (4). Conversely, chronic HBV does not significantly alter the progression of HIV infection, nor does it affect HIV suppression or CD4 T lymphocyte responses when initiating antiretroviral therapy (ART) (5,6). Post-initiation, numerous liver-related complications may arise due to immune modification (7,8). There is also a risk of HBV reactivation after discontinuing antiretroviral (ARV) treatments for both HIV and HBV (9-14). Consequently, a mathematical understanding of HBV is crucial for creating effective control strategies. mailto:pandiyanau@gmail.com mailto:statsathya@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 453 https://internationalpubls.com The human body possesses intrinsic immune capabilities, enabling it to endure a certain degree of antigenic diversity presented by the invading Hepatitis B virus (HBV). This specific degree of antigenic diversity is called the antigenic diversity threshold. When the antigenic diversity of an invading antigen exceeds this threshold, the human immune system fails. This antigenic diversity threshold is a variable that differs for everyone. In each interaction, a stochastic augmentation of the antigenic diversity exists. The anticipated duration required to navigate through the antigenic diversity, resulting from successive interactions within the time frame [0,t], is assessed utilizing the shock model and cumulative damage process methodology as articulated by Esary, Marshall, and Proschan (1973) (15). Sathiyamoorthi and Kannan (1998), Nirmala Ratchager (2003), and others have utilized this approach to estimate the anticipated time for seroconversion (16, 17). This paper presents a mathematical model that assumes Anti-Retroviral Therapy (ART) is administered to individuals identified as HBV positive. Given that Antiretroviral Therapy (ART) extends the duration of seroconversion and the manifestation of Hepatitis B Virus (HBV) symptoms, it is suggested that there is an elevation in the antigenic diversity threshold. Consequently, the threshold of antigenic diversity, beyond which the human immune system cannot withstand, can be characterized as the sum of two random variables. Based on this premise, the anticipated time to seroconversion is deduced. This new family of distribution functions is always positively skewed, and the skewness decreases as both the shape parameters increase to infinity. Interestingly, the new three-parameter generalized exponential distribution has increasing; decreasing, Uni-modal and bathtub shaped Hazard Functions. One can see for more detail in Sathiyamoorthy (1980), Pandiyan et.al., (2010), Subramanian et.al., (2011), Pandiyan et.al., (2012) about the expected time to cross the threshold level of the seroconversion. 