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We develop a dynamical model to understand the underlying dynamics of TUBERCULOSIS infection at population level. The model, which integrates the treatment of individuals, the infections of latent and recovery individuals, is rigorously analyzed to acquire insight into its dynamical features. The phenomenon resulted due to the exogenous infection of TUBERCULOSIS disease. The mathematical analysis reveals that the model exhibits a backward bifurcation when TB treatment remains of infected class. It is shown that, in the absence of treatment, the model has a disease-free equilibrium (DEF) which is globally asymptotically stable (GAS) and the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium (EEP), for a special case, whenever the associated reproduction threshold quantity exceeds unity. For a special case, the EEP is GAS using the central manifold theorem of Castillo-Chavez.

A differential equation which describes some physical process is often called a mathematical model of the process. Again a differential equation is a mathematical equation that rates some functions of one or more variables with its derivatives, differential equation arises whenever a deterministic relation involving some continuously varying quantities and their rates of change in space and time. These equations occupy the place at center stage of both pure and applied mathematics. For the mathematicians, mathematical modeling offers an important tool in the study of the evolution of diseases such as Tuberculosis, HIV, Hepatitis, Ebola, etc. Various epidemiology models such as SIR, SEIR, SIRS, SEIS, MSEIR, etc. can be built to analyze these types of diseases. Among them the SIR model is widely used in epidemiology and public health to compute number of individuals in each category of the population and to explain the change in the number of people needing medical attention during an epidemic as well as evaluate policies effectively during the endemic Tuberculosis [

Tuberculosis is an infectious disease usually caused by the Mycobacterium tuberculosis (MTB). Tuberculosis is spread through air, just like a cold or flu when people have active TB in their lungs, they are suffering from cough, spit, speak or sneeze. Tuberculosis generally affects the lungs but it can also be other parts of the body like brain and spine. Tuberculosis is contagious, but it is not easy to catch. It has slow intrinsic dynamics, the incubation period and the infectious period spam long term intervals in the order of years on average. Therefore, a mathematical model is needed to have a better insight on the dynamics of the disease [

Tuberculosis is not only a health problem, but also an economic problem of mankind as outbreaks usually lead to enormous expenditure on healthcare. Over 80% of all TB patients live in 22 countries, mostly in Sub-Saharan Africa and Asia [

Recent years have seen an increasing trend in the representation of mathematical models in publications in the epidemiological literature, from specialist journals of medicine, biology and mathematics to the highest impact generalist journals [

In this paper, we have formulated the transmission dynamics of Tuberculosis in the presence of treatment and investigated its role in the dynamics of the disease.

Following the classical assumptions, we formulate a deterministic, compact mental, mathematical model to describe the transmission dynamics of measles. The population is homogeneously mixing and reflecting the demography of a typical developing country, as it experiments an exponentially increasing dynamics. In other to describe the model equations, the total population (N) is divided into three classes: Susceptible (X), infected (Y) and Recovered (Z). Here we shall detail the transitions among these four classes as depicted in

The class X of susceptible is increased by birth or immigration at a rate Λ . It is decreased by infection following contact with infected individuals at a rate β, and diminished by natural death at a rate μ. The class Y is decreased by testing and therapy at a rate r, breakthrough into infected class at a rate β and diminished by natural death at a rate μ. The class Y of infected individuals is generated by breakthrough of individuals at a rate k. The class is decreased by recovery from infection at a rate β and diminished by natural death at a rate μ. The model assumes that both recovered exposed individuals and recovered infected individuals become permanently immune to the disease. This generates a class R of individuals who have complete protection against the disease.

The transitions between model classes can now be expressed by the following system of first order differential equations:

The description of Variables of the TB Model is shown in

d X d t = Λ − β ( Y + η Z ) X − μ X + r 1 Y + r 2 Z (1)

d Y d t = β ( Y + η Z ) X − k 1 Y (2)

d Z d t = k Y − k 2 Z (3)

Since the model monitors human population, all the associated parameters and state variables are non negative t ≥ 0. It is easy to show that the state variables of the model remain non-negative for all non-negative initial conditions. Consider the biological feasible region.

