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Adv Syst Sci Appl 2022; 04; 144-161 
Published online at http://ijassa.ipu.ru. 

 

Analysis and Dynamics of Tuberculosis Outbreak: 
A Mathematical Modelling Approach 

Festus Abiodun Oguntolu1, Olumuyiwa James Peter2,3*, Kayode Oshinubi4, 
Tawakalt Abosede Ayoola5, Asimiyu Olalekan Oladapo5, Mayowa M. Ojo6,7 
1)Department of Mathematics, Federal University of Technology Minna, Minna, Nigeria 
2)Department of Mathematical and Computer Sciences, University of Medical Sciences, 

Ondo City, Ondo State, Nigeria 
3)Department of Epidemiology and Biostatistics, School of Public Health, University of 

Medical Sciences, Ondo City, Ondo State, Nigeria. 
4)AGIES Research Unit, Universite Grenoble Alpes, Alpes, France 
5)Department of Mathematics, Osun State University Oshogbo, Nigeria 
6)Department of Mathematical Sciences, University of South Africa, Florida, South 

Africa 
7)Thermo Fisher Scientific, Microbiology Division, Lenexa, Kansas, USA 

E-mail: peterjames4real@gmail.com 

Abstract: Tuberculosis (TB) is an infectious disease caused by mycobacterium disease which 
causes major ill health in humans. Control strategies like vaccines, early detention, treatment and 
isolation are required to minimize or eradicate this deadly pandemic disease. This article presents 
a novel mathematical modelling approach to tuberculosis disease using Vaccinated-Susceptible-
Latent-Mild-Chronic-Isolated-Treated model. We examined if the epidemiology model is well 
posed and then obtained two equilibria points (disease free and endemic equilibrium). We also 
showed that TB disease free equilibrium is locally and globally asymptotically stable if 𝑅 < 1. 
We solved the model analytically using Homotopy Perturbation Method (HPM) and the graphical 
representations and interpretations of various effects of the model parameters in order to measure 
the impact for effective disease control are presented. The findings show that infected populations 
will be reduced when the isolation and treatment rates and their effectiveness are high.  

Keywords: tuberculosis; Homotopy Perturbation Method; infectious disease; basic reproduction 
number; vaccination 

1. INTRODUCTION  

Tuberculosis (TB) is the third greatest killer worldwide caused by an infectious agent [12]. 
According to World Health Organization (WHO), one-third of the world’s population is 
currently infected by the TB bacillus bacteria. Being a disease of poverty, the vast majority of 
TB deaths are in developing countries with more than half occurring in Asia. Furthermore, 
over 95% of these deaths occurred in low- and middle-income countries where the cost of 
diagnosis and treatment is high, and not readily accessible. 

Tuberculosis is a chronic bacteria infectious disease caused by Mycobacterium 
tuberculosis which poses a major health, social and economic burden globally, especially in 
low- and middle-income countries [5]. The surge in HIV-TB co-infection and the growing 
emergence of multidrug-resistant TB (MDR-TB) and extensively drug-resistant TB (XDR-TB) 
strains has further fueled TB epidemic. TB usually affects the lungs (pulmonary TB) but it can 
also affect other sites as well (extra-pulmonary TB). Tuberculosis is transmitted by tiny 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 145 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

 

airborne droplets which are expelled into the air when a person with active pulmonary TB 
coughs or talks [21,22].  

Diagnosis of latent TB infections (LTBI) and prompt treatment of active cases remains an 
important component of effective TB control as shown in previous studies. On the other hand, 
undetected TB infection and delay in the treatment of active TB cases leads to more severe 
disease conditions in the infected person which could result in wider disease spread in the 
community [3, 7, 10, 9].  

Some researchers have proposed some mathematical models to solve problems arising 
from TB models and we shall discuss some of them as follows: [8] presents a Susceptible-
Exposed-Infected-Recovered (SEIR) tuberculosis model which incorporated treatment of 
infectious individuals and chemoprophylaxis (treatment for the latently infected). The model 
assumed that the latently infected individuals develop the active disease as a result of 
endogenous re-activation, exogenous re-infection and disease relapse. [4] presents the 
mathematical model of a tuberculosis transmission dynamics incorporating first and second 
line treatment. [7] proposed a seven-compartment model which included diagnosed and 
undiagnosed infectious population and their result shows that high vaccination rate is required 
to eradicate TB. In [16], the authors develop a mathematical model for control of tuberculosis 
epidemiology by incorporating some control strategies, the findings of the study show that 
using multiple controls is the best way to control the spread of TB. Several studies have been 
conducted utilizing a mathematical model method in order to identify ways to control diseases 
in the population [1,13, 18-20]. [17] presented a deterministic SEIR model to described the 
transmission dynamics of TB in Ashanti region of Ghana and the results showed that the region 
has herd immunity against TB infection. In this study we consider some control strategies such 
as, treatment, isolation and vaccination into the mathematical formulation of TB outbreak with 
assumptions that people in each compartment have equal natural death rate and infection does 
not confer immunity to the treated and recovered individuals.  

The rest of the article is divided as follows: Method which includes model formulation and 
mathematical analysis of the model are described in “section 2. Next, section consists of the 
semi-analytic solution of the model formulated. Numerical simulation and graphical 
representation of results is given in 4, while the discussion of the results is presented in section 
5. Finally, in section 6, we have provided conclusions of this article. 