2. ASSUMPTIONS OF THE MODEL ➢ Hepatitis B infections are caused by sharing needles and sexual risk. ➢ The threshold of any individual is a random variable. ➢ Seroconversion happens and a person is identified as infected if the total harm surpasses a threshold level Y, which is a random variable in and of itself. ➢ The inter-arrival times between successive contacts, the sequence of damage and the threshold are mutually independent. 3. NOTATIONS 𝑋𝑖 ∶ A continuous random variable denoting the amount of contribution to the antigenic diversity due to the Hepatitis B virus transmitted in the ith contact, in other words the damage caused to the immune system in the ith contact, with p.d.f g (.) and c.d.f F (.). 𝑌1, 𝑌2 ∶ A continuous random variable denoting the threshold for two components which follows three parameter generalized exponential distribution 𝑔(. ) ∶ The probability density functions of 𝑋𝑖 𝑔∗(. ) ∶ Laplace transform of g (.) 𝑔𝑘(. ) ∶ The k- fold convolution of g (.) i.e., p.d.f. of ∑ 𝑋𝑖 𝑘 𝑗=1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 454 https://internationalpubls.com 𝑔𝑘 ∗(. ) ∶ Laplace transform of 𝑔𝑘(. ). 𝑈𝑖 ∶ A random variable denoting the inter-arrival times between contact with c.d.f. 𝐹𝑖(. ), 𝑖 = 1,2 3 … 𝑘. 𝑓(. ) ∶ p.d.f. of random variable denoting between successive contacts with the corresponding c.d.f. F (.) 𝐹𝑘(. ) ∶ The k-fold convolution functions of F (.) 𝑆(. ) ∶ The survivor function, i.e. 𝑃 [𝑇 > 𝑡] 𝐿(𝑡) ∶ 1 - S (t) 𝑉𝑘(𝑡) ∶ Probability that there are exactly k contacts. 4. MODEL DESCRIPTION 𝐻(𝑥) = {𝑒−(𝜆2𝑥−𝜆2𝜃2) + 𝑒−𝜆1(𝑥−𝜃1) − 𝑒−𝜆1(𝑥−𝜃1)−𝜆2(𝑥−𝜃2)} … (1) It can also be proved that 𝑃(𝑋𝑖 < 𝑌) = ∫ 𝑔𝑘(𝑥)𝐻(𝑥)𝑑𝑥 ∞ 0 = {[𝑔∗[𝜆2(1 − 𝜃2)]] 𝑘 + [𝑔∗[𝜆1(1 − 𝜃1)]] 𝑘 −[𝑔∗[𝜆1(1 − 𝜃1)] + [𝜆2(1 − 𝜃2)]] 𝑘 } … (2) 𝑃(𝑇 > 𝑡) = Probability that exactly k contact in (0,t] and the combined threshold level is not crossed 𝑃(𝑇 > 𝑡) = ∑ 𝑉𝑘(𝑡)𝑃(𝑋𝑖 < 𝑌) ∞ 𝑘=0 … (3) Therefore on simplification it can be shown that 𝐿(𝑡) = 1 − 𝑆(𝑡) Using convolution theorem for Laplace transforms, 𝐹0(𝑡) = 1 and on simplification, it can shown that, 𝐿(𝑇) = 1 − 𝑆(𝑡) Taking Laplace transformation 𝐿(𝑇), we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 455 https://internationalpubls.com 𝐿(𝑡) = 1 − {∑[𝐹𝑘(𝑡) − 𝐹𝑘+1(𝑡)][𝑔∗[𝜆2(1 − 𝜃2)]] 𝑘 ∞ 𝑘=0 + ∑[𝐹𝑘(𝑡) − 𝐹𝑘+1(𝑡)][𝑔∗[𝜆1(1 − 𝜃1)]] 𝑘 ∞ 𝑘=0 − ∑[𝐹𝑘(𝑡) − 𝐹𝑘+1(𝑡)] [𝑔∗[𝜆1(1 − 𝜃1)[𝜆2(1 − 𝜃2)]]] 𝑘 ∞ 𝑘=0 } 𝐿(𝑡) = 1 − {𝑇1 + 𝑇2 − 𝑇3} Then 𝑇1 = 1 − [1 − 𝑔∗𝜆2(1 − 𝜃2)] ∑ 𝐹𝑘(𝑡) ∞ 𝑘=1 [𝑔∗[𝜆2(1 − 𝜃2)]] 𝑘−1 … (4) Similarly 𝑇2 = 1 − [1 − 𝑔∗𝜆1(1 − 𝜃1)] ∑ 𝐹𝑘(𝑡) ∞ 