Ω = { ( S , I , R ) ∈ R + 3 : N → Λ μ }

From the model Equation (1) to (3) it will be shown that the region is positive. The total population of individuals is given by

Variables | Description |
---|---|

X | Susceptible class |

Y | Latently infected (exposed) class |

Z | Infected class |

Λ | Recruitment rate into the population |

μ | Natural death rate |

d | Death rate due to infection |

β | Probability rate of transmission |

r 1 | Treatment rate for exposed class |

r 2 | Treatment rate for infected class |

k | Infection rate for exposed individuals |

k 1 | Progression rate of exposed class |

k 2 | Progression rate of infected class |

η | Rate of infectiousness of infected class, where η > 1 |

N = S + I + R

Disease Free Equilibrium (DFE): The equilibrium points of the system can be obtained by equating the rate of changes of zero, given by ε 0 ,

∴ ε 0 = { X = Λ μ , Y = 0 , Z = 0 } (4)

The stability of the DFE will be analyzed using the next generation method [

F = ( β Λ μ β η Λ μ 0 0 )

and V = ( k 1 0 − k k 2 )

where, k 1 = r 1 + μ + k , k 2 = r 2 + μ + d .

The associated reproduction number, denoted by R 0 , is given by R 0 = ρ ( F V − 1 ) , where ρ denotes the spectral radius (dominant eigenvalue in magnitude) of the next generation matrix F V ′ . It follows that

V − 1 = ( 1 k 1 0 k k 1 k 2 1 k 2 )

∴ F V − 1 = ( β Λ ( k 2 + η k ) μ k 1 k 2 0 )

∴ R 0 = β Λ ( k 2 + η k ) μ k 1 k 2 (5)

Lemma: The disease free equilibrium ε 0 of the model (1), (2) and (3), is locally asymptotically stable (LAS) if R 0 < 1 and unstable if R 0 > 1 .

The threshold quantity, R 0 , is the reproduction number for the model. The epidemiological implication of Lemma 1 is that Tuberculosis spread can be effectively controlled in the community (when R 0 < 1 ) if the initial sizes of the populations of the model are in the basin of attraction of the disease free equilibrium ε 0 .

Since we have considered Tuberculosis model in some stages, are shown the backward bifurcation, where the stable DFE co-exists with a stable endemic equilibrium when the associated reproduction threshold ( R 0 ) is less than unity, it is instructive to determine whether or not the model also exhibits this dynamical feature. This is investigated below.

Theorem 1: The model (1), (2) and (3) undergoes a backward bifurcation at R 0 = 1 if the inequality holds.

Proof: The proof of theorem, which is based on the use of center manifold theory. The backward bifurcation phenomenon of the model is numerically illustrated below. It is convenient to let X = x 1 , Y = x 2 , Z = x 3 , so that N = x 1 + x 2 + x 3 . Further, by introducing the vector notation X = ( x 1 , x 2 , x 3 ) T , the model can be written in the form,

d x d t = F ( x ) ,

where F = ( f 1 , f 2 , f 3 ) T , as follows

d x 1 d t = f 1 = Λ − β ( x 2 + η x 3 ) x 1 − μ x 1 + r 1 x 2 + r 2 x 3 d x 2 d t = f 2 = β ( x 2 + η x 3 ) x 1 − k 1 x 2 d x 3 d t = f 3 = k x 2 − k 2 x 3 (6)

where, λ = β ( Y + η Z ) . The jacobian of the system at the DFE ( ε 0 ) is given by,

J ( ε 0 ) = [ − μ − β Λ μ − β η Λ μ 0 β Λ μ − k − μ β η Λ μ 0 k − d − μ ]