2. METHOD 

2.1. Model formulation 

In this section, the TB transmission model is formulated. Using a compartmental approach, 
the total host population can be partitioned into seven compartments according to their 
epidemiological status. The groups are the Vaccinated 𝑉(𝑡) , Susceptible 𝑆(𝑡) ,Latent 

𝐸(𝑡) , Mild TB 𝐼 (𝑡) , Chronic TB 𝐼 (𝑡) , isolated infectious 𝐽(𝑡)  and Treated 𝑇(𝑡)  
individual, where 𝑡 is the time variable. It is assumed that once the treatment of active TB 
cases is interrupted, there is no more treatment. Vaccination reduces the risk of infection by a 
factor 𝜃 ∈ (0,1) and the efficacy of the vaccine is 𝜔. Let a constant 𝜋 stands for the number 
of newborn babies into the population, then 𝜃𝜋 are the individuals in the vaccinated class while 
(1 − 𝜃)𝜋 are the susceptible individual. The susceptible class also increases with a waning 
rate of vaccine at 𝜔, due to fact that vaccine does not confer a total immunity. We assume 
that𝜇is per capital natural death rate and 𝑑 (𝑖 = 1,2,3) is the disease induced death rate in 
classes𝐼 (𝑡), 𝐼 (𝑡) and𝐽(𝑡) respectively. It is natural to assume that 𝑑 ≥ 𝑑 ≥ 𝑑  due to the 
treatment of active TB cases reducing the disease induced death rate are the transmission 
coefficients from class 𝑆(𝑡) , 𝐸(𝑡)  and 𝑇(𝑡)  respectively. We assume that 𝜆 > 𝜆 > 𝜆  
because the treatment of active TB cases reduces the infectivity of active TB cases. Take 
𝜌(0 < 𝜌 < 1)  as the fraction of the latent persons who have fast TB progression. The 



146            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

proportion 𝜙  of individual in the exposed class will progress to the chronic class via 
endogenous us reactivation. 𝜎is the reactivation rate from the latent persons to infected class. 
𝛾is the reactivation rate of the individual in the mild TB 𝐼 (𝑡)  to the chronic TB 𝐼 (𝑡) . 
The parameters are the recovery rates of the individual in the classes 𝐼 (𝑡) , 𝐼 (𝑡)  and 𝐽(𝑡) 
respectively. And the parameter is the rate of isolation of the individuals in chronic class 

𝐼 (𝑡) .  
In this article, the TB dynamic model describing the compartment is based on the following 

assumptions: 
That a proportion of the population of newborn is immunized against TB infection through 

vaccination. 
That the immunity conferred on individuals by treatment expires after some time at given 

rate. 
That people in each compartment have equal natural death rate of 𝜇 
That there are no immigrants and emigrants. The only way of entry into the population is 

through new-born babies and the only way of exit is through death from natural causes or death 
from TB related causes. 

That the infection does not confer immunity to the treated and recovered individuals and 
so they go back to the susceptible class at a given rates. 

That all newborns are previously uninfected by TB and therefore join either the immunized 
compartment or the susceptible compartment depending on whether they are vaccinated or not. 

We combine the basic assumptions, model parameters, variables and the TB infection 
processes to formulate a schematic diagram for TB infection as shown in Figure 1. The model 
equations are given as follow 

= 𝜋𝜃 − (𝜇 + 𝜔)𝑉 ,      (1) 

= 𝜋(1 − 𝜃) − 𝜆 𝑆 − 𝜇𝑆 + 𝜔𝑉,     (2) 

= 𝜆 𝑆 + 𝜆 𝑇 − 𝜆 𝐸 − (𝜇 + 𝜎)𝐸,     (3) 

= (1 − 𝜌)𝜎𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 + (1 − 𝜙)𝜆 𝐸,  (4) 

= 𝜌𝜎𝐸 + 𝛾𝐼 − (𝑟 + 𝑟 + 𝜇 + 𝑑 )𝐼 + 𝜙𝜆 𝐸,   (5) 

= 𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 ,     (6) 

= 𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 ,    (7) 

where 𝜆 = 𝛽(𝐼 + 𝜀 𝐼 + 𝜀 𝐽),𝜆 = 𝛼(𝐼 + 𝜀 𝐼 + 𝜀 𝐽) and 𝜆 = 𝛾(𝐼 + 𝜀 𝐼 + 𝜀 𝐽). 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 147 

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Figure 1: Schematic Representation of the Model 

Table 1. Notation and definition of other parameters 
Symbol Description 

𝑟  Treatment rate of individuals in mild class 

𝑟  The treatment rate for those is 𝐼 class 

𝑟  The progression rate from classes𝐼  to J 

𝑟  The treatment rate for those in isolated class 

𝛽 Transmission rate among the susceptible 

𝑑  Death rate for mild TB individual 

𝑑  Death rate for chronic TB individuals 

𝑑  
Death rate for isolated class 

𝛼 Effective transmission rate from latent class to infected class 

𝜀  Relative infectiousness of humans with mild TB compared to humans in the chronic class 

𝜀  Relative infectiousness of humans with TB in the isolated class compared to humans in the 
chronic class 

𝜀  Relative infectiousness of humans with mild TB due to endogenous reactivation compared 
to humans in the chronic class. 

𝜀  Relative infectiousness of humans on isolation after endogenous reactivation compared to 
humans in the chronic class 

𝜀  Relative infectiousness of humans with mild TB due to exogenous re-infection compared to 
humans in the chronic class 

𝜀  Relative infectiousness of humans on isolation after exogenous re-infection compared to 
humans in the chronic class 

𝜌 Fraction of the latent persons who have fast TB progression 

𝜎 Reactivation rate from the latent individuals 

𝜙 Proportion of individual in the exposed class 

𝛾 Reactivation rate of the individual in the mild TB class 

𝜇 Natural death rate 

2.2. The positive invariant region 

The entire population size 𝑁 can be determined from by adding Equations (1)–(7). Hence, 
𝑁(𝑡) = 𝑆(𝑡) + 𝑉(𝑡) + 𝐸(𝑡) + 𝐼 (𝑡) + 𝐼 (𝑡) + 𝐽(𝑡) + 𝑇(𝑡) then,  

= 𝜋 − 𝜇𝑁(𝑡) − 𝑑 𝑁 − 𝑑 𝑁 − 𝑑 𝑁    (8) 

In the absence of the disease (𝑑 = 𝑑 = 𝑑 = 0) then (8) gives 



148            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

= 𝜋 − 𝜇𝑁       (9) 

Theorem 1: 

The system (1) to (7) has solution which are contained in the feasible region Ω for all 𝑡 > 0. 
Proof. 
Let 𝛺 = (𝑆, 𝑉, 𝐸 , 𝐼 𝐼 , 𝐽, 𝑇) ∈ 𝑅  be any solution of the system (1)–(7) with non-negative 

initial condition. Using theorem of differential inequality, Equation (9) gives 

≤ 𝜋 − 𝜇𝑁, then 0 ≤ 𝑁 ≤ , Hence 𝜋 − 𝜇𝑁 ≥ 𝑘𝑒 , where 𝑘 is constant. Thus, the 

feasible set of the model is given by 

𝛺 = (𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇) ∈ 𝑅 : 𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇 ≥ 0, 𝑁 ≤ , 

which is positive invariant (i.e., solution remain positive for all time 𝑡) and the model is 
epidemiologically meaningful and mathematically well pose. □ 

2.3. Positivity of solutions 

Since Equations (1)–(7) represent the population in each compartment and all model 
parameters are all positive, then it lies in a region Ω defined by 

𝛺 = (𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇) ∈ 𝑅 : 𝑆, 𝑉, 𝐼 , 𝐼 , 𝐽, 𝑇 ≥ 0, 𝑁 ≤ . 

