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Stability and Sensitivity Analysis of Treatment and Distancing in a
Mathematical Model for Co-existing Drug-Sensitive and Drug-
Resistant Tuberculosis Strains in Bangladesh
M. A. Salek
1
, J. Nayeem
2
, M. Humayun Kabir
1
1
Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh
2
Department of Arts & Sciences, Ahsanullah University of Science and Technology, Tejgaon, Dhaka,
Bangladesh
DOI: https://dx.doi.org/10.51244/IJRSI.2025.1210000016
Received: 06 October 2025; Accepted: 12 October 2025; Published: 27 October 2025
ABSTRACT
This abstract provides a summary of a study that employs mathematical modeling to analyze a dual-strain
tuberculosis (TB) structure in Bangladesh, focusing on the identification of drug-resistant (DR) and drug-
susceptible (DS) strains. The model features a distinctive element known as "amplification," which illustrates
how insufficient treatment of DR TB can arise from the management of DS TB. Utilizing both analytical and
numerical techniques, the research investigates the disease's dynamics and its potential long-term implications.
The primary findings indicate that the long-term dynamics of TB within a population are influenced by the
basic reproduction numbers for each strain, referred to as
and
r. The disease tends to naturally decline
when there are fewer cases compared to a single growth rate. Conversely, if
exceeds both
and one, DR
TB will continue to exist while DS TB is eradicated. If
surpasses
and one, both strains will persist
together. Additionally, the research conducted an analysis of vulnerability to identify the key factors impacting
the disease's transmission rate. It was determined that the transmission rates (
and
) of both strains
significantly influence the progression of the illness. This underscores the necessity for public health initiatives
to focus on strategies that minimize interactions between infected and uninfected individuals, such as
educating patients on respiratory safety and enhancing ventilation systems. Another critical factor is the
treatment rate (
and
). The social implications of this study are considerable, suggesting that an effective
approach for nations like Bangladesh is to improve treatment accessibility by lowering costs through universal
healthcare. Timely and appropriate treatment of DS TB is the most effective method to mitigate resistance
development, while adequate management of DR TB is crucial to prevent its proliferation within the
population.
Keywords: Tuberculosis (TB), Mathematical Investigation, Drug Resistance, Amplification, Basic
Reproduction Number (
), Studying of Vulnerability.
INTRODUCTION
Globally, tuberculosis (TB) causes significant morbidity and mortality, making it a serious public health
concern (Syeda Mariam Riaz et. al.) [1]. The World Health Organization (WHO) confirmed that Tuberculosis
(TB) is the biggest factor behind passing away from only one spread of illness in 2019, with over 10.0 million
additional infections and 1.2 million mortality. Nearly 70% of all cases worldwide occur on the African and
Asian continents combined, indicating that the illness burden is disproportionately high in environments with
limited resources (Jiseon Lee et. al.) [2]. The disease, which is caused by the Mycobacterium tuberculosis
(MTB) bacteria, is mainly communicated through the respiratory system because an infected individual can
pass the bacteria to a vulnerable host by aerosolized droplets emitted while coughing or sneezing (Dheda,
Keertan et. al.) [3]. A key component of the epidemiology of tuberculosis is the intricate life cycle of the
bacillus within a human host (Bianca Sossen et. al.) [4]. Latent tuberculosis infection can result from the germs
going into a non-replicating, dormant stage after the initial inhalation. People with LTBI don't have any
symptoms and aren't a threat to others (Justin et. al.) [5]. The dormant bacteria may, however, reawaken and
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develop into an infectious disease in 5–15% of patients. In those with weakened immune systems, this rate is
noticeably higher. Eradication of tuberculosis is a long-term objective due to the crucial dynamic of latent and
active infection (Tunde T. Yusuf et. al.) [6]. A successful public health response must treat both active and
latent disease because the sizable pool of latent infections acts as a reservoir for subsequent epidemics (Nathan
Kapata et. al.) [7]. The rise of drug-resistant tuberculosis poses an obscure risk for worldwide TB control.
These strains, which can spread directly from person to person, are characterized by resistance to the main
first-line medications, isoniazid and rifampicin (Mchaki, Betty R et. al.) [8]. But a more common manner for
them to appear is when people with drug-susceptible (DS) TB receive insufficient or no treatment (P.G.C.