To analyze the dynamics of the model and we compute the eigenvalues of the jacobian of the equations at the disease free equilibrium (DEF). It can be shown that this jacobian has a left eigenvector is given by V = ( v 1 , v 2 , v 3 ) T where,

v 1 = 0

v 2 = free

and v 3 = β η Λ μ k 2 v 2

The right eigenvector is given by, W = ( w 1 , w 2 , w 3 ) T

w 1 = − β Λ ( k 2 + η k ) μ 2 k 2 w 2

w 2 = free and w 3 = k w 2 k 2

Theorem 2: (Castillo-Chavez and Song)

Consider the following general system of ordinary differential equations with a parameter φ .

d x d t = f ( x , φ ) , f : ℜ n × ℜ → ℜ and f ∈ C ( ℜ n × ℜ )

where 0 is an equilibrium of the system (i.e. f ( 0 , φ ) = 0 for all φ and assume

A1: A = D x f ( 0 , 0 ) = ( ∂ f i ∂ x j ( 0 , 0 ) ) is the linearization matrix of the system (6) around the equilibrium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts.

A2: Matrix A has a right eigenvector w and a left eigenvector v (each corresponding to the zero eigenvalue).

Let, f k be the kth component of f and

a = ∑ k , i , j = 1 n v k w i w j ∂ 2 f k ∂ x i ∂ x j ( 0 , 0 ) b = ∑ k , i = 1 n v k w i ∂ 2 f k ∂ x i ∂ φ ( 0 , 0 )

The local dynamics of the system around the equilibrium point 0 is totally determined by the sings of a and b. Particularly, a < 0 , b > 0 , the system does not show backward bifurcation at R 0 = 1 . In these cases, 0 < φ ≪ 1 , 0 becomes unstable and there exists a positive locally asymptotically stable equilibrium.

Computations of a and b:

a = ∑ k , i , j = 1 n v k w i w j ∂ 2 f k ∂ x i ∂ x j ( 0 , 0 ) = β v 2 w 1 ( w 2 + η w 3 ) < 0 (7)

b = ∑ k , i = 1 n v k w i ∂ 2 f k ∂ x i ∂ φ ( 0 , 0 ) = π v 2 ( η w 3 + w 2 ) μ > 0 (8)

This result is summarized below.

Theorem 3: The model (1), (2) and (3) exhibit backward bifurcation at R 0 = 1 whenever a < 0 , b > 0 . It should be noted that, in the absence of recovery exposed stage and infected stage the backward bifurcation co-efficient, a is given in below,

a = β v 2 w 1 ( η w 3 + w 2 ) < 0

since all the model parameters and the eigenvectors w i ( i = 2 , 3 , ⋯ ) and v i ( i = 1 , 2 , ⋯ ) are non-negative and 0 < ε < 1 . Thus, since the inequality does not hold in this case, the model (1), (2) and (3) will not undergo backward bifurcation in the absence of recovery exposed stage and infected stage. This result is summarized below.

Lemma: The model (1), (2) and (3) does not undergo backward bifurcation at R 0 = 1 in the absence of treatment ( r 1 = r 2 = 0 ). If we consider r 1 and r 2 exist then the coefficient of a maybe positive and b is also positive.

The backward bifurcation phenomenon of the model is numerically illustrated in below:

Simulations of the model shows that

Let,

Parameters | Values |
---|---|

Λ | 2000 [ |

μ | 0.02 [ |

d | 0.1 [ |

η | 0.08 (assumed) |

r 1 | 0.85 (assumed) |

r 2 | 0.9 (assumed) |

k | 0.7 [ |

d X d t = H ( X , Z ) d Z d t = G ( X , Z ) , G ( X , 0 ) = 0. (9)

where,

X = ( X , 0 ) and Z = ( Y , Z ) with the components of X ∈ R 1 denoting the uninfected population and the components of Z ∈ R 2 denoting the infected population.

