Hidden Markov Model of Evaluation of Break-Even Point of HIV patients: A Simulation Study

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Hidden Markov Model of Evaluation of Break-Even Point of HIV
patients: A Simulation Study
Mahalakshmi Rajendran1
, Senthamarai Kannan Kaliyaperumal2
, Balasubramaniam Ramakrishnan3
1, 3
Research Scholar, Department of Statistics, Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli, Tamil Nadu, India.
2
Professor of Statistics, Department of Statistics, Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli, Tamil Nadu, India.
.
Background: The HIV virus carries projection of significant global population with specific estimations of the mathematical results of
evolutionary methods which was presented in Tree Hidden Markov model (HMM).
Materials and Methods: Hidden Markov models used to model the progression of the disease among HIV infected people. The author
predicts a Baum Welch Algorithm method through HMM that can assess an unknown state of transition.
Results: The Tree HMM model predicts the break down point starts once patient is infected with the HIV virus as it affects the immune
system. The immune system drops more quickly in the initial inter arrival time when compared with the later time interval. The HIV virus
length in the nth
state within regrouping is uncertain to occur in each state of the given model. A simulation study was done to assess the
goodness of fit for the model.
Conclusion: The HIV virus length in the nth
state within regrouping is uncertain to occur in each state of the given model. The inter arrival
censoring between each state is essential in each infected HIV patients. The outcome of this works states that health care expert can use this
model for effective patient cares.
Keywords: expectation, hidden markov model, human immunodeficiency virus, immune system, transition
Introduction
Twenty-Six million people in 2020 June, were assessing the human immunodeficiency virus (HIV) antiretroviral therapy when compared to
2019 end an estimation of 25.4 million, an estimated 2.4% of increase was observed. Awareness among pregnant and breastfeeding women
have been increased around 85% who have received ART living with HIV, this avoids HIV transmission to their newborns and also ensures
their protective health. The 69th
World Wellbeing Gathering proposed a "Worldwide wellbeing area technique on HIV for 2016-2021”. [1]
The arrangement offered five vital headings, which are as per the following: data on designated activity of once pestilence and reaction,
counteraction, treatment, and care, and exploration. The impact of mediations on the administrations required, guaranteeing uniformity for
the populaces needing administrations, getting long haul subsidizing to pay the expenses of administrations, and speeding up the change to
a manageable future are immensely significant contemplations. [2] UNAIDS has set a 2030 cutoff time for the destruction of the HIV
pandemic, which will match with World Guides Day in 2014. As indicated by gauges, about 2.39 million individuals in India are tainted
with HIV, making it the third most crowded country on the planet. South India was the main region to be hit by the HIV pandemic since it
had the most noteworthy populace thickness at that point. [3]
Hidden Markov Model (HMM) is an extension of Markov model. Markov Model was named after Andrei Andreyevich Markov who lived
in the year (1856-1922). Markov Chain is a statistical model where the data describes in sequence form. HMM is an especially embedded
How to cite this article: Rajendran M, Kaliyaperumal SK, Ramakrishnan B. Hidden Markov Model of Evaluation of Break-Even Point of
HIV patients: A Simulation Study. Int J Med Sci and Nurs Res 2021;1(2):19-22
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Corresponding Author: Ms. Mahalakshmi Rajendran,
Research Scholar, Department of Statistics, Manonmaniam
Sundaranar University, Abishekapatti, Tirunelveli, Tamil Nadu, India.
Email ID: mahalakshmirajendran@gmail.com
International Journal of Medical Sciences and Nursing Research 2021;1(2):19-22 Page No:19
Abstract
Article Summary: Submitted:12-October-2021 Revised:16-November-2021 Accepted:24-December-2021 Published:31-December-2021

under the umbrella of stochastic process where each state holds the
Markov property. [4] The three main information to be observed in the
HIV affected immune system is the parameter space, state space and
state transition probability. [5]
Mathematical and Statistical models for infectious diseases commonly
in the process of looking forward in estimating the epidemic which helps
different public health sectors to plan optimally. Recent literature shows
large number of literatures on Mathematical Models for communicable
diseases. [6] A validated goodness of fit model (HMM) been used as an
investigative to expect the diseases progression outcomes in infected
cows. [7] Mathematical Modelling has been identified at the early stage
of HIV epidemiological research, also concluded that theoretical
research focuses on quantitative data on sequential changes in the
mathematical distribution of sexual partner change along with other
factors like variations in epidemiologic abundance in serum and
emissions. [8] Mathematical Modelling suggests the cost effectiveness
and time of HIV pandemic interventions, when given the right
information to experimental trials. As the HIV pandemic is being a silent
global threat since last four decades. [9] The HMM topology inference
model denotes its graphical figures including the number of states with
the association of symbols in each different state and state transitions
with non-zero probabilities. Assuming the HMM model always specify
the states prior to the information received. [10]
The Baum Welch Algorithm was published by Baum LE and along with
coauthors who worked through his articles, even the name “Welch”
appears as the coauthor that have been worked in developing this Baum
Welch Algorithm. This algorithm was an example of Expectation
Maximization (EM) algorithm. Mathematical methods associate to the
algorithm along with an explanation as how the Baum Welch Algorithm
fits the EM were also seen. [11 – 15]
We assume that the human immune system gets affected with HIV in a
future state when the present state is already affected with HIV. The
non-observable damage causing the immune system which leads to the
HMM is the one to observe in this article. When the human system gets
affected with HIV, it is represented by time t=1, which is the initial state
of the process. At every time interval the human system moves from the
current position to another position, i.e., t = (1, 2, 3, … …), the transition
probabilities are independent of the time t.
