Optimal Control of the Lost to Follow Up in a Tuberculosis Model

Abstract

This paper deals with the problem of optimal control for the transmission dynamics of tuberculosis (TB). A TB model that considers the existence of a new class (mainly in the African context) is considered: the lost to follow up individuals. Based on the model formulated and studied in the work of Plaire Tchinda Mouofo, (2009), the TB control is formulated and solved as an optimal control theory problem using the Pontryagin's maximum principle (Pontryagin et al., 1992). This control strategy indicates how the control of the lost to follow up class can considerably influence the basic reproduction ratio so as to reduce the number of lost to follow up. Numerical results show the performance of the optimization strategy.

1. Introduction

Cameroon has a high rate of tuberculosis endemic. It is estimated that in absence of effective epidemiology statistics, there are 100 new Cases for 100 000 habitants per year [1]. Like in many sub-Saharian African countries, the fight against tuberculosis (TB) in Cameroon is difficult due to the interaction with the Human Immunodeficiency Virus (HIV) [2] and particularly with the poor social-economic conditions.

In the literature, there are many TB mathematic models [3–5]. The study of those models has an impact in the control process of the disease. Most of those models are SEIR-models; for those models, one supposes that the population is subdivided in four epidemiological classes: Susceptible individuals, latently infected individuals (those who are infected but not yet infectious), infectious, and the recovered or cured individuals. The particularity of those type of models is that, the rate at which susceptible individuals become latently infected or infectious is a function of infectious individuals number in a population at that time.

In this paper, we study a TB model adapted in the African context in general, particularly for Cameroon. In this model, we take in account the susceptible low and fast progression to latently infected and infectious classes, respectively. We also take into account infectious individuals on chemoprophylaxis, and we introduce a constant rate to become a cured individual.

We note that, the statistic studies [6] prove that many infectious patients do not take their treatment until the end due to a brief relief or a long time for complete treatment. Otherwise, some of those individuals can transmit the disease without presenting any symptom. In this work, we call them lost to follow up individuals (those who have active TB but are not in a care center) [7]. In Cameroon, for example, for a national program of fight against TB, there is about 10% of infectious individuals who do not end their treatment and become lost to follow up individuals. The class of lost to follow up individuals has already been considered by some authors [5, 7]. Previous [5, 7] works clearly show that the progression toward the lost to follow up class has a negative effect on the host population. For that, lost to follow up individuals are very dangerous for human health because they are able to transmit the disease very quickly and discreetly. For our knowledge, the previous authors did not address the question of controlling the evolution to the lost to follow up class. In this paper, we address the question to do so. Our control policy based on decreasing the number of people going to the class of lost to follow up individuals. We first formulate a mathematical model taking into account our control functions. Then, we perfect a mathematical analysis of the model where we compute the basic reproduction ratio of the controlled system. We then define a cost function so that we could deduce the optimal control function. A huge part of this work is to compute the solutions numerically and then draw a conclusion about the efficiency of the control.

2. The Model

We present a tuberculosis model that incorporates the essential biological and epidemiological features of the disease such as exogenous reinfection and chemoprophylaxis of latently infected individuals.

