A Within-host Tuberculosis Model Using Optimal Control

Received : 05-01-2021 Revised : 17-03-2021 Accepted : 20-03-2021 Online : 16-04-2021 In this paper, we studied a mathematical model of tuberculosis with vaccination for the treatment of tuberculosis. We considered an in-host tuberculosis model that described the interaction between Macrophages and Mycobacterium tuberculosis and investigated the effect of vaccination treatments on uninfected macrophages. Optimal control is applied to show the optimal vaccination and effective strategies to control the disease. The optimal control formula is obtained using the Hamiltonian function and Pontryagin's maximum principle. Finally, we perform numerical simulations to support the analytical results. The results suggest that control or vaccination is required if the maximal transmission of infection rate at which macrophages became infected is large.


A. INTRODUCTION
Tuberculosis (TB) is the world's leading infectious disease caused by Mycobacterium tuberculosis (MTb). Many attempts have been proposed to explain the progression of TB disease by considering the influencing factors, including the mechanisms of bacterial response and immunity. However, the factors that influence the development of TB are not currently known with certainty. Motivated by this, we consider a within-host tuberculosis model in (Adi & Thobirin, 2020) with three components: uninfected macrophages, infected macrophages ( ), and MTb bacteria, denoted by , , and , respectively. The model is given in the following form of ordinary differential equation system (1.1) In this model (1.1), Λ and are a constant production rate and a natural death rate of uninfected macrophages, respectively. Parameter is the maximal transmission of infection rate at which macrophages became infected at a saturated incidence rate of 1+ , whit 1 1+ is an inhibition effect. Parameter and are the average numbers of the MTb bacteria released by infected macrophages and the rate of macrophages burst, respectively. The parameters is the MTb bacteria death rate by uninfected macrophages, and is MTb bacteria natural death rate. The infected macrophages die due to an adaptive immune response modeled in a density-dependent term, 1+ , whereas is the maximum killing rate, and is a halfsaturation constant. In many works of literature of tuberculosis models, it is usually considered drug administration or vaccination, as in (Baba et al., 2020;Blaser et al., 2016;Brooks-Pollock et al., 2014;Byrne et al., 2015;Kim et al., 2018;Kuddus et al., 2020;Kyu et al., 2018). Many of the TB models use optimal control theory regarding their treatment strategies (Agusto & Adekunle, 2014;Baba et al., 2020;Bowong, 2010;Choi et al., 2015;Emvudu et al., 2011;Fatmawati et al., 2020;Gao & Huang, 2018;Moualeu et al., 2015). In a within-host tuberculosis model, the immune response to bacteria is generally assumed, as in (Adi & Thobirin, 2020;Zhang, 2020;Zhang et al., 2020). This paper will apply the optimal control theory to a within-host tuberculosis model by considering the vaccination that affects macrophages.
The purpose of this study is to determine the treatment strategies in a within-host tuberculosis model. We apply optimal control theory to determine optimal treatment that will maximize tuberculosis prevention efforts, such as vaccinations.
(2.2) In this system, the solution ( ) depends on the control ( ). The corresponding response of the system is a trajectory that corresponds to the control ( ). In system (2.2), the control may be arbitrary, so the problem does not have a solution. Therefore, we need to find the best control to minimize or maximize the performance measure as an objective function. In a disease-control model, we need to find the control to minimize the cost of controlling the disease. For that purpose, a payoff functional is defined as follows [ ] = ∫ 0 ( ( ), ( )) , (2.3) where ( ) solves (2.2) for the specified control ( ), with the given function : ℝ × → ℝ and terminal time as well. The function is called the running payoff. Now, introduce the admissible controls Ω = { ( ) ∈ 1 (0, ): ( ) ∈ }.
(2.4) The optimal control problem is to find a control * ( ) ∈ Ω that minimizes or maximizes the payoff functional (2.3). A corresponding solution together with the optimal control gives the pair of optimal control ( * , * ).
If such a control ( ) exists, it is called the optimal control. For maximum problem, the solution ( * , * ), if exits, can be found by Pontryagin's maximum principle. According to the constraint in the Lagrangian problem, the time-varying Lagrange multiplier ( ) was introduced. The function ( ) is usually called an adjoint variable of the system. The comparable function, in this case, is the Hamiltonian function , defined for all ∈ [0, ] by ( ( ), ( ), ( )) = ( ( ), ( )) + ∑ =1 ( ) ( ( ), ( )). (2.5) The Pontryagin maximum principle is precisely formulated as follows . Theorem 1. Let * ( ) be a piecewise control defined on [0, ] and * ( ) be the associated trajectory. Then there exists a nonzero adjoint vector function * ( ) that is a solution to the adjoint system Thus, the necessary conditions for optimizing the Hamiltonian are: (2.8) Please refer to Pontryagin's book (Kaufman, 1964) and some extensions book, such as (Becker et al., 1989;Seierstad & Sydsaeter, 1977) for more details.