{Nayanathara Thathsarani Pilapitiya}et. al.) [9]. Often referred to as "amplification," this process enables
surviving germs to become resistant due to patient non-adherence, inappropriate prescription practices, or
irregular access to medication (Geoffrey P. Dobson et. al.) [10]. Compared to treating DS TB, treating DR TB
is far more challenging and resource-intensive, requiring longer, more toxic, and substantially more costly
regimens (Kai Ling Chin et. al.) [11]. This imposes a significant strain on healthcare systems, especially in
developing countries, and highlights the critical necessity to comprehend the fundamental factors contributing
to the emergence and spread of DR TB (Xinyue Wang et. al.) [12]. A useful framework for analyzing the
intricate interactions between biological and social elements that control infectious disease epidemics, such as
tuberculosis, is provided by mathematical modeling (Ram Singh et. al.) [13]. These models have long been
used by researchers to predict the possible effects of different intervention measures and to obtain insights into
the epidemiology of tuberculosis. Murphy et al. (2002), for example, looked into how biological and
socioeconomic variables affect the prevalence of tuberculosis using an SEI (Susceptible-Exposed-Infected)
model. Other modeling attempts have looked at certain epidemiological phenomena, like how seasonal
changes and exogenous reinfection affect the prevalence of tuberculosis (Eka D.A. Ginting et. al.) [14].
(Khristine Kaith S. Lloren et. al.) [15] Evaluated the combined impacts of vaccination, reinfection, and de
novo resistance using a more complex 10-compartment model, emphasizing the critical role that reinfection
plays in the development of disease. The body of research demonstrates the effectiveness of mathematical
models in representing the complex dynamics of tuberculosis transmission (V.M. Mbalilo et. al.) [16]. The
application of mathematical models to inform public health policy is of paramount importance. A theoretical
foundation for assessing the efficacy of prophylaxis and treatment was established by early, which showed that
reducing treatment failure rates is essential for managing epidemics, particularly in developing nations (Sirwan
Khalid Ahmed et. al.) [17]. Multi-strain models have been created more recently to examine the intricacies of
medication resistance (Sirwan Khalid Ahmed et. al.) [18]. Using a two-strain model, (Sudipa Chauhan et. al.)
[19] determined the best combinations of control measures and came to the conclusion that integrated
approaches are more economical and effective. Similar to this, (Md Abdul Kuddus et. al.) [20] examined
disease equilibrium states using a two-strain model and found that the disease will endure if the fundamental
procreation rates for both strains are greater than the other but will die out if they are less than one.
This paper presents a novel two-strain mathematical model, building on previous research and driven by the
ongoing danger of tuberculosis and the increasing prevalence of medication resistance in Bangladesh. The
public health issue of acquired drug resistance—where a drug-susceptible infection might become a drug-
resistant one due to treatment failure—is explicitly incorporated into our Susceptible-Latent-Infectious-
Removed-Susceptible (SLIRS) framework. The model can be used to forecast the future course of the
epidemic and offers a strong platform for investigating the competitive dynamics between DS and DR TB
strains (Mumbu et. al.) [21].
This paper's remaining sections are organized to offer a thorough examination of the suggested model. Using
the next-generation matrix approach, we start with a thorough analytical study to obtain estimates for the basic
fertilization rates of the DS and DR strains (Maryam Zarean et. al.) [22]. Next, from a mathematical and
biological perspective, we determine the criteria required for the maintenance of healthy, single-present, and
adjoining pandemic equilibria.
Numerical simulations are used to figure out the mathematical formulas and demonstrate how the system
behaves dynamically under various scenarios, so complementing the analytical conclusions (Chinmay Saha et.
al.) [23]. To identify the factors that have the biggest effects on we investigate a higher incidence of DS, DR,
and total TB infections, as well as vulnerability. The ultimate objective of this study is to produce useful
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insights into the evolution of TB in this country and aid in the creation of efficient, empirically supported
control measures (Most Amina et. al.) [24].
Objectives are:
To present a novel two-strain mathematical model for the transmission dynamics of drug-sensitive (DS) and
drug-resistant (DR) tuberculosis in Bangladesh.