𝑘=1 [𝑔∗[𝜆1(1 − 𝜃1)]] 𝑘−1 … (5) 𝑇3 = 1 − [1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)] ∑ 𝐹𝑘(𝑡)[𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)]𝑘−1 ∞ 𝐾=1 … (6) 𝐿(𝑡) = 1 − {[1 − 𝑔∗[𝜆2(1 − 𝜃2)]] ∑ 𝐹𝑘(𝑡)[𝑔∗[𝜆2(1 − 𝜃2)]] 𝑘−1 + 1 ∞ 𝐾=1 − [1 − 𝑔∗[𝜆1(1 − 𝜃1)]] ∑[𝑔∗[𝜆1(1 − 𝜃1)]] 𝑘−1 − 1 ∞ 𝐾=1 + [1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)] ∑ 𝐹𝑘(𝑡)[𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)]𝑘−1 ∞ 𝐾=1 } By taking Laplace-Stieltjes transform, it can be shown that 𝑙∗(𝑠) = [1 − 𝑔∗𝜆2(1 − 𝜃2)] 𝑓∗(𝑠) [1 − 𝑔∗𝜆2(1 − 𝜃2) 𝑓∗(𝑠)] + [1 − 𝑔∗𝜆1(1 − 𝜃1)] 𝑓∗(𝑠) [1 − 𝑔∗𝜆1(1 − 𝜃1) 𝑓∗(𝑠)] − [1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)] 𝑓∗(𝑠) [1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2) 𝑓∗(𝑠)] … (7) Let the random variable 𝑈 denoting inter arrival time which follows exponential with parameter c. Now 𝑓∗(𝑠) = ( 𝑐 𝑐+𝑠 ), substituting in the (7.7) equation we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 456 https://internationalpubls.com 𝑙∗(𝑠) = [1 − 𝑔∗𝜆2(1 − 𝜃2)] ( 𝑐 𝑐 + 𝑠 ) [1 − 𝑔∗𝜆2(1 − 𝜃2) ( 𝑐 𝑐 + 𝑠 )] + [1 − 𝑔∗𝜆1(1 − 𝜃1)] ( 𝑐 𝑐 + 𝑠 ) [1 − 𝑔∗𝜆1(1 − 𝜃1) ( 𝑐 𝑐 + 𝑠 )] − [1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)] ( 𝑐 𝑐 + 𝑠 ) [1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2) ( 𝑐 𝑐 + 𝑠 )] = [1 − 𝑔∗𝜆2(1 − 𝜃2)]𝑐 [𝑐 + 𝑠 − 𝑔∗𝜆2(1 − 𝜃2)𝑐] + [1 − 𝑔∗𝜆1(1 − 𝜃1)]𝑐 [𝑐 + 𝑠 − 𝑔∗𝜆1(1 − 𝜃1)𝑐] − [1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)]𝑐 [𝑐 + 𝑠 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2) 𝑐] … (8) 𝐸(𝑇) = − 𝑑 𝑑𝑠 𝑙∗(𝑠) given s = 0 = 𝑐[1 − 𝑔∗𝜆2(1 − 𝜃2)] 𝑐2[1 − 𝑔∗𝜆2(1 − 𝜃2)]2 + 𝑐[1 − 𝑔∗𝜆1(1 − 𝜃1)] 𝑐2[1 − 𝑔∗𝜆1(1 − 𝜃1)]2 − 𝑐[1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)] 𝑐2[1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2) 𝑐]2 = 1 𝑐[1 − 𝑔∗(𝜆2 − 𝜆2𝜃2)] + 1 𝑐[1 − 𝑔∗(𝜆1 − 𝜆1𝜃1)] − 1 𝑐 [1 − 𝑔∗(𝜆2 − 𝜆2𝜃2 + 𝜆1 − 𝜆1𝜃1) ] … (9) 𝐸(𝑇2) = 𝑑2 𝑑𝑠2 𝑙∗(𝑠) given s = 0 We know that = 2 𝑐2[1 − 𝑔∗𝜆2(1 − 𝜃2)]2 + 2 𝑐2[1 − 𝑔∗𝜆1(1 − 𝜃1)]2 − 2 𝑐2[1 − 𝑔∗𝜆1(1 − 𝜃1) + 𝜆2(1 − 𝜃2)]2 … (10) Now,𝑔∗(. )~ exp(𝜇), 𝑔∗(𝜆)~ exp ( 𝜇 𝜇+𝜆 ), 𝑔∗(𝜆𝜃)~ exp ( 𝜇 𝜇+𝜆𝜃 ), 𝐸(𝑇) = 1 𝑐 [1 − ( 𝜇 𝜇 + 𝜆2 − 𝜇 𝜇 + 𝜆2𝜃2 )] + 1 𝑐 [1 − ( 𝜇 𝜇 + 𝜆1 − 𝜇 𝜇 + 𝜆1𝜃1 )] − 1 𝑐 [1 − ( 𝜇 𝜇 + 𝜆2 − 𝜇 𝜇 + 𝜆2𝜃2 + 𝜇 𝜇 + 𝜆1 − 𝜇 𝜇 + 𝜆1𝜃1 )] On simplification we get, 𝐸(𝑇) = 𝜇2 + 𝜆2𝜇 𝜃2 + 𝜆2𝜇 + 𝜆2 2𝜃2 𝑐 [𝜇2 + 2 𝜆2𝜇 + 𝜆2 2𝜃2] + 𝜇2 + 𝜆1𝜇 𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1 𝑐 [𝜇2 + 2𝜆1𝜇 + 𝜆1 2𝜃1] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 457 https://internationalpubls.com − (𝜇2 + 𝜆2𝜇 𝜃2 + 𝜆2𝜇 + 𝜆2 2𝜃2)(𝜇2 + 𝜆1𝜇𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1) 𝑐[𝜇4 + 2𝜇3𝜆1 + 𝜇2𝜆1 2𝜃1 + 2𝜇3𝜆2 + 𝜆1𝜆2𝜇2𝜃1 + 3𝜆1𝜆2𝜇2 + 𝜆2 2𝜃2𝜇2 + 2𝜆1𝜆2 2 𝜇 𝜃2 + 𝜆1 