The disease free equilibrium is now denoted as,

E 0 = ( X * , 0 , 0 ) , X * = { Λ μ , 0 , 0 }

Now, d X d t = H ( X , 0 ) , X * is globally asymptotically stable (GAS)

G ( X , Z ) = P z − G ^ ( X , Z ) , G ^ ( X , Z ) ≥ 0 for ( X , Z ) ∈ Ω . (10)

where, P = D Z G ( X * , 0 ) is an M-matrix (the off diagonal elements of P are non negative) and Ω is the region where the model makes biological sense. If the system (9) satisfies the conditions of (10) then the theorem below holds.

Theorem 4: The fixed point ε 0 ( X * , 0 ) is a globally asymptotically stable equilibrium of system (9) provided that R 0 < 1 and the assumptions in (10) are satisfied.

Proof:

From the system (1) and (2) we have,

H ( X , 0 ) = ( Λ − μ X + r 1 Y + r 2 Z 0 )

G ( X , Z ) = P ( Z ) − G ^ ( X , Z ) ⇒ G ^ ( X , Z ) = P ( Z ) − G ( X , Z ) (11)

where,

P ( Z ) = ( − k 1 0 k − k 2 )

and G ( X , Z ) = ( β Y X + β η Z X − k 1 Y k Y − k 2 Z )

Putting values P ( Z ) and G ( X , Z ) in (11) no equation and we obtain,

G ^ ( X , Z ) = ( G ^ 1 ( X , Z ) G ^ 2 ( X , Z ) ) = 0 (12)

It is clear that G ^ ( X , Z ) = 0 for all ( X , Z ) ∈ Ω we also note that matrix P is an M-matrix since its off diagonal elements are non-negative.

Let, ε 1 = ( X * * , Y * * , Z * * ) represents any arbitrary endemic equilibrium of the Tuberculosis model. Solving the Equations (1)-(3), the model has the following endemic equilibrium points (EEP),

X * * = Λ μ R 0 Y * * = Λ k 2 ( R 0 − 1 ) R 0 { μ r 2 + ( μ + d ) ( μ + k ) } Z * * = Λ k ( R 0 − 1 ) R 0 { μ r 2 + ( μ + d ) ( μ + k ) }

Existence of Endemic Equilibrium Point (EEP): special case

In this section, the possible existence and stability of endemic (positive) equilibria of the model (1), (2) and (3) (i.e. equilibria where at least one of the infected of the model is non-zero) will be consider for the special case where recovery rate from exposed stage and infected stage does not occur (i.e. r 1 = r 2 = 0 ).

Let, ε 1 = ( X * * , Y * * , Z * * ) represents any arbitrary endemic equilibrium of the model (1), (2) and (3) with r 1 = r 2 = 0 . Solving the equations of the system at steady-stage gives,

X * * = − { β ( y + η z ) } − μ Y * * = β ( y + η z ) Z * * = 0 (13)

The expression for λ , defined in (1), (2) and (3) at the endemic steady-state is given by

λ * * = β ( y + η z ) (14)

For mathematical convenience, the expression in (14) is re-written,

λ * * = β ( Λ k 2 ( R 0 − 1 ) R 0 { μ r 2 + ( μ + d ) ( μ + k ) } + η k Λ ( R 0 − 1 ) R 0 { μ r 2 + ( μ + d ) ( μ + k ) } )

And we get,

λ * * = β Λ ( R 0 − 1 ) ( η k + k 2 ) R 0 { μ r 2 + ( μ + d ) ( μ + k ) } > 0 (15)

The components of the unique endemic equilibrium ε 1 can be obtained by substituting the unique value of λ * * , given into the expression in (14). Thus, the following has been established.

Lemma: The model with recovery rate from exposed stage and infected stage r 1 = r 2 = 0 has a unique endemic equilibrium, given by ε 1 , whenever R 0 > 1 , λ * * > 0 .