Theorem 2: 

Let the initial value for the model equation be given as 

𝑆(0), 𝑉(0) , 𝐸(0), 𝐼 (0) , 𝐼 (0), 𝐽(0), 𝑇(0) ∈ 𝑛 

Then the solution set {𝑆(𝑡), 𝑉(𝑡) , 𝐸(0), 𝐼 (𝑡) , 𝐼 (𝑡), 𝐽(𝑡), 𝑇(𝑡)}  of the system (1)–(7) is 
positive for all 𝑡 > 0. 

Proof. 
From Equation (2) 

= 𝜋(1 − 𝜃) + 𝜔𝑉 − (𝜆 + 𝜇)𝑆 then, ≥ −(𝜆 + 𝜇)𝑆 

On Integrating it gives ∫ ≥ − ∫(𝜆 + 𝜇) 𝑑𝑡. 

Hence, 𝑆(𝑡) ≥ 𝑆(0)𝑒 ( ) . 
Applying the same approach to other equations in the model equations, we have: 

𝑉(𝑡) ≥ 𝑉(0)𝑒 ( ) , 𝐸(𝑡) ≥ 𝐸(0)𝑒 ( ) , 𝐼 (𝑡) ≥ 𝐼 (0)𝑒 ( ) , 

𝐼 (𝑡) ≥ 𝐼 (0)𝑒 ( ) ,𝐽(𝑡) ≥ 𝐽(0)𝑒 ( ) , 𝑇(𝑡) ≥ 𝑇(0)𝑒 ( ) . Therefore, 
the solution to the model equations is positive for all 𝑡 > 0.□ 

2.4. Equilibrium points of the model 

The equilibrium state is the point in which there is zero disturbance on the system under 
consideration. That is, the rate of change of the model variables with time is zero. Thus, at 
equilibrium, 

= = = = = = = 0    (10) 

Let 𝐸∗ = (𝑉, 𝑆, 𝐸, 𝐼 , 𝐼 , 𝐽, 𝑇) = (𝑉∗, 𝑆∗, 𝐸∗, 𝐼 ∗, 𝐼 ∗, 𝐽∗, 𝑇∗) be arbitrarily equilibrium point 
Substituting Equations (10) into Equations (1)–(7) gives: 

𝜋𝜃 − (𝜇 + 𝜔)𝑉 = 0,      (11) 

𝜋(1 − 𝜃) − 𝜆 𝑆 − 𝜇𝑆 + 𝜔𝑉 = 0,    (12) 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 149 

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𝜆 𝑆 + 𝜆 𝑇 − 𝜆 𝐸 − (𝜇 + 𝜎)𝐸 = 0,    (13) 

(1 − 𝜌)𝜎𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 + (1 − 𝜙)𝜆 𝐸 = 0,  (14) 

𝜌𝜎𝐸 + 𝛾𝐼 − (𝑟 + 𝑟 + 𝜇 + 𝑑 )𝐼 + 𝜙𝜆 𝐸 = 0,   (15) 

𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 = 0,      (16) 

𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 = 0.     (17) 

From Equation (11) we have 

𝑉 =
( )

.      (18) 

Substitute Equation (18) into equation (12) we have 

𝜋(1 − 𝜃) + = 𝑆.     (19) 

From Equation (16) we have 𝐽 = 𝐾 𝐼  where 𝐾 = .    

From (15) we have [(1 − 𝜌)𝜎 + (1 − 𝜙)𝜆 ]𝐸 = [𝑟 + 𝑟 + 𝜇 + 𝑑 ]𝐼 ⇒ 𝐸 = 𝐾 𝐼  

where 𝐾 =
( ) ( )

, from Equation (16) we have (𝑟 + 𝑟 + 𝜇 + 𝑑 )𝐼 = 𝜌𝜎𝐸 +

𝜙𝜆 𝐸 + 𝛾𝐼 , then 𝐼 = 𝐾 𝐼  where 𝐾 =
( )

 further substitution gives 𝐽 = 𝐾 𝐾 𝐼  

and from Equation (17) we have: 𝑇 = . 

then 𝑇 = 𝐾 𝐼  where 𝐾 =
( )

, 

𝜆 = 𝛽(𝐼 + 𝜀 𝐼 + 𝜀 𝐽)

𝜆 = 𝛼(𝐼 + 𝜀 𝐼 + 𝜀 𝐽)

𝜆 = 𝛾 (𝐼 + 𝜀 𝐼 + 𝜀 𝐽)
 and  

𝐽 = 𝐾 𝐾 𝐼

𝐼 = 𝐾 𝐼
 by combining both equations: 

𝜆 = 𝛽𝐼 (𝐾 + 𝜀 𝐾 𝐾 + 𝜀 )

𝜆 = 𝛼𝐼 (𝐾 + 𝜀 𝐾 𝐾 + 𝜀 )

𝜆 = 𝛾 𝐼 (𝐾 + 𝜀 𝐾 𝐾 + 𝜀 )⎭
⎪
⎬

⎪
⎫

 

Further substitution gives [𝛽𝐾 𝑆 + 𝛾 𝐾 − 𝛼𝐾 𝐾 𝐼 − (𝜇 + 𝜎)𝐾 ]𝐼 = 0.  
Therefore, 𝐼 = 0 or 𝛽𝐾 𝑆 + 𝛾 𝐾 − 𝛼𝐾 𝐾 𝐼 − (𝜇 + 𝜎)𝐾 = 0  

where  
𝐾 = 𝐾 + 𝜀 + 𝜀 𝐾 𝐾
𝐾 = 𝐾 + 𝜀 + 𝜀 𝐾 𝐾
𝐾 = 𝐾 + 𝜀 + 𝜀 𝐾 𝐾

 and  

 

𝜆 = 𝛽𝐾 𝐼

𝜆 = 𝛼𝐾 𝐼

𝜆 = 𝛾 𝐾 𝐼 ⎭
⎪
⎬

⎪
⎫

.     (20) 

2.5. Disease free equilibrium (D.F.E) 

The disease-free equilibrium state is the point at which there exist no infection in the given 
population.  