We consider a population of 𝑁 people. We assume that latently infected individuals (inactive TB) have a variable (typically long) latency period. At any given time, an individual is in one of the following four states: susceptible, latently infected (i.e., exposed to TB but are not infectious), infectious (i.e., has active TB but is in a care center), and lost to follow up (i.e., has active TB but is not in a care center). We will denote these states by 𝑆, 𝐸, 𝐼, and 𝐿, respectively. Any recruitment is into the susceptible class and occurs at a constant rate Λ. The transmission of tuberculosis occurs following an adequate contact between a susceptible and infectious or lost to follow up. We assume that a fraction 𝛿 of the lost to follow up are still infectious and can transmit the disease to susceptible individuals (some of them could die or recover). On an adequate contact with infectious or lost to follow up, a susceptible individual becomes infected but not yet infectious. This individual remains in the latently infected class for some latent period. We use the standard mass balance incidence expressions 𝛽𝑆𝐼 and 𝛽𝛿𝑆𝐿 to indicate successful transmission of TB due to nonlinear contact dynamics in the population by infectious and lost to follow up, respectively. The fractions 𝑝₁ and 𝑝₃ of the newly infected individuals are assumed to undergo fast progression directly to the infectious and lost to follow up classes, respectively. The remainders 𝑝₂=1−𝑝₁−𝑝₃ are latently infected and enter the latent class. After receiving an effective therapy, individuals leave the infectious class 𝐼 to the latently infected class 𝐸 at a rate 𝑟₂. We assume that chemoprophylaxis of latently infected individuals reduces their reactivation at a constant rate 𝑟₂. We also assume that individuals leave the lost to follow up class 𝐿 to the latently infected class 𝐸 with a constant rate 𝛾₂. This can be due to the response of the immune system or traditional treatment (via a traditional practitioner). Another assumption is that among the fraction 1−𝑟₂ of infectious who did not recover, some of them who had begun their treatment would not return to the hospital for the examination of sputum at a constant rate 𝜙 and enter the class of lost to follow up 𝐿. After some times, some of them will continue to suffer from the disease and will return to the hospital at a constant rate 𝛾₁. We assume that the chemoprophylaxis of latently infected individuals 𝐸 reduces their reactivation at rate 𝑟₁. Thus, a fraction (1−𝑟₁)𝐸 of infected individuals who do not receive effective chemoprophylaxis become infectious and lost to follow up with a constant rate 𝐾₁ and 𝑘₂, respectively (low progression of the disease). The constant rate for non-disease-related death is 𝜇, thus 1/𝜇 is the average lifetime. Infectious and lost to follow up have additional death rates due to TB-induced mortality with constant rates 𝑑₁ and 𝑑₂, respectively.

Thus, the corresponding transfer diagram is [7] illustrated in Figure 1.

Figure 1 

Flow diagram of the model without control.

We have 𝑁=𝑆+𝐸+𝐼+𝐿 individuals. And the not listed parameter in the previous paragraph is as follows.

𝛽: Transmission Rate
The above scheme leads to the following differential system: 𝐸̇=𝛽𝑝₂𝑆̇=Λ−𝜇𝑆−𝛽(𝐼+𝛿𝐿)𝑆,(𝐼𝐼̇+=𝛿𝐿𝛽𝑝)₁𝑆+𝛾₂𝐿+𝑟₂𝐼−𝜇+𝑘₁+𝑘₂1−𝑟₁𝐸,−𝑟₂(𝐼+𝛿𝐿)𝑆+𝑘₁1−𝑟₁𝐸+𝛾₁𝐿𝐿̇=𝛽𝑝₃+𝜇+𝑑₁+Φ1−𝑟₂𝐼,−𝛾(₁𝐼+𝛿𝐿)𝑆+𝑘₂1−𝑟₁𝐸+Φ1−𝑟₂𝐼+𝛾₂+𝜇+𝑑₂𝐿.(1)

2.1. The Control and Its Policy

The aim of the control is to decrease the total number of the lost sight patients during a period of time 𝑡_𝑓. The strategy of control we adopt consists of introducing two control parameters 𝑢₁(𝑡) and 𝑢₂(𝑡) representing the following.𝑢₁:The effort made to take systematically the infectious patients in a health center in charge.𝑢₂:The effort made to take systematically the latently infected people declared infectious in charge.

Having introduced the functions 𝑢_𝑖(𝑡); 𝑖=1,2, we obtain the following compartmental model.