C. RESULT AND DISCUSSION 1. Optimal Control Problem
This section reformulates and analyzes an optimal control problem for the model (1.1) to determine the optimal trajectories of uninfected macrophages, infected macrophages, and MTb bacteria in response to the optimal strategy. The control is chosen basis on the significant parameter used as the bifurcation parameter (Adi & Thobirin, 2020). We introduce a control function ( ), which represents the effort of tuberculosis prevention, such as vaccination. The control model is given as follows where ( ) represents a control strategy that cures a fraction of uninfected macrophages and reduces the rate at which macrophages leaves uninfected class towards the infected class. The control is bounded between 0 and . From a medical point of view, it is realistic to assume that < 1, since the vaccination is not completely effective. Let us define the set of admissible control as Then, optimal control theory is applied to determine the optimal treatment administration that will maximize the effort on tuberculosis prevention measures and the cost associated with this support. We define the set of state variables ( ) = ( ( ), ( ), ( )) and the objective functional as [ ] = ∫ 0 ( ( ) − ( ) − 2 ( )) .
( 3.3) which consider the fraction of the uninfected macrophages ( ) and the infected macrophages ( ) and the cost associated with the support of transmission measure ( ).
This completes the proof.

Numerical Simulations
This section gives some numerical simulations by using the ode45 solver in MATLAB to demonstrate the previous section's theoretical results. We compute numerically the Theorem 2 by implementing a forward-backward fourth-order Runge-Kutta method, as described in (Campos et al., 2020). The iterative method is starting with a guess on a control variable over the time interval [0, ] using a forward scheme. Then, using the transversality conditions ( ) = 0, the co-state equation (3.7) are solved by a backward scheme. Furthermore, we update the control by using the state's new values and the value from (3.8). The iterative processes are stopped if the values reach convergence. For the simulation, we consider a set of parameter values obtained from the literature (Adi & Thobirin, 2020;Zhang et al., 2020), with the unit volume in milliliters and time in days as follows Λ = 3300, = 0.01, = 0.01, = 0.01, = 0.1, = 10, = 100, = 0.125 × 10 −8 , = 0.05, (4.1) with variation of parameter . The initial value for the uninfected macrophage, infected macrophage, and MTb bacteria are taken as (0) = 300000, (0) = 20, and (0) = 500, respectively. Now, we consider the case of low transmission of infection rate at which the uninfected macrophages became infected and choose parameter = 1.5 × 10 −8 . Figure 1 shows that the infected macrophages and the MTb bacteria population are reduced, and in this case, almost no macrophages will be infected by the MTb bacteria, and the MTb bacteria become extinct. This means, in cases of very low transmission, vaccination or control is not needed, and the Mtb bacteria and infected macrophages will disappear from the bodyFigure 1 shows that infected macrophages and MTb bacteria disappear from the body in about a year (after about the 300th day). This result is very different from the model without control in (Adi & Thobirin, 2020), where the value of the parameter = 2.28 × 10 −8 which is smaller than 1.5 × 10 −8 will results in the stable disease equilibrium point.  Now, we vary with a much higher transmission rate, which is about ten times as much, and keep all other parameter values as in (4.1). Since vaccination is not completely effective, in this simulation, we set = 0.85. Figure 2-3 show comparison trajectories for uninfected macrophages, infected macrophages, and MTb bacteria with and without control. Meanwhile, Figure 4 shows control variate over time. With the control, the number of uninfected macrophage populations is higher than without control. The increase in the number of uninfected macrophages is proportional to the decrease in the number of infected macrophages. As shown in Figure 3, with the initial condition of infected macrophages 20, without control, this number will increase to its maximum level in less than two years. Likewise, MTb bacteria's population continued to grow until it was at a constant level for less than two years. Meanwhile, if given control, both population infected macrophages and MTb Bacteria will decrease and disappear from the body in about two years. Appropriate control is shown in Figure 4, and a control policy is obtained; in this case, the complete vaccination is given for approximately 900 days (30 months) then decreases within 100 days later.

D. CONCLUSION AND SUGGESTIONS
This paper has studied an optimal control problem for a within-host tuberculosis model describing the interaction between Microbacterium tuberculosis and macrophages. We determine the existence of optimal control analytically and characterize them using Pontryagin's maximum principle. The results suggest that control or vaccination is required if the parameter is large. We found that with = 0.85, the complete vaccination should be given for approximately 900 days (30 months) then decreased within 100 days later. Generally speaking, we suggest that for the tuberculosis disease to be successfully eradicated, it is necessary to optimize the treatment or vaccination. In other words, there is still a need to improve medical methods and technology. From the point of view of mathematical modeling, it is still necessary to develop a more realistic in-host TB model that considers the most relevant treatment methods and uses actual data to help doctors determine the right treatment for TB patients. Our future research will learn more about this.

ACKNOWLEDGEMENT
The author would like to thank the reviewers for their valuables comments. The author would also like to thank Ahmad Dahlan University for supporting this work.