At Disease Free Equilibrium, we let 

(𝑉, 𝑆, 𝐸, 𝐼 , 𝐼 , 𝐽, 𝑇) = 𝐸∗ = (𝑉∗, 𝑆∗, 𝐸∗, 𝐼∗ , 𝐼∗, 𝐽∗, 𝑇∗). 

Lemma 1: 



150            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

The D.F.E of the model exists and is given by 

𝐸 = (𝑉∗, 𝑆∗, 𝐸∗, 𝐼∗ , 𝐼∗, 𝐽∗, 𝑇∗) =
𝜋𝜃

(𝜇 + 𝜔)
,
𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)
, 0,0,0,0,0 . 

Proof. 

Suppose 𝐼 = 0. Then Equation (20) becomes 
𝜆 = 0
𝜆 = 0
𝜆 = 0

 and also  

𝐽∗ = 0
𝐸∗ = 0
𝐼∗ = 0
𝑇∗ = 0

 . 

From (19) 

𝑆∗ =
𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)
. 

Thus, the lemma is proved and  

⎝

⎜
⎜
⎜
⎜
⎜
⎛

𝑉∗

𝑆∗

𝐸∗

𝐼 ∗

𝐼 ∗

𝐽∗

𝑇∗ ⎠

⎟
⎟
⎟
⎟
⎟
⎞

=

⎝

⎜
⎜
⎜
⎜
⎜
⎛

( )

( )

( )

0
0
0
0
0 ⎠

⎟
⎟
⎟
⎟
⎟
⎞

.

      (21) 

The above equation is the Disease-Free Equilibrium.  

2.7. Effective reproduction number 

In biomathematics, the basic reproduction number (R0) is the average number of infected 
contacts per infected individual. It is one of the fundamental concepts to determine the future 
of an epidemics in a population. When 𝑅 < 1 The infection will die out in the long run, but 
if 𝑅 > 1. The infection will be able to spread in a population. In this model, the spectral radius 
of the equation is given as largest eigenvalue given as 𝑅 = 𝜌𝑓𝑣  

Following the procedure in [14, 15], the next generation matrix operator is used to estimate 
the effective reproduction number such that the Jacobian matrices for the new infection terms 
and the remaining transfer terms are obtained below 

𝐹 =

𝜆 𝑆 + 𝜆 𝑇

(1 − 𝜙)𝜆 𝐸

(𝜙𝜆 )𝐸
0

 

𝑉 =

⎝

⎛

(𝜇 + 𝜎)𝐸
(𝛾 + 𝑟 + 𝑑 + 𝜇)𝐼 − (1 − 𝜌)𝜎𝐸
(𝑟 + 𝑟 + 𝑑 + 𝜇)𝐼 − 𝛾 𝐼 − 𝜌𝜎𝐸

(𝜇 + 𝑑 + 𝑟 )𝐽 − 𝑟 𝐼 ⎠

⎞ 

𝐹(𝐷𝐹𝐸) =

⎝

⎜
⎛

0
𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)

𝛽𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)

𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)
0 0 0 0
0 0 0 0
0 0 0 0 ⎠

⎟
⎞

 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 151 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

 

𝑉 =

⎝

⎛

(𝜇 + 𝜎) 0 0 0

−(1 − 𝜌)𝜎 (𝛾 + 𝜇 + 𝑟 + 𝑑 ) 0 0

−𝜌𝜎 −𝛾 (𝑟 + 𝑟 + 𝑑 + 𝜇) 0

0 0 −𝑟 (𝜇 + 𝑟 + 𝑑 )⎠

⎞ 

Let  𝑉 =

𝑄 0 0 0
−𝐶 𝑄 0 0
−𝐶 −𝛾 𝑄 0

0 0 −𝑟 𝑄

 

where 

𝑄 (𝜇 + 𝜎), 
𝑄 = 𝛾 + 𝜇 + 𝑟 + 𝑑 , 
𝑄 = 𝑟 + 𝑟 + 𝑑 + 𝜇, 

𝑄 = 𝑑 + 𝑟 + 𝜇, 

therefore
 

𝑉 =

⎝

⎜
⎜
⎜
⎜
⎜
⎛

1

(𝜇 + 𝜎)
0 0 0

(1 − 𝜌)𝜎

𝑄 𝑄

1

𝑄
0 0

(1 − 𝜌)𝛾𝜎 + 𝜌𝜎𝑄

𝑄 𝑄 𝑄

𝛾

𝑄 𝑄

1

𝑄
0

𝑟 [(1 − 𝜌)𝛾𝜎 + 𝜌𝜎𝑄 ]

𝑄 𝑄 𝑄 𝑄

𝑟 𝛾

𝑄 𝑄 𝑄

𝑟

𝑄 𝑄

1

𝑄 ⎠

⎟
⎟
⎟
⎟
⎟
⎞

 

𝐹𝑉 = 

⎝

⎜
⎛

𝛽𝜀 𝜎𝑥(1 − 𝜌)

𝑄 𝑄
+

𝛽𝑥𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ]

𝑄 𝑄 𝑄
+

𝛽𝜀 𝑥𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ]

𝑄 𝑄 𝑄 𝑄

𝛽𝜀 𝑥

𝑄
+

𝛽𝑥𝛾

𝑄 𝑄
+

𝛽𝜀 𝑥𝑟 𝛾

𝑄 𝑄 𝑄

𝛽𝑥

𝑄
+

𝛽𝜀 𝑥𝑟

𝑄 𝑄

𝛽𝜀 𝑥

𝑄

0 0 0 0
0 0 0 0
0 0 0 0 ⎠

⎟
⎞

, 

(22) 

where 𝑥 =
( )

( )
. 