To explicitly incorporate the public health issue of "amplification," where drug-susceptible infections can
become drug-resistant due to treatment failure.
To forecast the future course of the TB epidemic and investigate the competitive dynamics between DS and
DR TB strains.
To perform a thorough analytical study to derive the basic reproduction numbers for the DS and DR strains
using the next-generation matrix approach.
To determine the conditions for the existence and stability of disease-free, single-present, and co-existent
endemic equilibria from both a mathematical and biological perspective.
To identify the key parameters that have the biggest effects on the incidence of DS, DR, and total TB
infections through a sensitivity analysis.
To produce useful insights to aid in the creation of efficient, empirically supported control measures for the
evolution of TB in Bangladesh.
Techniques and resources
In order to examine his study proposes a deterministic mathematical representation for the epidemiological
patterns of drug-sensitive (DS) and drug resistance (DR) tuberculosis organisms. The population is divided
into the following mutually exclusive compartments under the model:
At risk: People who have not yet contracted tuberculosis.
Potentially contaminated (
): Refers to individuals considered sick but are not experiencing a severe
illness. Drug-sensitive and drug-resistance to TB both are indicated by the subscripts D and R, respectively.
(whereas the initials i = s, r indicates the proportions related to the TB infection that are susceptible to drugs (s)
and resistance to drugs (r).)
Infectious (
): People who have active tuberculosis and are able to spread the infection. (whereas the
initials i = s, r indicates the proportions related to the TB infection that are susceptible to drugs (s) and
resistance to drugs (r).)
Recovered (R): Individuals who have received effective treatment and are momentarily immune.
One of the model's main assumptions is that drug resistance (DR) might be directly conveyed after insufficient
treatment for drug-susceptible (DS) TB. Additionally, the process of transient immunity loss is included in the
model, wherein recovered people can revert to the susceptible state at a consistent per-capita rate. The entire
population Growth N(t) is determined by
. (1)
While active TB cases in
incur disease-related death at a rate of
, individuals in the
other compartments die naturally as its similar consistent amount . Newborn deaths take place in the
vulnerable compartment to maintain a stable population size. At an amount that changes with time
, where
is the rate of propagation among susceptible as well as infected people, a prevalent MTB
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organism can harm people in the S area . People migrate to the latently infected compartment
after
contracting the strain
. Autonomous proliferation of dormant bacteria at an amount
causes a
percentage of people with latent infections to become active TB. The rate of treatment for individuals with
drug-sensitive (DS) active TB,
, is
. Some of them, denoted by , will recover well after treatment and join
the recovered group. Unfortunately, drug resistance develops in the complementary proportion
, which
is the leftover fraction. This may occur as a result of improper medication adherence or insufficient treatment.
They are transferred to the drug-resistant (DR) compartment
as a result. In the same way, patients with DR-
active TB receive treatment at their own pace,
.
Table 1: Measurement evaluation and modeling.
Parameters
Description
Estimated
value
Population in 2015
159,000,000
29
Birth/death
rate
per year
30
Transmission rate for DS TB
1.57 × 10
−8
Fitted
Transmission rate for DR TB
6.25 × 10
−9
Fitted
Progression
rate
from
to
0.129
per
year
16
Progression
rate
from
to
0.129
per
year
16
Recovery rate for DS TB
0.287
per
year
12
Recovery rate for DR TB
0.12
per
year
12
Proportion of treated patients who amplify
0.07
per
year
11
Disease related death rate for DS TB
0.37
over
3
years
16
Disease related death rate for DR TB
0.37
over
3
years
16
Treatment rate for DS TB
0.94
per
year
20
Treatment rate for DR TB
0.78
per
year
20
Rate of losing immunity
0.10
per
year
12
The World Health Organization's (WHO) yearly incidence data for drug-sensitive (DS) and drug-resistant
(DR) tuberculosis (TB) are compared with the results of our suggested model in Figure 2. The model's best fit
to the data is represented by the green solid curve, while the WHO-reported data points are displayed as blue
dots. This comparison is displayed for DS TB in the graph on the left and for DR TB in the graph on the right.
This graphic illustrates how closely the model's predictions match actual data.