2𝜆2 2𝜃1𝜃2 + 𝜆1𝜆2𝜇2𝜃2] 𝑉(𝑇) = 𝐸(𝑇2) − [𝐸(𝑇)]2 On simplification 𝑉(𝑇) = ( 𝜇2 + 𝜆2𝜇 𝜃2 + 𝜆2𝜇 + 𝜆2 2𝜃2 𝑐 [𝜇2 + 2 𝜆2𝜇 + 𝜆2 2𝜃2] ) 2 + ( 𝜇2 + 𝜆1𝜇 𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1 𝑐 [𝜇2 + 2𝜆1𝜇 + 𝜆1 2𝜃1] ) 2 − ( (𝜇2 + 𝜆2𝜇 𝜃2 + 𝜆2𝜇 + 𝜆2 2𝜃2)(𝜇2 + 𝜆1𝜇𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1) 𝑐[𝜇4 + 2𝜇3𝜆1 + 𝜇2𝜆1 2𝜃1 + 2𝜇3𝜆2 + 𝜆1𝜆2𝜇2𝜃1 + 3𝜆1𝜆2𝜇2 + 𝜆2 2𝜃2𝜇2 + 2𝜆1𝜆2 2 𝜇 𝜃2 + 𝜆1 2𝜆2 2𝜃1𝜃2 + 𝜆1𝜆2𝜇2𝜃2] ) 2 −2 ( 𝜇2 + 𝜆2𝜇 𝜃2 + 𝜆2𝜇 + 𝜆2 2𝜃2 𝑐 [𝜇2 + 2 𝜆2𝜇 + 𝜆2 2𝜃2] ) ( 𝜇2 + 𝜆1𝜇 𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1 𝑐 [𝜇2 + 2𝜆1𝜇 + 𝜆1 2𝜃1] ) + 2 ( 𝜇2 + 𝜆1𝜇 𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1 𝑐 [𝜇2 + 2𝜆1𝜇 + 𝜆1 2𝜃1] ) ( (𝜇2 + 𝜆2𝜇 𝜃2 + 𝜆2𝜇 + 𝜆2 2𝜃2)(𝜇2 + 𝜆1𝜇𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1) 𝑐[𝜇4 + 2𝜇3𝜆1 + 𝜇2𝜆1 2𝜃1 + 2𝜇3𝜆2 + 𝜆1𝜆2𝜇2𝜃1 + 3𝜆1𝜆2𝜇2 + 𝜆2 2𝜃2𝜇2 + 2𝜆1𝜆2 2 𝜇 𝜃2 + 𝜆1 2𝜆2 2𝜃1𝜃2 + 𝜆1𝜆2𝜇2𝜃2] ) +2 ( 𝜇2 + 𝜆1𝜇 𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1 𝑐 [𝜇2 + 2𝜆1𝜇 + 𝜆1 2𝜃1] ) ( (𝜇2 + 𝜆2𝜇 𝜃2 + 𝜆2𝜇 + 𝜆2 2𝜃2)(𝜇2 + 𝜆1𝜇𝜃1 + 𝜆1𝜇 + 𝜆1 2𝜃1) 𝑐[𝜇4 + 2𝜇3𝜆1 + 𝜇2𝜆1 2𝜃1 + 2𝜇3𝜆2 + 𝜆1𝜆2𝜇2𝜃1 + 3𝜆1𝜆2𝜇2 + 𝜆2 2𝜃2𝜇2 + 2𝜆1𝜆2 2 𝜇 𝜃2 + 𝜆1 2𝜆2 2𝜃1𝜃2 + 𝜆1𝜆2𝜇2𝜃2] ) Figure – 1: The Chart for infected person’s Expected time - 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1 2 3 4 5 6 7 8 9 10 E(T) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 458 https://internationalpubls.com Figure – 2: The Chart for infected person’s Variance time 5. CONCLUSIONS When 𝜇 is kept fixed with other parameters 𝜆1, 𝜆2, 𝜃1 and 𝜃2 the inter-arrival time ′𝑐′, which follows Exponential distribution, is an increasing parameter. Therefore, the value of the expected time 𝐸(𝑇) to cross the threshold of seroconversion is decreasing, for all cases of the parameter value 𝜇. when the value of the parameter 𝜇 increases, the expected time is also found increasing, this is observed in Figure a and the same case is found in variance 𝑉(𝑇) which is observed in Figure 2. Mathematical models have significantly advanced our understanding of HBV epidemiology and played a vital role in public health decision-making. By informing the design and implementation of effective control and prevention strategies, these models have helped shape responses to HBV at both national and global levels. As we continue to refine these models with new data and insights, we move closer to the goal of reducing the global burden of HBV infection, as well as its associated morbidity and mortality. Continued investment in research and surveillance is essential to deepen our understanding of HBV transmission dynamics and to optimize intervention strategies aimed at mitigating this persistent public health threat. References 1. Spradling PR, Richardson JT, Buchacz K, Moorman AC, Brooks JT. Prevalence of chronic hepatitis B virus infection among patients in the HIV Outpatient Study, 1996–2007. J Viral Hepat. 