Figure 2 leads us to the following differential system:𝐸̇=𝛽𝑝₂𝑆̇=Λ−𝜇𝑆−𝛽(𝐼+𝛿𝐿)𝑆,−𝜇(+𝐼+𝑘𝛿₁𝐿)𝑆+𝛾₂𝐿+𝑟₂𝐼𝐼̇=𝛽𝑝₁+𝑘₂1−𝑢₂(𝑡)1−𝑟₁𝐸,−𝑟₂(𝐼+𝛿𝐿)𝑆+𝑘₁1−𝑟₁+𝛾₁𝐿𝐿̇=𝛽𝑝₃+𝜇+𝑑₁+Φ1−𝑢₁(𝑡)1−𝑟₂𝐼,+Φ(1𝐼+−𝛿𝑢₁𝐿)𝑆+𝑘₂1−𝑢₂(𝑡)1−𝑟₁𝐸(𝑡)1−𝑟₂𝐼−𝛾₁+𝛾₂+𝜇+𝑑₂𝐿.(2) With initial conditions (𝑆(0);𝐸(0);𝐼(0);𝐿(0))∈ℝ⁴⁺.

Figure 2 

Flow diagram of the model with control.

We set∶𝑆=̇;𝐸𝑔̇;₁𝐼̇;𝐿̇(𝑆,𝐸,𝐼,𝐿);𝑔₂(𝑆,𝐸,𝐼,𝐿);𝑔₃(𝑆,𝐸,𝐼,𝐿);𝑔₄(𝑆,𝐸,𝐼,𝐿),(3) where the functions 𝑔₁, 𝑔₂, 𝑔₃, and 𝑔₄ are defined by the right-hand side of the system (2).

Remark 1. The functions 𝑢_𝑖(𝑡); 𝑖=1,2 are assumed to be integrable in the sense of Lebesgue, bounded with (0≤𝑢_𝑖(𝑡)≤1). When the functions of control are near to 1, the control is very strict.

3. Mathematical Analysis of the Model with Control

System (2) can be written in the following compact form:𝑌𝑆̇=̇=𝑆𝜑⟨𝜂(𝑆,𝑌)⟩−𝐵𝑆+⟨𝜂𝐴,𝑌(𝑡⟩)𝑌,,(4) where 𝑆 is a state representing the compartment of susceptible individuals and 𝑌=(𝐸,𝐼,𝐿)^𝑇 is the vector representing the state compartment of different infected individuals (latently infected individuals, infectious, lost to follow up individuals). 𝜑(𝑆)=Λ−𝜇𝑆 is a function that depends on 𝑆∈ℝ₊, 𝜂=(0,𝛽,𝛽𝛿)^𝑇, 𝐵=(𝑝₂,𝑝₁,1−𝑝₁−𝑝₂), ⟨,⟩ is the usual scalar product in ℝ³, and 𝐴 is a Metzler [8] 3×3 nonconstant matrix defined as𝐴(𝑡)=−𝑎¹¹(𝑡)𝑟₂𝛾₂𝑘₁1−𝑟₁−𝑎₂₂(𝑡)𝛾₁𝑎₃₁(𝑡)𝑎₃₂(𝑡)−𝑎₃₃(𝑡)(5) with𝑎𝑎₁₁₂₂(𝑡)=𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁,𝑎₃₁(𝑡)=𝑟₂+𝜇+𝑑₁+𝜙1−𝑢₁(𝑡)1−𝑟₂,𝑎₃₂(𝑡)=𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁,𝑎₃₃(𝑡)=𝜙1−𝑢₁(𝑡)1−𝑟₂,(𝑡)=𝛾₁+𝛾₂+𝜇+𝑑₂.(6)

Remark 2. The dynamic of the susceptibles is asymptotically stable. In other words, for the system 𝑆̇=𝜑(𝑆),(7) there exists a unique equilibrium 𝑆₀=Λ/𝜇 such that 𝜑(𝜑𝑆()𝑆>)<00forfo0r<𝑆₀𝑆<𝑆₀,<𝑆.(8)

3.1. Positive Invariance of the Nonnegative Orthant

We have the following result.

Proposition 3. The nonnegative orthant ℝ⁴⁺ is positively invariant for the system (4).