Additionally, people with active tuberculosis can go from the
compartment to the
compartment by
recovering naturally at a rate of
. The flow diagram in Figure 1 illustrates this process as well as other
population movements. Table 1 contains the precise values for every parameter utilized in this model,
including the recovery rate
. Deterministic systems of chaotic basic mathematical problems control the
overall dynamics of drug-sensitive (DS) and drug-resistant (DR) tuberculosis transmission:
(2)
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(3)
(4)
(5)
(6)
(7)
It is simple to demonstrate that every state variable in the differential equation system stays non-negative for
all times
when the initial conditions are non-negative. We can also calculate the overall population size,
, using equations (2) through (7) requires
= constant
It seems sense that the size of each individual compartment, such as
, etc.
must likewise be restricted since
the initial conditions are non-negative while the overall individuals remain steady. These factors lead to the
following viable zone for the system of equations (2) through (7):
{
s
s
r
r
s
s
r
r
Simulation of prediction of dimensions
The majority of the modeling variables were determined based on current global literature or local evaluations,
according to reports from the National TB Control Program (NTP) and the World Health Organization
(Table
1)
. Bangladesh's total population in 2015 was estimated to be roughly 159,000,000. Biological life expectancy,
symbolized by
(
),
was determined as the opposite of Bangladesh's lifespan of 70 years. In Bangladesh,
treatment success rates for drug-sensitive (DS) and drug-resistant (DR) tuberculosis (TB) were approximately
94% and 78%, correspondingly, according to WHO data from 2020 (Burhan et. al.) [25]. We fitted the method
we used was applied to the DS and DR TB mortality statistics for Bangladesh, which were acquired from
WHO and NTP reports from 2000, to 2018, respectively, in order to estimate the two remaining important
parameters,
and
. In order to reduce a mean discrepancy among both estimated and actual yearly
incidence of either DS or DR TB, the least-square optimization method was used to determine the most
appropriate (see Fig. 2) variables,
and
. Information is provided in the additional materials.
Studying of vulnerability
A global studying of vulnerability was performed to determine their own partial rank correlation coefficients
(PRCCs) data variable and different outcome measures using Latin Hypercube Sampling (LHS) (Md Afsar Ali
et. al.) [26]. This was accomplished by giving each variable (i.e.
and) a
uniform distribution between half and four times its baseline value (Table 1). After that, 10,000,000 simulated
draws were made for each variable, giving the study a solid dataset. The modeling procedure was run for each
of the 10,000,000 essential variable configurations and the results, including the incidence and prevalence of
the disease, were captured. We specifically studied the equilibrium prevalence for three important metrics:
overall TB
rates, which is the sum of drug-sensitive (DS) and drug-resistant DR TB
denoted by
DS and DR, respectively, and drug-sensitive DS TB
represented by DS. The method outlined
can be
used to determine the vulnerability rates for each model variables using the analytical formulas for
and
.
The rate in which a variable varies in response to a change in a specific variable can be identified through these
coefficients. Regarding
, for instance, we all have:
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We can change
, the vulnerability rates for the fundamental reproduction number
are represented by
.
Basis for the authorization, The study was carried by using aggregated surveillance data for drug-sensitive
(DS) and drug-resistant (DR) TB from the National TB Control Program (NTP) in Bangladesh and the World
Health Organization (WHO) (Sarker et. al.) [27]. Since all analyses were conducted at an aggregate level, no
private information was used. The James Cook University human ethics approval board accepted the research
protocol, H7300, and all procedures were conducted in accordance with it.
RESULTS
Fundamental evolution of rate,
There are two healthy states and and four unhealthy states
in
the configuration. There are no active cases at the healthy constant state, where
,
consequently
. We applied the methods to figure out the fundamental production amounts for the DS and
DR TB strains (Karmakar et. al.) [28]. The linearized infection components obtained from equations (2) through
(7) are the primary goal of this method.
(8)
(9)
(10)
, (11)
The total rates at which people are taken out of the active DS and DR TB infection states are represented by
the values
and
. The fluctuations within these infected states
around the non-infectious optimum are described by the ordinary differential equations (ODEs) from (8) to
(11). This is predicated on the idea that the infection-induced decline in the vulnerable population is negligible
enough to be disregarded.