From (22), we calculate the eigen-values to determine the basic reproduction number, 𝑅  
by taking the spectral radius (dominant eigenvalue) of the matrix 𝐹𝑉 , This is computed by 
|𝐽 − 𝜆𝐼| = 0, hence the matrix becomes 

|𝐽 − 𝜆𝐼| =

𝑇 − 𝜆 𝑇 𝑇 𝑇
0 −𝜆 0 0
0 0 −𝜆 0
0 0 0 −𝜆

= 0 

 

where 

𝑇 =
𝛽𝜀 𝜎𝑥(1 − 𝜌)

𝑄 𝑄
+

𝛽𝑥𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ]

𝑄 𝑄 𝑄
+

𝛽𝑥𝜀 𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ]

𝑄 𝑄 𝑄 𝑄
 

𝑇 =
𝛽𝜀 𝑥

𝑄
+

𝛽𝑥𝛾

𝑄 𝑄
+

𝛽𝜀 𝑥𝑟 𝛾

𝑄 𝑄 𝑄
, 𝑇 =

𝛽𝑥

𝑄
+

𝛽𝜀 𝑥𝑟

𝑄 𝑄
, 𝑇 =

𝛽𝜀 𝑥

𝑄
 

and 



152            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

𝑥 =
𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)
. 

This implies that 

𝜆 = 𝑇 =
𝛽𝜀 𝜎𝑥(1 − 𝜌)

𝑄 𝑄
+

𝛽𝑥𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ]

𝑄 𝑄 𝑄
+

𝛽𝜀 𝑥𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ]

𝑄 𝑄 𝑄 𝑄
 

𝜆 = 0, 𝜆 = 0, 𝜆 = 0. 

Therefore 

𝑅 =
𝛽𝜎𝑥[(1 − 𝜌)(𝜀 𝑄 + 𝛾 ) + 𝜌𝑄 ]

𝑄 𝑄 𝑄
+

𝛽𝜀 𝑥𝑟 𝜎[(1 − 𝜌)𝛾 + 𝜌𝑄 ]

𝑄 𝑄 𝑄 𝑄
 

=
𝛽𝜎𝜋(𝜇 + 𝜔 − 𝜇𝜃)([(1 − 𝜌)(𝜀 𝑄 + 𝛾 ) + 𝜌𝑄 ]𝑄 + 𝜀 𝑟 [(1 − 𝜌)𝛾 + 𝜌𝑄 ])

𝜇(𝜇 + 𝜔)(𝜇 + 𝜎)𝑄 𝑄 𝑄
 

 

where 

𝑄 = 𝛾 + 𝜇 + 𝑟 + 𝑑 , 
𝑄 = 𝑟 + 𝑟 + 𝑑 + 𝜇, 

𝑄 = 𝑑 + 𝑟 + 𝜇. 

2.8. Local stability of disease-free equilibrium  

Theorem 3:  

The Disease Equilibrium of the model equations (1)–(7) is locally asymptotically Stable 
(LAS) if 𝑅 < 1. 

Proof. 
Using Jacobian stability techniques, the Jacobian matrix at D.F.E is given by: 

𝐽(𝐸 )

=

⎝

⎜
⎜
⎜
⎜
⎜
⎛

−(𝜇 + 𝜔) 0 0 0 0 0 0
𝜔 −𝜇 0 0 0 0 0

0 0 −(𝜇 + 𝜎)
𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)

𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)

𝛽𝜀 𝜋(𝜇 + 𝜔 − 𝜇𝜃)

𝜇(𝜇 + 𝜔)
0

0 0 (1 − 𝜌)𝜎 −(𝛾 + 𝜇 + 𝑑 + 𝑟 ) 0 0 0

0 0 𝜌𝜎 𝛾 −(𝑟 + 𝑟 + 𝜇 + 𝑑 ) 0 0

0 0 0 0 𝑟 −(𝜇 + 𝑑 + 𝑟 ) 0
0 0 0 𝑟 𝑟 𝑟 −𝜇⎠

⎟
⎟
⎟
⎟
⎟
⎞

. 

Let 

𝑄 = (𝜇 + 𝜎),

𝑄 = (𝛾 + 𝜇 + 𝑑 + 𝑟 ),

𝑄 = −(𝑟 + 𝑟 + 𝜇 + 𝑑 ),

𝑄 = (𝜇 + 𝑑 + 𝑟 ),

𝑄 = (𝜇 + 𝜔),

𝐵 =
( )

( )
,

𝐵 =
( )

( )
,

𝐵 =
( )

( )
,

𝐶 = (1 − 𝜌)𝜎 ⎭
⎪
⎪
⎪
⎪
⎬

⎪
⎪
⎪
⎪
⎫

,    (23) 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 153 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

 

𝐽(𝐸 ) =

⎝

⎜
⎜
⎜
⎛

−𝑄 0 0 0 0 0 0
𝜔 −𝜇 0 0 0 0 0

0 0 −𝑄 𝐵 ∗ 𝐵 𝐵 0
0 0 𝐶 𝑄 0 0 0
0 0 𝜌𝜎 𝛾 −𝑄 0 0
0 0 0 0 𝑟 −𝑄 0
0 0 0 𝑟 𝑟 𝑟 −𝜇⎠

⎟
⎟
⎟
⎞

. 

Then,  

|𝐽 − 𝜆𝐼|

=

⎝

⎜
⎜
⎜
⎛

−𝑄 − 𝜆 0 0 0 0 0 0
0 −𝜇𝑄 − 𝜆 0 0 0 0 0
0 0 −𝑄 − 𝜆 𝐵 𝐵 𝐵 0
0 0 0 𝑄 − 𝜆 𝐶 𝐵 𝐶 𝐵 0
0 0 0 0 𝑄 − 𝜆 𝑄 0
0 0 0 0 0 −𝑄 − 𝜆 0
0 0 0 0 0 0 −𝑄 𝑄 − 𝜆⎠

⎟
⎟
⎟
⎞

 

and 

𝑄 = 𝐶 𝐵 − 𝑄 𝑄 , 
𝑄 = (𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 , 
𝑄 = (𝐶 𝐵 − 𝑄 𝑄 )𝜌𝜎𝐵 − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 , 
𝑄 = −([(𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ]𝑄 ), 
+𝑟 [(𝐶 𝐵 − 𝑄 𝑄 )𝜌𝜎𝐵 − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ], 
𝑄 = 𝜇(𝐶 𝐵 − 𝑄 𝑄 )[(𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ] 

where 

−

𝑄 − 𝜆 = 0,
−𝜇𝑄 − 𝜆 = 0,
−𝑄 − 𝜆 = 0,
𝑄 − 𝜆 = 0,
𝑄 − 𝜆 = 0,
−𝑄 𝑄 − 𝜆 = 0⎭

⎪
⎬

⎪
⎫

.     (24) 