Theorem 5: The unique EEP

ε 1 = { X * * , Y * * , Z * * } = { Λ μ R 0 , Λ k 2 ( R 0 − 1 ) R 0 { μ r 2 + ( μ + d ) ( μ + k 1 ) } , Λ k ( R 0 − 1 ) R 0 { μ r 2 + ( μ + d ) ( μ + k 1 ) } } of the model with r 1 = r 2 = 0 is globally asymptotically stable (GAS), whenever R 0 > 1 .

Proof:

Let, r 1 = r 2 = 0 and R 0 > 1 , so the EEP, ε 1 exists. Consider the following non-linear Lyapunov function,

L = ( X − X * * − X * * ln X X * * ) + ( Y − Y * * − Y * * ln Y Y * * ) + a 1 ( Z − Z * * − Z * * ln Z Z * * ) (16)

With Lyapunov derivative is given by,

L ˙ = ( 1 − X * * X ) X ˙ + ( 1 − Y * * Y ) Y ˙ + β η X * * k 2 ( 1 − Z * * Z ) Z ˙

Now,

L ˙ = β Y * * X * * + β η Z * * X * * + 2 μ X * * − μ X − β Y * * ( X * * ) 2 X − β η Z * * ( X * * ) 2 X − μ ( X * * ) 2 X + β Y X * * − μ Y − k Y − β X Y * * − β η Y * * Z Y + β Y * * X * * + β η X * * Z * * (17)

And

β η X * * k 2 ( 1 − Z * * Z ) ( k Y − k 2 Z ) = β η X * * Y Y * * Z * * − β η X * * Z − β η X * * Z * * Z * * Z Y Y * * + β η X * * Z * * (18)

Adding (17) and (18) and we get,

= μ X * * [ 2 − X X * * − X * * X ] + β X * * Y * * [ 2 − X X * * − X * * X ] + β η X * * Z * * [ 3 − X * * X − Y Y * * Z * * Z − X X * * Y * * Y Z Z * * ]

Since the arithmetic mean exceeds the geometric mean, it follows then that

2 − X X * * − X * * X ≤ 0 3 − X * * X − Y Y * * Z * * Z − X X * * Y * * Y Z Z * * ≤ 0 (19)

So that L ˙ ≤ 0 for R 0 > 1 . Hence, L is a Lyapunov function of the system with r 1 = r 2 = 0 on Ω . In other words, lim t → ∞ ( X , Y , Z ) = ( X * * , Y * * , Z * * ) .

Thus, by the Lyapunov function L and LaSalle’s Invariance Principle every solution to the equation in the model, with r 1 = r 2 = 0 approaches ε 1 as t → ∞ for R 0 > 1 .

The effect of the TB transmission dynamics is monitored by simulating the model with the parameters from

In

We rigorously analyzed (mathematically and numerically) the dynamics of TB in the model. Some mathematical and epidemiological findings of the study are given below:

1) The model has a disease free equilibrium (DFE) which is asymptotically stable R 0 < 1 and unstable if R 0 > 1 . The model is also globally asymptotically stable for a special case when r 1 = r 2 = 0 .

2) When R 0 = 0.8544 < 1 , the rate of infected individuals increases and after a certain time it smoothly decreases.

3) And lastly, the prevalence is very high when R 0 = 4.7466 > 1 .

A deterministic model for the transmission dynamics of TB in the population level is designed and rigorously analyzed. Some of the main findings of the study include the following:

1) The model exhibits a phenomenon of backward bifurcation, when DFE is locally asymptotically stable.

2) The model has an EEP which is globally asymptotically stable for special case (i.e. r 1 = r 2 = 0 ).

The model does not undergo backward bifurcation in the absence of treatment stage.

The authors declare no conflicts of interest regarding the publication of this paper.

Nayeem, J. and Sultana, I. (2019) Mathematical Analysis of the Transmission Dynamics of Tuberculosis. American Journal of Computational Mathematics, 9, 158-173. https://doi.org/10.4236/ajcm.2019.93012