Assume that is the vector of infected states, expressed as
, where is the transpose. We
can now represent the infection component using this notation as follows:
(12)
In order to analyze the model, the matrices
and
are essential. The transition matrix, , records any other
changes in the infected states, whereas the transmission matrix,
, depicts the process by which new infections
are created. The set of ordinary differential equations (8) – (11) is the source of these matrices. The rate at
which people in infected state cause new illnesses in infected state is indicated by an entry in these matrices
with
, such as
. This method aids in distinguishing the onset of new infections from other
occurrences, such as healing, passing away, or changing into a different disease state. Therefore, for the
subsystem (8) − (11), we have,
and
The following generation matrix, represented as , as described in (the negative sign is a critical part of the
equation and must not be forgotten)
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The principal coefficients of the subsequent generation matrix, , are the essential expansion numbers for
drug-sensitive (DS) and drug-resistant (DR) tuberculosis. These figures, which are represented by the notation
, DS and
, DR, indicate the typical number of secondary infections that an infected individual would
produce. Thus, the following are the fundamental procreation numbers for DS and DR TB:
(a)
and
(b)
In this context,
and
represent as two key estimations. These represent the likelihood that the drug-
sensitive (DS) and drug-resistant (DR) strains will migrate from the latent compartment to the infectious
compartment, respectively. The average amount of time infectious people spend in states
and
is likewise
indicated by them. The numbers
and
stand for the anticipated number of drug-resistant (DR) and
secondary drug-sensitive (DS) TB cases, respectively, that a single infectious individual would produce if they
were placed in a community that is completely susceptible. As noted, it is an astounding discovery that the
fundamental expansion numbers for DS TB (
) and DR TB (
) are regardless of the expansion portion, .
In the supplemental resources, particularly in the sections on the existence of equilibria and stability analysis,
we present an exhaustive examination of the suggested TB model, defined by equations (2) – (7). This analysis
examines the existence and stability of three important optimum points: healthy (
), single-present (
), and
jointly transmitted (
). To put it simply, the healthy optimum is universally linearly reliable if and only if
max [
,
] < 1. In the stability analysis part of the supplementary materials, we demonstrate that the
single-existence equilibrium is generally steady when
> max [1,
]. We used the Monte Carlo approach
to examine the co-existence optimum and determine if the criterion
> max [
] holds. We estimated the
real part of the Jacobian matrix's eigenvalues and assessed them at the co-existing an endemic balance (Ef).
The steady state coordinates can be described using twelve model parameters:
. Table 1 shows the initial values for these variables. To analyze the
model's behavior, a sample pool
was established. This pool is a Cartesian product of
twelve closed intervals,
, where each
represents the initial value
of a parameter from Table 1, and sets the variation range. Our research included 10,000 random
situations,
, with each
iselected from a homogeneous distribution
where there is no correlation among variables. For this study, we picked a 20% variation from the baseline
values (). Figure 3 demonstrates that the mixed prevalent balance is permanent (yellow dots) when the
criterion
> max [
] is met. Otherwise, it is unstable (shown by the blue spots). Figure 4 depicts three
points of balance and their specific equilibrium areas in relation to fundamental reproduction numbers for
drug-susceptible
and drug-resistant (
) tuberculosis (TB). The healthy optimum is depicted by
the magenta-colored region and is bounded by the criterion max [
,
] < 1. The green colored area
represents the single-present optimum , which occurs when
> max [1,
]. Finally, the co-existing
optimum (
) is displayed by the yellow-colored area, which corresponds to the requirement
> max
[
].
