Stochastic Modeling of Viral Replication and Lysing CD4 T Cells in the HIV Infection

Received: 03/Aug/2018, Accepted 23/Aug/2018, Online 31/Oct/2018 Abstract-The existing models of HIV infection are non-linear system of differential equations. Solving system of differential equations is very difficult task and also drawing inference is not easy. Therefore, an attempt has been made to estimate the HIV replication periodically using Markov processes in the condition of decay of CD4 T cells. The proposed model is illustrated in this paper. KeywordsCD4 + T cell, HIV and Markov processes


I. INTRODUCTION
In the viral dynamic study, the HIV infection spread in the human being is very vast in the world. It is a major epidemic in the world now days.HIV infection is naturally distory the human the immune system, in particularly by depleting the CD 4 + T cells.
The HIV transmission system has biologic and social determinants. Biologic determinants include characteristics of the pathogen, the host, and biomedical interventions. Social determinants include individual-level, pair wise and community-level processes that affect behavior, and thus the structure and dynamics of the transmission networks. In the 1990s, the time to development of AIDS after initial infection with the virus is approximately 10 to 12 years. In the mid 1980s, however, the average time from infection to AIDS was 8 to 10 years (Klatt, 1998). This improvement in time to development of AIDS is due, in part, to improved diagnosis, increased use of antiretroviral therapy and improved management of opportunistic infections.At this stage of infection, viral load in an individual may be extremely high, around one million copies/ml, although individual variation is significant. Although CD 4 + T cell counts may also vary, individuals with CD 4 + counts below 200 cells/mm3 are at the greatest risk of developing opportunistic infections (note that CD 4 + T cell counts of healthy individuals are usually above 1000 cells/mm3) (Chibatamoto, 1996).The overall effect of infection with HIV and its interaction with the body's natural response mechanisms is severe damage to the immune system, destroying by the means which the human body naturally defends itself against infections. Following entry into the host is disseminated via the blood and circulatory system to different tissues in the body. From this moment of infection, the virus is replicating at extremely rapid rates. As the virus replicates and spreads throughout the body. Effectively, the virus has now hijacked the host cell's own replication system. As a result, when the cellular DNA is transcribed, so the viral DNA to form an RNA transcript. Further processing of this RNA into messenger RNA (mRNA) and genomic viral RNA occurs. The viral mRNA is then translated into viral proteins, which along with the genomic RNA, are assembled into new virus particles. This last stage requires the viral enzyme, protease (Marr, 1998). Finally, the new viral particles are released from the infected cell and go on to infect other cells in the body.
This paper concentrated to the periodically viral replication of infected persons. The stochastic models are designed for the viral replication in the CD 4 + T cells and lysing CD 4 + T cells count and illustrated.
HIV dynamics that included combined therapeutic treatment and intracellular delay between the infection of a cell and the emission of viral particles. This is model included dynamics of three compartments the number of healthy CD4 cells, the number of infected CD4 cells, and the HIV virons and it described HIV infection of CD4 T cells before and during therapy. Let as assume that Succeeding period viral load {X t, s } is random variable which is distributed as exponential with parameter = and Transition probability of succeeding periods is given by The State space is viral replication ever the n period is given by The Time space has n-period is denoted by HIV replication is treated as the markov chain, the sequence of random variable the time space and state space is finite. Time space-is treated as every period of (every three months) monitoring viral level.
= 1,2, … , State space-is treated as viral load per period say 1 , 2 , … Viral replication is random variable which is considered as rapid growth. It is monotonically increasing function and also infect to rapidly to = 1,2, … , and ′ state. , is said to be the viral replication in the blood plasma of HIV infected person. The probability matrix of the "N" infected persons and their n period viral replication is given by. Where, ∩ +1, = 1,2, … , , = 1,2, … and 0 < < 1, > 0.
is said to be continuous time markov chain and its time space is stationary steady state (every three month).but the state space is the increasing order of the exponential growth 1 < 2 < 3 … < … < .The transition probability of the j th patient i th period viral replication is denoted by . →is j th patient's probability that to replicate the virus in the 4 + cell. is viral load per period. ( 0 < < 1, > 0) Let as assume that viral replication is towards constant rate for successions periods, = 10 2 .
The average viral replication of the I st patient is denoted by, The average replication of N th patients first period average replication is given. The following table-I explain the exponential nature of the viral replication for the future period is illustrated. Graph I illustrate that viral load is increasing and probability of infection increases by assumption of exponential growth rate of replication.   The state space A and time space ∈ . , ∈ , ∈ 0, ∞ for every three months. Interval, is markov process, and 0 < 1 < 2 < ⋯ < . Then

Viral replication in a individual
Where, The expected number of replication at +1 is same as the is called as martingale process. The stationary probability matrix of succeeding time periods of 4 + cells.
The transition stationary probability matrix is illustrated below.
The distribution of lysing 4 + cells is given in the following matrix over the period of time.  The above graph II illustrate that the variation of viral load follows the normal distribution from this we identify the probability of infection is increased.

IV. CONCLUSION
The model which is estimated here to the HIV replication periodically as the Markov processes under the condition of decay of CD 4 + Tcells. This idea will help to the physician those who me treated HIV infected patients to suggest a proper treatment for infected patients in advance. When data in the large size viral load of the infected patients follows normal is distribution in future and the graph II explain the variation of viral load when the probability infection is increased.