Materials and Methods:
Hidden Markov Model: [10] A continuous process to develop
model parameters in the transition state to explain the respective time
point in the infected patients. A Hidden Markov Model (HMM) is
usually represented by 𝐻𝑀𝑀: 𝜇 = (𝐴, 𝐵, 𝜋). This model tells us; the
state transition probability, observational probability, probability of
starting in a particular state. The Baum-Welch algorithm also known as
EM-algorithm to emphasis on parameter estimation built on direct
numerical maximum likelihood estimation. To maximize and find the
posterior estimation of the hidden variables of HIV infected patients. The
estimation depends on the assumption of the independent observations
Tree HMM as seen in Figure-1. Transition variables defined as;
𝑝𝑡(𝑖, 𝑗), 1 ≤ 𝑡 ≤ 𝑇, 1 ≤ 𝑖, 𝑗 ≤ 𝑁
Figure–1 Hidden random variable shown with Tree HMM
𝑎𝑖𝑗
,
=
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛𝑠 𝑓𝑟𝑜𝑚 𝑠𝑡𝑎𝑡𝑒 𝑖 𝑡𝑜 𝑗
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛𝑠 𝑓𝑟𝑜𝑚 𝑠𝑡𝑎𝑡𝑒 𝑖
=
∑ 𝑝𝑡 (𝑖, 𝑗)
𝑇
𝑡=1
∑ 𝛾𝑖(𝑡)
𝑇
𝑡=1
𝑎𝑖𝑗
,
=
∑ 𝛼𝑖(𝑡)𝑎𝑖𝑗𝑏𝑗(𝑂𝑡+1)
𝑇
𝑡=1 𝛽𝑗(𝑡 + 1)
∑ 𝛼𝑖(𝑡)𝛽𝑖(𝑡)
𝑇
𝑡=1
… … … (1)
𝑃𝑟(𝑖𝑗) = 𝑃𝑟(𝑆𝑡=𝑖,𝑆𝑡+1=𝑗/𝑂, 𝜇)
=
𝑃𝑟(𝑆𝑡=𝑖,𝑆𝑡+1=𝑗/𝑂, 𝜇)
𝑃(𝑂/𝜇)
… … … (2)
=
𝛼𝑖(𝑡)𝑎𝑖𝑗𝑏𝑖𝑗𝑜𝑡𝛽𝑗(𝑡 + 1)
∑ ∑ 𝛼𝑚(𝑡)𝑎𝑚𝑛𝑏𝑚𝑛𝑜𝑡
𝑁
𝑛=1
𝑁
𝑚=1 𝛽𝑛(𝑡 + 1)
Equation (2) observes the probability of being at state 𝑖 at time 𝑡, and
at state 𝑗 at time 𝑡 + 1, given the model 𝜇 and the observation 𝑂.
Then, define 𝛾𝑖(𝑡) this is the probability of being at state 𝑖 at time 𝑡,
given the observation 𝑂 and the model 𝜇, as seen in equation (3),
𝛾𝑖(𝑡) = 𝑃𝑟 (
𝑆𝑡=𝑖
𝑂
, 𝜇) = ∑ 𝑃𝑟(𝑆𝑡=𝑖,𝑆𝑡+1=𝑗/𝑂, 𝜇)
𝑁
𝑗=1
… … … (3)
= ∑ 𝑃𝑟(𝑖, 𝑗)
𝑁
𝑗=1
The above equation (3) holds because 𝛾𝑖(𝑡) is the expected number of
transitions from state 𝑖 and 𝑝𝑡(𝑖, 𝑗) is the expected number of transitions
from 𝑖to 𝑗. Given the above definitions we begin with an initial model
𝜇 and simply it for different states.