Mathematical models were performed in MATLAB to back up the analytical findings and to see how
propagation rates and DS TB therapy effect overall as well as DR-TB rates. We investigated various beginning
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circumstances on each DS and DR TB in every groups. The findings validated a stable state of the model's
equilibria. Simulations indicate that tuberculosis will be removed from the population when max [
,
] <
1. If
> max [1,
], DS TB will die off but DR TB will survive. Additionally, the requirement
> max
[
] shows that DS and DR, TB will coexist in the population. Additional information and visual
representations are included in the additional data, notably Figures S1, S2, and S3. We can mimic the
consequences of public health actions by altering a few factors in our model. Reducing contact between
individuals can be indicated by reduced transmission rates (
or
). Similarly, increased treatment rates (
or
) can be used to identify and treat infectious individuals. The effective reproduction number is the new
reproduction number that comes from these interventions or the non-susceptibility of a major proportion of the
population. As depicted in Figure 5, the growth rate
, has a significant effect on the frequency of both drug-
sensitive (DS) and drug-resistant (DR) tuberculosis. In the first region, where , DS TB is the
dominating strain. When , DR TB becomes more common. The expansion route enables DR TB to
thrive even when its basic reproduction number is lower than that of the vulnerable variation has a gestation
quantity fewer instead of a single as long as the requirement
> max [
] is met. This study implies that
medication treatment may inadvertently cause the onset of DR TB. Our research also demonstrates that the
spread of drug-resistant (DR) tuberculosis can be halted if the treatment rate for drug-sensitive (DS)
tuberculosis is high enough to eradicate it from the community. In order to attain a viable growth rate below a
choice, the DS TB treatment rate needs to be sufficiently high. The effects of DS TB medication ratio (
) and
propagation rate () on overall TB and DR TB rate with fixed pathogenic ratios
,
are depicted in
Figures 6 and 7. Propagation raises the rate of both total TB and DR TB. According to Figure 7, as the DS TB
medication level rises from zero to roughly 0.8-0.9, a high progression rate initially results in an increase in the
prevalence of DR TB. Following this peak, the prevalence of DR TB declines and levels out at a steady level.
The percentage of DR TB only increased to a shared average level for low propagation rates. This point
depicts the permanent DR TB- just balance which occurs when the basic growth number of DR TB surpasses
the actual fertilization rate of DS TB, provided the latter is larger than one. Numerical research has
revealed that when growth is particularly high and drug-sensitive (DS) TB is treated, the prevalence of drug-
resistant (DR) TB might rise above its natural equilibrium level. This finding emphasizes the vital need of
effective therapy for DS TB patients. If not managed properly, insufficient or inadequate therapy can result in
the formation of new DR TB cases, allowing the DR strain to become the dominant form of the illness.
Figure 3. illustrates the stability of the co-existing endemic equilibrium as a function of
(defined by
the eigenvalues of the system Jacobian matrix). Random parameter drawings that result in eigenvalues with
only negative real parts (stable) are colored yellow, while those that result in eigenvalues with at least one
positive real part (unstable) are colored blue. All stable points (yellow) sit above the line
, whereas
all unstable points (blue) lie below.
The model trajectories and reported WHO incidence data are shown together to validate model behaviour. The
simulated annual incidence curves for drug-sensitive and drug-resistant tuberculosis are plotted and were
compared with surveillance points. Close agreement between modelled and reported points is indicated in
years where treatment and transmission parameters were fitted. The influence of amplification on raising drug-
resistant prevalence when treatment of drug-sensitive cases is imperfect is illustrated by the divergent trends.
We conducted studying of vulnerability to investigate the quantitative link between the model's parameters and
its outputs, such as the prevalence of DS, DR, and total TB. Figure 8 depicts the relationship among the co-
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existing suitable number of DS TB prevalence (
) and the model's twelve parameters:
and. These correlations are detected under the condition that
> max
[
]. The figure shows that the rate of DS TB (
) is positively correlated with
and. This
indicates that increasing any of these characteristics will result in an increase in DS TB rates. Our analysis
showed that the amount of data transmitted (
) reached the highest favorable impact on DS TB rates (
)
among all variables. Reproduction is directly proportional to
, therefore this is expected. The most important
public health measure is to reduce exposure between diseased and uninfected persons. This action is more
successful than enhancing treatment, lowering recurrence rates, or limiting amplification. As a result, public
health resources should be directed toward individuals with high exposure, such as those living in the same
household as someone with infectious tuberculosis. The variables
andhave a negative
connection with DS TB rates (
). Increasing any of these criteria will lower
predominance. Our data
revealed that medical care velocity
is the subsequent essential factor in determining DS TB rates. This
outcome is consistent with WHO recommendations and other studies, all of which emphasize the importance
of early treatment and curing infected cases in breaking the chain of tuberculosis transmission within a
geographic region. According to our findings, the most effective measures for preventing tuberculosis include
swift detection of prospective instances, prompt investigation, and the initiation and profitable conclusion of
therapy. Furthermore, we discovered that propagation has a detrimental influence on the spread of DS TB due
to individuals with the DS strain become DR TB due to inadequate therapy. Figure 9 illustrates the relationship
between DR TB prevalence and the model's twelve parameters (
and)
when
> max [
]. The parameters
,
,
,
have positive Partial Rank Correlation
Coefficients (PRCCs), while
,
,
,
,
and
have negative PRCCs. This is consistent with the
observations. At the co-exist equilibrium, amplification
(
)
has a beneficial connection with DR TB rate, as
expected given its role in the progression from DS TB to DR TB.