2010. Available at: https://www.ncbi.nlm.nih.gov/pubmed/20158604. 2. Kim J, Newcomb CW, Carbonari DM, et al. Hepatitis B care cascade among people with HIV/HBV coinfection in the North American AIDS Cohort Collaboration on Research and Design, 2012– 2016. PLoS One. 2023;18(9):e0290889. Available at: https://pubmed.ncbi.nlm.nih.gov/37656704. 3. Platt L, French CE, McGowan CR, et al. Prevalence and burden of HBV co-infection among people living with HIV: a global systematic review and meta-analysis. 2019. Available at: https://onlinelibrary.wiley.com/doi/10.1111/jvh.13217. 4. Thio CL, Seaberg EC, Skolasky RJ, et al. HIV-1, hepatitis B virus, and risk of liver-related mortality in the Multicenter Cohort Study (MACS). 2002;360(9349):1921-1926. Available at: https://www.ncbi.nlm.nih.gov/pubmed/12493258. 5. Konopnicki D, Mocroft A, de Wit S, et al. Hepatitis B and HIV: prevalence, AIDS progression, response to highly active antiretroviral therapy and increased mortality in the EuroSIDA cohort. 2005;19(6):593-601. Available at: https://www.ncbi.nlm.nih.gov/pubmed/15802978. 0 0.5 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 7 8 9 10 Chart Title https://www.ncbi.nlm.nih.gov/pubmed/20158604 https://pubmed.ncbi.nlm.nih.gov/37656704 https://onlinelibrary.wiley.com/doi/10.1111/jvh.13217 https://www.ncbi.nlm.nih.gov/pubmed/12493258 https://www.ncbi.nlm.nih.gov/pubmed/15802978 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 459 https://internationalpubls.com 6. Hoffmann CJ, Seaberg EC, Young S, et al. Hepatitis B and long-term HIV outcomes in coinfected HAART recipients. 2009;23(14):1881-1889. Available at: https://www.ncbi.nlm.nih.gov/pubmed/19550291. 7. Yoshikawa S, Yoshio S, Yoshida Y, et al. Impact of immune reconstitution-induced hepatic flare on hepatitis B surface antigen loss in hepatitis B Virus/human immunodeficiency virus-1 coinfected patients. J Infect Dis. 2021;223(12):2080-2089. Available at: https://pubmed.ncbi.nlm.nih.gov/33073291. 8. Iannetta M, Crea AMA, Di Lorenzo A, et al. Hepatitis B-related hepatic flare during immune reconstitution syndrome after antiretroviral treatment initiation in an HBV surface antigen-positive patient with HIV: viroimmunological and histological characterization. Open Forum Infect Dis. 2022;9(9):ofac451. Available at: https://pubmed.ncbi.nlm.nih.gov/36092833. 