From Equation (24) 𝜆 = −𝑄 , 

𝜆 = −
[(𝐶 𝐵 − 𝑄 𝑄 )(𝜌𝜎𝐵 − 𝑄 𝑄 ) − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ]𝑄

+𝑟 [(𝐶 𝐵 − 𝑄 𝑄 )𝜌𝜎𝐵 − (𝛾𝑄 − 𝜌𝜎𝐵 )𝐶 𝐵 ]
. (25) 

Substitute equation (23) into equation (25), we have 

= −𝛽𝛿𝜋(𝜇 + 𝜔 − 𝜇𝜃)
(1 − 𝜌)(𝜀 𝑄 + 𝛾)𝑄 + 𝜌𝑄 𝑄 + 𝑟 𝜀 (1 − 𝜌)𝛾 + 𝜌𝑄

𝜇(𝜇 + 𝜔)𝑄 𝑄 𝑄 𝑄
. 

Therefore, 

𝑅 = 𝛽𝛿𝜋(𝜇 + 𝜔 − 𝜇𝜃)
( )( ) ( )

( )
.  (26) 



154            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

The DFE is locally asymptotically stable since all the eigenvalues of (26) are negatives for 
𝑅 < 1, hence the proof is established. □ 

3. NUMERICAL METHOD 

3.1. Homotopy Perturbation Method (HPM)  

The fundamental of Homotopy Perturbation Method (HPM) was first proposed by [11]. 
The Homotopy Perturbation Method, which provides analytical approximate solution, is 
applied to various linear and non-linear equations. [2, 18] used Homotopy Perturbation Method 
to solve a Susceptible-Infected-Recovered (SIR) model of infectious diseases. The Homotopy 
Perturbation Method is a series expansion method used in the solution of nonlinear partial 
differential equations [18]. 

To show the simple concepts of this method, we consider the following non-linear 
differential equation given as equations according to [2].  

𝐴 (𝑈) − 𝑓(𝑟) = 0, 𝑟 ∈ 𝛺     (27) 

Subject to the boundary condition  

𝐵 𝑈, = 0, 𝑟 ∈ 𝛤     (28) 

Where A3 is a general differential operator, B3 a boundary operator, 𝑓(𝑟) is a known 
analytical function and Γ is the boundary of the domain 𝛺. The operator A3 can be divided into 
two parts L and N, where L is the linear part, and N is the nonlinear part. Equation (27) can be 
written as: 

𝐿(𝑈) + 𝑁(𝑈) − 𝑓(𝑟) = 0, 𝑟 ∈ 𝛺.   

The Homotopy Perturbation structure is shown as follows 

𝐻(𝑉, ℎ) = (1 − ℎ)[𝐿(𝑉) − 𝐿(𝑈 )] + ℎ[𝐴(𝑉) − 𝑓(𝑟)] = 0   (29) 

where 

𝑉(𝑟, 𝑃): 𝛺 ∈ [0,1] → 𝑅.     (30) 

In equation (29) 𝑃 ∈ [0,1] is an embedding parameter and 𝑈 is the approximation that 
satisfies the boundary condition. It can be assumed that the solution of the equation (30) can 
be written as power series in h given as equations (31) to (32): 

𝑉 = 𝑉 + ℎ𝑉 + ℎ 𝑉 +. ..     (31) 

And the best approximation for the solution is: 𝑈 = 𝑙𝑖𝑚 𝑣 = 𝑣 + ℎ𝑣 + ℎ 𝑣 + ⋯ 

ℎ → 1.       (32) 

The series (31) is convergent for most cases. However, the convergent rate depends on the 
nonlinear operator A (V) 

3.2. Solution of the model equations using HPM 

From differential equations 1 to 7  

+ (𝜇 + 𝜆 )𝑆 − 𝜔𝑉 − 𝜋(1 − 𝜃) = 0,    (33) 

+ (𝜇 + 𝜔)𝑉 − 𝜃𝜋 = 0,      (34) 

+ (𝜎 + 𝜆 + 𝜇)𝐸 − 𝜆 𝑇 − 𝜆 𝑆 = 0,    (35) 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 155 

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+ (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 − (1 − 𝜌)𝜎𝐸 − (1 − 𝜙)𝜆 𝐸 = 0, (36) 

+ (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝐼 − 𝜌𝜎𝐸 − 𝛾𝐼 − 𝜙𝜆 𝐸 = 0,  (37) 

+ (𝜇 + 𝑑 + 𝑟 )𝐽 − 𝑟 𝐼 = 0,     (38) 

+ (𝜇 + 𝜆 )𝑇 − 𝑟 𝐽 − 𝑟 𝐼 − 𝑟 𝐼 = 0.    (39) 

With the initial condition given as  

𝑆(0) = 𝑆 , 𝑉(0) = 𝑉 , 𝐸(0) = 𝐸 , 𝐼 (0) = 𝐼 , 𝐼 (0) = 𝐼 , 𝐽(0) = 𝐽 , 𝑇(0) = 𝑇 .
 (40) 

Let 

𝑆 = 𝑠 + ℎ𝑠 + ℎ 𝑠 +. . .,      (41) 

𝑉 = 𝑢 + ℎ𝑢 + ℎ 𝑢 +. . .,      (42) 

𝐸 = 𝑣 + ℎ𝑣 + ℎ 𝑣 +. . .,      (43) 

𝐼 = 𝑤 + ℎ𝑤 + ℎ 𝑤 + ⋯,     (44) 

𝐼 = 𝑥 + ℎ𝑥 + ℎ 𝑥 +. . .,      (45) 

𝐽 = 𝑦 + ℎ𝑦 + ℎ 𝑦 +. . .,      (46) 

𝑇 = 𝑧 + ℎ𝑧 + ℎ 𝑧 +. …      (47) 

Applying HPM into equation (33) 

(1 − ℎ) + ℎ + (𝜇 + 𝜆 )𝑆 − 𝜔𝑉 − 𝜋(1 − 𝜃) = 0.   (48) 

Substitute equation (41) and (42) into (48) 

(𝑠 + ℎ𝑠 + ℎ 𝑠 +. . . ) + ℎ
(𝜇 + 𝜆 )(𝑠 + ℎ𝑠 + ℎ 𝑠 +. . . ) − 𝜔

(𝑢 + ℎ𝑢 + ℎ 𝑢 +. . . ) − 𝜋(1 − 𝜃)
= 0. 