Figure 4 depicts the model's equilibrium points' regions of existence as well as regional stability. The disease-
free equilibrium is represented in magenta, while the mono-existent equilibrium is indicated in green. The co-
existent equilibrium is in the yellow-shaded area.
Figure 5 depicts the impact of PRCC amplification
(
)
on DS and DR TB prevalence. All additional values are
based on their initial values presented in Table 1.
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Figure 6 illustrates the impact of DS TB treatment rate (
) on overall TB prevalence when
and
remain
constant. All additional variables are based on the initial values presented in Table 1.
The evaluation of sensitivity was also performed to investigate the overall TB prevalence
as well as
DR TB incidences in connection with any respective variables. This was performed under two conditions:
when
> max [
] and when
> max [
]. The findings of this study are included in the
supplemental materials, specifically the studying of vulnerability section and Figures S4 and S5. Table 2
shows the sensitivity indices for
and
with regard to each parameter. Transmission rates (
for DS
TB and
for DR TB) have a substantial impact on these indicators. The sensitivity indices are
and
, indicating that changing the transmission rates (
and
) by a specific percentage will result in
the same change in reproduction numbers (
and
).
Figure 7 illustrates the impact of DS TB treatment rate (
) on DR TB prevalence when
and
are constant.
All additional variables are based on the baseline values presented in Table 1.
Figure 8 shows the relationship among model output
and parameters (transmission rate), (progression
rate), (recovery rate), (disease-related death rate), (treatment rate), and (amplification rate) when
> max
[
]. Subscripts represent DS and DR amounts, correspondingly.
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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Page 155
Figure 9 illustrates the relationship among the result of the model
and variables
and when
>
and
> 1.
The spiral sensitivity plot was produced to illustrate parameter influence on tuberculosis dynamics. Drug-
sensitive and drug-resistant PRCC values were displayed as distinct spiral curves, enabling direct comparison.
Transmission coefficients, treatment rates, and amplification probability were identified as dominant drivers of
outcomes. The results highlighted that imperfect treatment played a critical role in sustaining drug-resistant
prevalence.
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According to earlier research, if the fundamental growth number of antibiotic-resistant microorganisms is over
a certain threshold, even if it is less than one, and the fundamental growth number of sensitive microorganisms
is larger than one, both types of bacteria can persist (Tanvir Mahtab Uddin et. al.) [29]. This is not what we
found. Our findings demonstrate that the fundamental the increased rate of antibiotic-resistant microorganisms
does not need to rise above a certain threshold in order for both strains to coexist. This is a direct result of the
amplification pathway, which gives the resistant strain another way to survive and proliferate. The most
efficient method of lowering the total burden of the disease is to combine diverse therapeutic techniques,
according to several TB modeling studies. We have created a six-compartmental, two-strain SLIRS model
specifically for Bangladesh in this study, which incorporates a feature known as amplification. The coupling
between the two strains, which illustrates how infected people may become resistant to drugs while undergoing
treatment, is a crucial component of our model. We made a number of important assumptions in our
investigation that were absent from previous analyses. We believed that naturally occurring genetic alterations
in TB bacteria, when exposed to insufficient therapy, are the primary cause of amplification, or the
development of drug resistance. In addition, our model incorporates treatment and natural recovery factors that
were not taken into account in those earlier investigations. Furthermore, we have included a factor that was not
included in previous studies: people who have recovered from tuberculosis may lose their immunity and re-
enter the vulnerable group. According to our concept, the fundamental reproduction numbers,
and
are
specifically impacted by the transmission rates
(
)
rates of advancement
(
)
rates of recovery
(
)
,
disease-related mortality rates
(
)
, as well as treatment rates
(
).