Rajendran M et al. Hidden Markov Model of Evaluation of Break-Even Point of HIV patients

𝜋𝑖
,
= 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑏𝑒𝑖𝑛𝑔 𝑎𝑡 𝑠𝑡𝑎𝑡𝑒 𝑖 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 = 1; = 𝛾𝑖(𝑡)
𝑎𝑖𝑗
,
=
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛𝑠 𝑓𝑟𝑜𝑚 𝑠𝑡𝑎𝑡𝑒 𝑖 𝑡𝑜 𝑗
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛𝑠 𝑓𝑟𝑜𝑚 𝑠𝑡𝑎𝑡𝑒 𝑖
=
∑ 𝑃𝑟(𝑖, 𝑗)
𝑇
𝑡=1
∑ 𝛾𝑖(𝑡)
𝑇
𝑡=1
… … … (4)
𝑏𝑖𝑗𝑛
,
=
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛𝑠 𝑓𝑟𝑜𝑚 𝑖 𝑡𝑜 𝑗 𝑤𝑖𝑡ℎ 𝑛 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛𝑠 𝑓𝑟𝑜𝑚 𝑖 𝑡𝑜 𝑗
=
𝑡:𝑂𝑡=𝑛,1≤𝑡≤𝑇
𝑇
𝑡=1
… … … (5)
Results and Discussion
The three states are defined as; First state the initial state of HIV infection
identified and under treatment (i.e., the person identified as HIV positive
starting from the initial time period); Second State identified as the person
infected under HIV after some period of initial time period; Third state
observes the later time period of the infected person (i.e., the HIV infected
persons are not aware of the diseases in them and identified it very lately).
A simulation study was done to assess the goodness of fit for the model.
The simulation was carried out using Mathcad Software and the graphical
representation was figured through Minitab software.
Table–1 HIV infected patients risk observed in the three states
as time increases
Time Per
Week
First State Second
State
Third
State
1 2 3 4
2 1.5 1.5 2
3 1.33 1 1.333
4 1.25 0.75 1
5 1.2 0.6 0.8
6 1.16 0.5 0.667
7 1.14 0.429 0.571
8 1.12 0.375 0.5
9 1.11 0.333 0.444
10 1.1 0.3 0.4
20 1.05 0.15 0.2
30 1.03 0.1 0.133
40 1.02 0.075 0.1
50 1.02 0.06 0.08
The Tree HMM model predicts the break down point starts once
patient is infected with the HIV virus as it affects the immune
system. As the infected patient passes from one state to another the
likelihood of high risk is more in the HIV patient as observed in
Table-1 and Figure-2. The hidden nature of the virus is clearly
observed in Table-1, stating the infected patient has a very less
chance of survival as and when the time increases. The immune
system drops more quickly in the initial inter arrival time when
compared with the later time interval. The model finally concludes
that, assessing the HIV patients at the initial time and state the
likelihood of risk is less. As the time and state increases the
likelihood of risk increases compared to the previous state.
Figure–2 Three states of HIV infected patient’s risk
This simulation study attempts to make predictions of HIV patients
and assess the performance of the model. For this, the dataset had
taken from the World Health Organization Website. [2] The dataset
had categorized into three subparts and renamed by states. The
states are: S1 also known as the first state, is the initial state of HIV
infection identified and under treatment. In this way S2, second
state is the person infected under HIV; S3 is the state observes the
later time period of the infected person.
Using the three states, the risk for the patients in the above states in
every week was estimated and tabulated as shown in Table-1. The
same estimated values were visualized using a three-dimensional
graph as shown in Figure-2. Thus, the Hidden Markov Model was
trained and the prediction was made using the Baum Welch
Algorithm. [13, 14] The performance of the trained model was
assessed. The risk of the patients in the three states also discussed.
Conclusion
The HIV virus carries projection of significant global population
with specific estimations of the mathematical results of
evolutionary methods which was presented in Tree HMM model.
Our model assumes that the HIV infected patients are possibly of
high risk in after state one. This HIV infected patients are of a single
controlling strain in each state of the Tree HMM model. The HIV
virus length in the nth
state within regrouping is uncertain to occur

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in each state of the given model. The inter arrival censoring between
each state is essential in each infected HIV patients. The outcome of
this works states that health care expert can use this model for effective
patient cares.
Authors’ contributions: MR, SKK, BR: Conception and Study
design; MR: Acquisition of Data; MR, SKK, BR: Data processing,
Analysis and Interpretation of Data; All authors MR, SKK, BR were
drafting the article, revising it for intellectual content; All authors were
checked and approved of the final version of the manuscript.
Here, MR – Mahalakshmi Rajendran; SKK – Senthamarai Kannan
Kaliyaperumal, BR – Balasubramaniam Ramakrishnan
Source of funding: None
Conflict of interest: None
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