Collecting the coefficient of power of h, we have 

ℎ : 𝑠 = 0,      (49) 

ℎ : 𝑠 + (𝜇 + 𝜆 )𝑠 − 𝜔𝑢 − 𝜋(1 − 𝜃) = 0,   (50) 

ℎ : 𝑠 + (𝜇 + 𝜆 )𝑠 − 𝜔𝑢 = 0.    (51) 

From Equation (49) 𝑠 = 0 , integrating both side 𝑠 = 𝐷  and applying initial condition  
𝑠 (0) = 𝑆 = 𝑠 , 𝐷 = 𝑆  and 𝑠 = 𝑆 .  
From Equation (51) 

𝑠 = 𝜋(1 − 𝜃) + 𝜔𝑣 − (𝜇 + 𝜆 )𝑠 . 

Integrate both sides and applying the initial condition we get:  

𝑠 (𝑡) = (𝜋(1 − 𝜃) + 𝜔𝑣 − (𝜇 + 𝜆 )𝑠 )𝑡.    (52) 

Substitute Equations (41) and (42) into (52) 

𝑠 (𝑡) = (𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )𝑡.    (53) 

Applying HPM to Equation (34) 

(1 − ℎ) + ℎ + (𝜇 + 𝜔)𝑉 − 𝜋𝜃 = 0.     (54) 

Substitute Equation (42) into (54) 



156            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

(𝑢 + ℎ𝑢 + ℎ 𝑢 +. . . ) + ℎ[(𝜇 + 𝜔)(𝑢 + ℎ𝑢 + ℎ 𝑢 +. . . ) − 𝜋𝜃] = 0. 

Collecting the coefficient of power of ℎ, we have 

ℎ : 𝑢 = 0,      (55) 

ℎ : 𝑢 + (𝜇 + 𝜔)𝑢 − 𝜋𝜃 = 0,    (56) 

ℎ : 𝑢 + (𝜇 + 𝜔)𝑢 = 0.      (57) 

From Equation (56) 

𝑢 = 𝜋𝜃 − (𝜇 + 𝜔)𝑢 . 

Integrate both sides and applying the initial condition we have: 

𝑢 (𝑡) = (𝜋𝜃 − (𝜇 + 𝜔)𝑢 )𝑡.    (58) 

Substitute Equation (42) into (60) 

𝑢 (𝑡) = (𝜋𝜃 − (𝜇 + 𝜔)𝑉 )𝑡.    (59) 

From Equation (51) 

𝑠 = 𝜔𝑢 − (𝜇 + 𝜆 )𝑠 .     (60) 

Substitute Equation (59) and (53) into (60) 

𝑠 = 𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 )𝑡 − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )𝑡, 

𝑠 = 𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 ) − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 ) 𝑡. 

Integrating both sides and applying the initial condition 

𝑠 (𝑡) = 𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 ) − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 ) . (61) 

Substitute the initial condition and equation (53) and (61) into (41) 

𝑆(𝑡) = 𝑠 + ℎ𝑠 + ℎ 𝑠 +. . ., 

𝑆(𝑡) = lim
{ → }

(𝑠 + ℎ𝑠 + ℎ 𝑠 + ⋯ ), 

𝑆(𝑡) = 𝑠 + 𝑠 + 𝑠 +. . ., 

hence, 

𝑆(𝑡) = 𝑆 + (𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )𝑡 + 

𝜔(𝜋𝜃 − (𝜇 + 𝜔)𝑉 ) − (𝜇 + 𝜆 )(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )
𝑡

2
. 

Following the same process for other equations, we have: 

𝑉(𝑡) = 𝑉 + (𝜋𝜃 − (𝜇 + 𝜔)𝑉 )𝑡 + (𝜇 + 𝜔)(𝜋𝜃 − (𝜇 + 𝜔)𝑉 )
𝑡

2
, 

𝐸(𝑡) = 𝐸 +
𝜆 𝑇 + 𝜆 𝑆 −
(𝜎 + 𝜆 + 𝜇)𝐸

𝑡

+

𝜆 (𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 ) + 𝜆
(𝜋(1 − 𝜃) + 𝜔𝐸 − (𝜇 + 𝜆 )𝑆 )

−(𝜆 + 𝜇 + 𝜎)(𝜆 𝑇 + 𝜆 𝑆 − (𝜎 + 𝜆 + 𝜇)𝐸 )

𝑡

2
, 

𝐼 (𝑡) = 𝐼 + (1 − 𝜌)𝜎𝐸 + (1 − 𝜙)𝜆 𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 𝑡 + 

(1 − 𝜌)𝜎 + (1 − 𝜃) (𝜆 𝑇 + 𝜆 𝑆 − (𝜎 + 𝜆 + 𝜇)𝐸 ) − (𝛾 + 𝑟 + 𝑑 + 𝜇)

(1 − 𝜌)𝜎 + (1 − 𝜙)𝜆 𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼

𝑡

2
, 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 157 

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𝐼 (𝑡) = 𝐼 + (𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼 )𝑡 + 
(𝜌𝜎 + 𝜙𝜆 )(𝜆 𝑇 + 𝜆 𝑆 − (𝜎 + 𝜆 + 𝜇)𝐸 ) +

𝛾 (1 − 𝜌)𝜎𝐸 + (1 − 𝜙)𝜆 𝐸 − (𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼 −

(𝜇 + 𝑑 + 𝑟 + 𝑟 )(𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼 )

𝑡

2
, 

𝐽(𝑡) = 𝐽 +
𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )

𝐽
𝑡

+
𝑟 (𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 − (𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼 )

−(𝜇 + 𝑑 + 𝑟 )(𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 )

𝑡

2
, 

𝑇(𝑡) = 𝑇 + (𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼 − (𝜇 + 𝜆 )𝑇 )𝑡

+

⎝

⎜
⎜
⎜
⎜
⎛

𝑟 (𝑟 𝐼 − (𝜇 + 𝑑 + 𝑟 )𝐽 ) +

𝑟
𝜙𝜆 𝐸 + 𝜌𝜎𝐸 + 𝛾𝐼 −
(𝜇 + 𝑑 + 𝑟 + 𝑟 )𝑥𝐼

+

𝑟
(1 − 𝜌)𝜎𝐸 + (1 − 𝜙)𝜆 𝐸 −
(𝛾 + 𝜇 + 𝑟 + 𝑑 )𝐼

−(𝜇 + 𝜆 )
𝑟 𝐽 + 𝑟 𝐼 + 𝑟 𝐼

−(𝜇 + 𝜆 )𝑇 ⎠

⎟
⎟
⎟
⎟
⎞

𝑡

2
. 

