Abstract
A deterministic model was formulated and employed in the analysis of the dynamics of tuberculosis with a keen emphasis on vaccination and drug resistance as the first line of treatment. It was assumed that some of the susceptible population were vaccinated but with temporal immunity. This is due to the fact that vaccines do not confer permanent immunity. Moreover, part of the infected individual after treatment grows resistance to the drug. Infective immigrants were also considered to be part of the population. The basic reproductive number for the model is estimated using the next-generation matrix method. The equilibrium points of the TB model and their local and global stability were determined. It was established that if the basic reproductive number was less than unity , then the disease free equilibrium is stable and unstable if . Furthermore, we investigated the optimal prevention, treatment, and vaccination as control measures for the disease. As the objective functional was optimised, there have been a significant reduction in the number of infections and an increase in the number of recovery. The best control measure in combating tuberculosis infections is prevention and vaccination of the susceptible population.
1. Introduction
Respiratory disease can be described as an infection which can be treated with time. The commonest respiratory infections include pneumonia, tuberculosis, and flu. Chronic conditions such as asthma and chronic bronchitis are persistent and sometimes long-lasting [1].
Tuberculosis is among the most ancient diseases worldwide. It is very contagious. The causative organism, Mycobacterium tuberculosis, was discovered by the German microbiologist Robert Koch in 1882 [2]. The motivation behind this study is to use a deterministic model to analyse the dynamics of the infection and suggest the best optimal control measure in combating the disease.
Through coughing, singing, and sneezing, pulmonary tuberculosis is spread from a sick TB patient as a droplet infection. Inhalation by an uninfected individual of these droplets may cause infection. With the frequency and duration of contact with people who have the disease, the risk of contracting TB rises.
In 1993, the WHO decreed TB a global epidemic [3]. It is estimated that the risk of contracting active TB after coming into contact with an infected person is between 5% and 10%, with a greater proportion of the disease playing a crucial role which happens in the very first few years after the initial infection with the arrival of HIV [4].
Biological models usually explain the transmission dynamics of infectious diseases and can determine the status of the disease in a population with time. The basic reproduction number is the threshold value that determines the persistence of a disease in a population [5–7].
Optimal control theory is usually employed in biological models to determine the best optimal control strategy in combating infections in a population [8–10].
2. Model Description and Formulation
The model partitions the entire populace into six compartments according to their epidemiological status. We define , , , , , and as the number of susceptible individuals, vaccinated individuals, exposed individuals, infectious individuals, individuals with resistance to treatment, and recovered individuals, respectively, at time .
Tables 1 and 2 show the variables and parameters used in the tuberculosis model. Figure 1 shows the tuberculosis (TB) model transmission dynamics.
The following differential equations were obtained from the model flow diagram:
Thus, the total population is given as with initial conditions:
3. Tuberculosis Model Analysis
The tuberculosis (TB) model is about human population; hence, model state variables ought to be nonnegative and limited for all . In this section, we demonstrate that the TB model is numerically and epidemiologically sensible.
3.1. Positivity of Solution
We prove the positivity of the variables in the model. Based on the concept of derivative of a function, the behavior of the function at a known point can be established.
Theorem 1. Let the initial set be , , , , , and and be nonnegative; then, the solution set of of equation (1) is positive and bounded for all , wherever they exist.
Proof. From equation (1), we can state that At , since Also, At , since Also, At , since Also, At , since Also, At , since Also, At , since
3.2. Boundedness of the System
The region in which solutions of the tuberculosis (TB) model system are uniformly bounded is the proper subset, and it is given by
Proof. So as , .
Therefore, is a positive invariant.
3.3. Existence of Disease-Free Equilibrium Point
The disease-free equilibrium of the dynamical system (1) is obtained by setting , and since there is no disease ,
Therefore, the disease-free equilibrium of the dynamical system (1) is
3.4. Basic Reproductive Number
The basic reproductive number can be computed utilizing the cutting edge matrix approach. The basic reproduction number determines the state of a disease with time in a dynamical system [11, 12]. It is utilized to predict the stability of the disease equilibrium. The basic reproductive number is characterized as the quantity of secondary infections that one tainted person can create in a completely susceptible population [13, 14]. According to [13, 15], the next-generation matrix is defined as and , where denotes the spectral radius of .
Using the next-generation matrix, we consider only the infectious compartments in the system of differential equation in (1).
Let be the count of emerging infection moving into the system and be the count of infections exiting the system.