9. Bellini C, Keiser O, Chave JP, et al. Liver enzyme elevation after lamivudine withdrawal in HIV- hepatitis B virus co-infected patients: the Swiss HIV Cohort Study. HIV Med. 2009;10(1):12-18. Available at: https://www.ncbi.nlm.nih.gov/pubmed/18795964. 10. Hall SAL, Burns GS, Mooney BJ, et al. Hepatitis B virus flares after nucleot(s)ide analogue cessation are associated with activation of toll-like receptor signaling pathways. J Infect Dis. 2022;227(1):123-132. Available at: https://pubmed.ncbi.nlm.nih.gov/36108079. 11. Mican R, Busca Arenzana C, Vasquez J, Daroca G, Perez-Valero I, Martin-Carbonero L. Hepatitis B reactivation after tenofovir withdrawal in an HIV-infected patient with history of cured hepatitis B virus infection and poor immunological status. 2021;35(10):1707-1708. Available at: https://pubmed.ncbi.nlm.nih.gov/34270493. 12. Adachi E, Sedohara A, Arizono K, et al. Hepatitis B virus reactivation after switch to cabotegravir/rilpivirine in patient with low hepatitis B surface antibody. Emerg Infect Dis. 2024;30(8):1668-1671. Available at: https://pubmed.ncbi.nlm.nih.gov/39043430. 13. Pintado C, Delaugerre C, Molina JM. Acute hepatitis B infection after a switch to long-acting cabotegravir and rilpivirine. Open Forum Infect Dis. 2020;7(9):ofaa367. Available at: https://pubmed.ncbi.nlm.nih.gov/33005698. 14. Abdullahi A, Fopoussi OM, Torimiro J, Atkins M, Kouanfack C, Geretti AM. Hepatitis B virus (HBV) infection and re-activation during nucleos(t)ide reverse transcriptase inhibitor-sparing antiretroviral therapy in a high-HBV endemicity setting. Open Forum Infect Dis. 2018;5(10):ofy251. Available at: https://pubmed.ncbi.nlm.nih.gov/30377627. 15. Esary, J.D., Marshall. A.W., and Proschan. F (1973). Shock models and wear processes. Ann. Probability, 627-649. 16. Nirmala, P. Ratchagar, Vijaya. S., Sathiyamoorthy.R. and Kannan. R., (2003). A Stochastic Model for The Estimation of Time to Seroconversion of HIV Infected Using Order Statistics. Proceedings of the national conference on “Mathematical and Computational Models” at PSG college of Technology, coimbatore. 17. Sathiyamoorthi. R and R. Kannan (1998). On the Time to Seroconversion of HIV Patients under Correlated Inter Contact Times, Pure and Applied Mathematika Science vol. XLVIII. No. 1-2, pp. 75 -87. https://www.ncbi.nlm.nih.gov/pubmed/19550291 https://pubmed.ncbi.nlm.nih.gov/33073291 https://pubmed.ncbi.nlm.nih.gov/36092833 https://www.ncbi.nlm.nih.gov/pubmed/18795964 https://pubmed.ncbi.nlm.nih.gov/36108079 https://pubmed.ncbi.nlm.nih.gov/34270493 https://pubmed.ncbi.nlm.nih.gov/39043430 https://pubmed.ncbi.nlm.nih.gov/33005698 https://pubmed.ncbi.nlm.nih.gov/30377627