4. RESULTS 

4.1. Numerical parameters 

In this section, we give the values and source of the parameters used for simulating the 
model. We used the following initial values for 𝑆(𝑡) = 160,840,589, 𝐸(𝑡) = 1,700,000, 𝐼 (𝑡) 
= 90000, 𝐼 (𝑡) = 10,400,000, 𝐽(𝑡) = 1,000,000, 𝑇(𝑡) = 1,109,000 and 𝑉(𝑡) = 8,000,000. 𝑁 =
𝑆(𝑡) + 𝐸(𝑡) + 𝐼 (𝑡) + 𝐼 (𝑡) + 𝐽(𝑡) + 𝑇(𝑡) + 𝑉(𝑡) = 206,139,589. 

Table 2. The parameters value used for the model 
Parameters Value Source 

𝜋 2,895,131 Estimated 

𝜇 0.018 Estimated 

𝑑  0.0365 Assumed 

𝑑  0.68 [21] 

𝑑  0.1 [22] 

𝑟  0.02 Assumed 

𝑟  0.02 Assumed 

𝑟  0.00375 [21] 

𝜃 0.020 [22] 

𝜌 0.075 [22] 

𝜙 0.3 Assumed 

𝜆
 

0.2 Assumed 

𝜆
 

0.2 Assumed 

𝜎
 

0.01 Assumed 

4.2. Graphical representation of solutions of the model equation 

The graphical representations are from the analytical solutions of the model equations. 
They are plotted using MAPLE software. 



158            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

 
(a) 

 
(b) 

 
(c) 

 
(d) 

 
(e) 

 
(f) 

 

 
(g) 

Fig. 2. (a) Effect of waning rate of vaccine on the chronic class. (b) Effect of waning rate of vaccine on 
vaccinated class. (c) Effect of isolation rate on the isolated compartment. (d) Treated individual against time for 

different values of recovery rate for chronic class. (e) Effect of recovery rate for those in isolated class. (f) 
Impact of effective interaction between susceptible and infected classes. (g) Mild TB individual against time for 

different values of transmission rate. 



              ANALYSIS AND DYNAMICS OF TUBERCULOSIS OUTBREAK 159 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

 

5. DISCUSSION  

Figure 2a is the graph of chronic infected individuals against time for different values of 
waning rate of vaccine. We carried out simulations by varying waning rate of vaccine as 0.2, 
0.4 and 0.6. It could be observed that different level of waning rate does not have effect on the 
infected population. For different level of waning rate, the TB infection continues to persist in 
the given population. 

Figure 2b is the graph of vaccinated individuals against time for different values of waning 
rate of vaccine. It was observed that the vaccinated population decreases with increase in the 
vaccination rate. Therefore, high waning rate of vaccine reduces the vaccinated population and 
thus puts them at the risk of contracting the disease.  

Figure 2c is the graph of isolated infectious individuals against time at different values of 
Progression rate from chronic TB class to isolated infected class. It was observed that the 
number of isolated Individuals increases as Progression rate from chronic TB class to isolated 
infected class Increases. 

Figure 2d is the graph of treated TB individuals against time. It was observed that the 
number of treated TB individual increases as the recovery rate among the chronic TB 
individual increases. This implies that increase in the progression rate will lead to increase in 
number of individuals with chronic TB disease. 

Figure 2e is the graph of recovered individual against time. The lower the treated rate the 
lower the number of recovered individuals. The lowest percentage almost decrease to zero. 
This shows that as the recovered are treated, they move to Chronic TB population. 

Figure 2f is the graph of exposed individual against time for different values of contact rate. 
We can observe that infected individuals increase as contact rate increases. The figure 
illustrates the great influence of effective contact rate on the exposed population. 

Figure 2g is the graph of mild TB individual against time. It was observed that the number 
of mild TB individual increases as the transmission rate from the exposed to the chronic 
individuals increases.  

6. CONCLUSION 

In this study, a mathematical model of tuberculosis transmission dynamics incorporating 
treatment, isolation and vaccination using the system of first order ordinary differential 
equations was developed and analyzed. It was discovered that the model has two equilibria. 
The equilibrium states were obtained and analyzed for their stability relatively to the effective 
reproduction number. The result shows that, the disease-free equilibrium was stable. We are 
able to show that the tuberculosis infectious free equilibrium is locally and globally 
asymptotically stable if 𝑅 < 1 . The analytical solution was obtained using Homotopy 
Perturbation Method and effective reproduction number was computed in order to measure the 
relative impact for individual or combined intervention for effective disease control.  

The graphs illustrate the impact of a combined effect of contact rate, waning rate of vaccine 
and rate of isolation. One can observe that this combines effects reduce the size of infected 
compartments. Thus, the simultaneous increase of effectiveness of vaccination rate, isolation 
rate and treatment rate are effective control measures against TB infection.  

The model shows that the spread of tuberculosis infection depends largely on the contact 
rate, hence the ministry of health and other health workers should emphasize on the 
improvement in early detection of tuberculosis infection cases, so that transmission can be 
minimized. Infected individuals should be isolated and treated immediately and individuals 
infected with tuberculosis should be given antiretroviral drugs immediately. In future work, 
we intent to incorporate optimal control strategy into the model for greater insight into the 
dynamics. 



160            F.A. OGUNTOLU, O.J. PETER, K. OSHINUBI, T.A. AYOOLA, A.O. OLADAPO, M.M. OJO 

Copyright ©2022 ASSA                                                                                    Adv. in Systems Science and Appl. (2022) 

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