The Jacobian matrix of and are obtained by
But .
From the relation , the inverse of can be calculated:
Computing the product of ,
By selecting the dominant eigenvalue of , the basic reproductive number is
At the disease-free equilibrium, we substitute into the basic reproductive number, .
This therefore implies that
3.5. Local Stability of the Disease-Free Equilibrium
Theorem 2. The disease-free equilibrium point of the dynamical system (1) is locally asymptotically stable if and unstable .
Proof. The Jacobian matrix of the dynamical system (1) at the DFE point
is given by
The corresponding characteristic equation for the eigenvalues is .
This is
According to the Routh-Hurwitz criterion, since , , and , , , and will have negative real part as roots.
Also,
The roots, and , of this characteristic polynomial will have negative real part if and only if
Therefore, is asymptotically stable since and unstable if .
3.6. Global Stability of the Disease-Free Equilibrium
Theorem 3. The disease-free equilibrium point of the dynamical system (1) is globally asymptotically stable in if and unstable .
Proof. Using the Perron eigenvector to prove the global stability of the disease free equilibrium as in [16–18], we apply the matrix-theoretic method. In the dynamical system, the disease compartment is and the nondisease compartment is .
Taking the same path as [16, 18], let us set
Then, the equation of the disease compartment can be written as
Theorem 4. Let be defined as in equation (11). Then, the threshold property holds for system (1).
Proof. Using the condition outlined in Theorem 3, we set the Lyapunov function for the disease-free equilibrium (DFE).
We first find (the left eigenvector of the nonnegative matrix ):
, and any multiple of this becomes our eigenvector.
From equation (40),
That is, ,
where at the disease-free equilibrium.
Therefore, , and this satisfies the demand of Theorem 4. The Lyapunov function is
But
Since , , and ,
From the derivative of the Lyapunov function, when , which satisfies the condition that the disease-free equilibrium is asymptotically stable and unstable when .
3.7. Existence of the Endemic Equilibrium
The endemic equilibrium point is acquired by mounting the right-hand side of the dynamical system (1) equal to zero and solving them simultaneously [19, 20]. The endemic equilibrium point is , where
is the positive root of , that is, .
We have three possibilities of getting the value of : (1)If , then there is no endemic equilibrium state(2)If , then again, the endemic equilibrium point does not exist(3)If , then the endemic equilibrium point exists when where ,
3.8. Local Stability of the Endemic Equilibrium
Theorem 5. The positive endemic equilibrium point of the system (1) is locally asymptotically stable if .
Proof. The Jacobian matrix of the system of equation (1) at the endemic point is
where , , , , , , , , and .
The corresponding characteristic equation is denoted by and is given as
The matrix is a strictly column diagonally dominant matrix. Again, all the diagonal entries are negative. Hence, all eigenvalues of have negative real part. Now applying the Gershgorin circle theorem [21], is locally asymptotically stable if , , , , , and .
3.9. Global Stability of the Endemic Equilibrium
Theorem 6. The dynamical system (1) is said to have an endemic equilibrium if , and it is globally asymptotically stable.
Proof. Consider the Lyapunov function defined by
Computing the derivative of along the solution of the dynamical system in (1) directly,
where and .
Imposing the condition that , the derivative of the Lyapunov function with respect to time is less than or equal to zero.
If , then .
But if and only if , , , , , and .
Therefore, the endemic equilibrium point is globally asymptotically stable in if .
The largest invariant set in is a singleton, where is the endemic equilibrium point.
4. TB Model Extension to Optimal Control
An analysis of the optimal controls to ascertain its effects on the model is been conducted. The optimal control problem is obtained by integrating the undermentioned control functions into the tuberculosis model (1) and introducing an objective functional that desires to minimize the controls , where is the vaccination of the susceptible population as a control measure, is the treatment of the infected individuals as a control measure, and is the education/sensitization of the exposed population as a control measure.
By inserting the various controls, the system with the optimal controls becomes
Let the optimal levels of the control set be , which is Lebesgue measurable and defined as
The quadratic nature of the control efforts as a result of the assumption that costs is generally nonlinear in nature. Our objective is to minimise the number of infections and reduce the cost of treatment.
The problem is to find a control and its associated state variables , , , , , and to minimize the objective functional given by
That is, subject to the differential equation system (57), where , , , , and are the weight constants to balance the terms in the integrals to abstain the ascendance of one over the others.