Additionally, sensitivity analysis shows
that the transmission rates
(
)
are the most important factors, and the treatment rates
(
).
Therefore, the
main goal of effective management and eventual eradication of DS TB and DR TB infections should be to
lower the contact rates
(
)
with contagious people. We can reduce contact rates in a number of ways. One
method is personal respiratory protection, which involves masks for patients to wear while they cough, sneeze,
yawn, or talk to stop the spread of TB bacteria. Patients should also be instructed in basic infection control
practices, such as covering their mouth and nose while coughing or sneezing and appropriately discarding used
tissues in covered bins. Environmental measures, which include optimizing air exchange and dilution and
decontaminating the air in high-risk regions where adequate ventilation is not feasible, are also essential.
Regular maintenance and monitoring of any ventilation system is also essential (Iasmin Lourenço Niza et. al.)
[30]. Lastly, disease diagnosis-focused public health programs are also required to reduce transmission.
Increasing the treatment rates
(
)
is the second most important tactic among contagious people.
Furthermore, large amplification levels
(
)
, especially when combined with high
depict situations of
insufficient or poorly managed therapy that encourage the formation of new DR TB cases, and so contribute to
a higher prevalence of DR TB. Therefore, in order to effectively address the challenges presented by DR TB,
we advise that evaluations of the amplification risk
(
)
be carried out in conjunction with assessments of the
reproduction numbers for both DS and DR TB. This will ensure that treatment levels are optimized to
minimize the development and spread of DR TB. The most efficient method of preventing medication
resistance in underdeveloped countries like Bangladesh is to provide first-line treatment for DS TB that is
properly delivered. Preventing an increase in the prevalence of DR TB requires early detection of the disease
and the provision of appropriate second-line medication regimens. However, because long-term therapy is
expensive, it is difficult to segregate infectious persons. Therefore, increasing treatment rates by lowering
treatment costs and implementing universal healthcare is the most sensible and effective way to eradicate both
DS TB and DR TB in Bangladesh.
DISCUSSION AND CONCLUSION
This research introduces and evaluates a new TB simulation using two forms of bacteria as well as
propagation. This model distinguishes between drug-sensitive (DS) and drug-resistant tuberculosis. In this
approach, amplification is defined as the procedure whereby a person with DS TB acquires a type of resistance
following a rejection of their first-line medication treatment.
Our model identifies three equilibrium points: healthy, single-present, or mixed. We used the forthcoming
matrix approach to obtain the basic number of replications for each strain (
for DS TB and
for DR TB
).
Table 1 shows the estimated values for these reproduction numbers, as well as the model's other biological
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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Page 157
parameters, based on available data. We also used dynamical systems analysis to study Particular constancy of
healthy and single-present equilibria. This research confirms that stability depends on two important threshold
values: fundamental reproduction numbers for DS TB (
) and DR TB (
). If
max [
,
]
, the
disease-free equilibrium is globally asymptotically stable, implying the disease will eventually die out. The
model indicates that if
max [
,
]
, DS TB will die out, but DR TB will endure. If
max [
]
,
both DS and DR TB will survive in the community. This approach aids in identifying certain portions of the
variable area in which the different exponential levels are either steady or unreliable. This feature enables us to
forecast the long-term behavior of DS and DR-TB dynamics. The data can help the Bangladesh National TB
Control Program lower infectiousness until the condition
max [
,
]
is reached, when the disease will
be eradicated. In previous modeling studies on the transmission patterns of sensitive and resistant tuberculosis
bacteria, the existence and stability of both within-host and between-host models were investigated. Studies
indicate that when the fundamental reproduction numbers for both sensitive antimicrobial-resistant
microorganisms are fewer instead of a single, both strains are eliminated. Assuming the fundamental growth
rate of antimicrobial-resistant microorganisms (
) exceeds one and those of vulnerable microorganisms
(
) is fewer instead of a single and solely antibiotic-resistant microorganism will survive.
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