Also, , , and are the cost associated with the infected individuals and the individuals with resistance to treatment, respectively, while , , and are the cost associated with vaccination, treatment, and sensitization as preventive measures. is the period of the intervention.
The purpose of inserting the controls is to minimize the number of infections and at the same time reduce the cost of treatment.
Our task at this point is to find the optimal functions: , , and such that , where is referred to as the control set.
4.1. Pontryagin’s Maximum Principle
Consider the Lagrangian function:
The Pontryagin maximum principle provides the essential condition that the optimal must satisfy. This changes the system of the differential equation into minimization problem pointwise Hamiltonian with respect to .
Hence, the Hamiltonian becomes where , , , , , and are disjoint variables. considering the relation
By taking partial derivatives of the Hamiltonian function with respect to and negating each of them, the following costate variables are the solutions of the adjoint systems.
The above satisfy the transversality condition:
Moreover, the characterization of the optimal control is obtained by solving where .
Theorem 7. Given the optimal control vector and the solutions of the corresponding state system (34) and (35) that minimise the objective functional over , then there exist adjoint variables , , , , , and , where where , , , , , and are the solutions of equations (28) and (30).
Proof. The presence of optimal control is as an aftereffect of the convexity of the integral of regarding , , and , the Lipschitz property of the state system concerning the state factors from the earlier boundedness of the state arrangements [22, 23].
The differential conditions administering the adjoint factors are acquired by separation of the Hamiltonian work, assessed at the ideal control. By standard control contentions including the limits on the control, we conclude
where , , and .
5. Numerical Results
The state systems, adjoint equations, and the transversality terms are solved simultaneously to get the optimal strategies. The optimal problem is a two-point boundary value problem with two abstracted boundary conditions at initial times and , where months. This represents the period at which preventive strategies and treatment are expected to be stopped. The numerical simulation was conducted by solving the state equations, the adjoint equations, and the transversality conditions using the Runge-Kutta fourth-order scheme by guessing the controls over a simulated time. We then use the current iteration of the state equation, the adjoint equations, and the transversality conditions by a backward method. Further iterations are done until values of the unknown variables at the previous iteration are very close to those at the present iteration [18, 24, 25].
Table 3 shows the various parameter values used in the TB model simulations.
5.1. Strategy 1: Treatment, Prevention, and Vaccination of the Susceptible Population
Objective functional was optimised by using treatment, prevention, and vaccination as control measures. As a result of these control measures, there have been significant reduction of infections and an increase in the number of recovered populations as shown in Figures 2 and 3.
5.2. Strategy 2: Prevention and Treatment of the Infected Population
Objective functional was optimised by using prevention, vaccination, and treatment as control measures. The outcome of these control measures indicates a reduction of the population infected and increased recoveries, indicating that these variables have greatly impacted in the combat of the spread of infections as shown in Figures 4 and 5.
5.3. Strategy 3: Vaccination and Treatment of the Infected Population
Objective functional was optimised by using treatment, vaccination, and prevention of the susceptible population as control measures. Figures 6 and 7 show the effects of treatment and vaccination, respectively: an increase in the recovery population, a decrease in the infectious population, and a decrease in the number of population susceptible.
6. Conclusion
A deterministic model for tuberculosis was formulated and analysed. The basic reproductive number for the TB model is estimated using the next-generation matrix method.
The equilibrium points of the TB model and their local and global stability were determined. It was established that if the basic reproductive number was less than unity , then the disease-free equilibrium is stable and unstable if . Furthermore, we investigated the optimal prevention, treatment, and vaccination as control measures for the disease.
Objective functional was optimised by using treatment, prevention, and vaccination as control measures. As a result of these control measures, there have been significant reduction of infections and an increase in the number of recovered populations as shown in Figures 2 and 3.
Objective functional was optimised by using prevention, vaccination, and treatment as control measures. The outcome of these control measures indicates a reduction of the population infected and increased recoveries as shown in Figures 4 and 5, indicating that these variables have greatly impacted in the combat of the spread of infections.
Objective functional was optimised by using treatment, vaccination, and prevention of the susceptible population as control measures. An increase in the recovery population, a decrease in the infectious population, and a decrease in the number of population susceptible are shown in Figures 6 and 7.
It was established that the best control measure in combating tuberculosis infections is prevention and vaccination of the susceptible population.
Data Availability
Some of the parameter values are assumed, and others are taken from published articles and are cited in this paper. These published articles are also cited at relevant places within the text as references.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
The authors acknowledged the support of faculty members in their respective departments.