Proof. The system (4) can be written as 𝑌𝑆̇̇==𝜑𝑆(𝐵𝑆𝜂)_𝑇−𝑆⟨𝜂,𝑌⟩,+𝐴(𝑡)𝑌.(9) The fist equation of system (9) implies that 𝐾𝑆(𝑡)=𝐾𝑆₀𝑒^{−𝐾(𝑡−𝑡0)}+Λ1−𝑒^{−𝐾(𝑡−𝑡0)}(10) for 𝑡≥𝑡₀.
With 𝐾=𝜇+𝛽(𝐼+𝛿𝐿). For 𝐼≥0, 𝐿≥0, and 𝑆₀≥0, it comes that 𝑆(𝑡)≥0 for all 𝑡≥𝑡₀. As a consequence, ℝ₊ is invariant for the system 𝑆̇=𝜑(𝑆)−𝑆⟨𝜂,𝑌⟩. For 𝑆≥0, the matrix (𝑆𝐵𝜂^𝑇+𝐴(𝑡)) is a Metzler matrix. Since it is well known that linear Metzler matrices let invariant the nonnegative orthant, this proves the positive invariance of the nonnegative orthant ℝ⁴⁺ for the system (4).

3.2. Boundedness of Trajectories

Adding all equations of model (2), one has𝑁̇(𝑡)=Λ−𝜇(𝑆+𝐸+𝐼+𝐿)−𝑑₁𝐼−𝑑₂𝐿.(11) Thus, one can deduce that𝑁̇(𝑡)≤Λ−𝜇𝑁(𝑡).(12) After integration, using the constant variation formula, we have𝑁(𝑡)≤Λ𝜇+𝑒^−𝜇𝑡𝑁(0).(13) It then follows thatlim_𝑡→+∞𝑁(𝑡)≤𝑆₀.(14) It is straightforward to prove that for 𝜖>0 the simplexΩ_𝜖=(𝑆,𝐸,𝐼,𝐿)∈ℝ⁴⁺;𝑁(𝑡)≤Λ𝜇+𝜖(15) is a compact invariant set for the system (2) and that for 𝜖>0 this set is absorbing. So, we limit our study to this simplex.

3.3. Basic Reproduction Ratio

Basic reproduction ratio is the average number of secondary cases produced by a single infective individual which is introduced into an entirely susceptible population.

We are going to compute the basic reproduction ratio of the system with control, and then deduce the basic reproduction ratio of the system without control.

Proposition 4. The basic reproduction ratio 𝑅₀(𝑢) of system (2), with control 𝑢=(𝑢₁,𝑢₂), is given by 𝑅₀(𝑢)=𝛽𝑆⁰𝑅_0,3(𝑢)𝑅^0,1(𝑢)+𝛿𝑅_0,2(𝑢),(16) where 𝑅_0,1(𝑢)=𝑝₂𝑘₁1−𝑟₁𝛾₂+𝜇+𝑑₂+𝑝₁𝛾₂×𝜇+𝑘₁1−𝑟₁+𝑝₂𝛾₁+×𝑝₁𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁×𝜇+𝛾₁+𝑘₁𝜇+𝑑₂+𝑝₃+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁𝛾₂𝑘₁1−𝑟₁+𝑝₃𝛾₁𝑅×𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁_0,2,(𝑢)=+𝑝₃Φ𝑟₂1𝜇−+𝑢₁𝜇+𝑑₁𝜇+𝑘₁1−𝑟₁×𝑘₁(𝑡)1−𝑟₂+𝜇Φ+1𝑘−₂𝑢−₁𝑘₂𝑢₂(𝑡)1−𝑟₁+𝑘₂(𝑡)1−𝑟₂𝑝₁+𝑝₂×𝑟₂1−𝑢₂(𝑡)1−𝑟₁𝑅+𝜇+𝑑₁𝑝₃+𝑝₂_0,3,(𝑢)=×𝜇𝜇++𝑑₂𝑘₁𝜇+𝑑₁+Φ1−𝑢₁(𝑡)1−𝑟₂+𝛾₂+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁+𝑟₂𝜇𝑟₂+𝜇+𝑑₁+Φ1−𝑢₁(𝑡)1−𝑟₂+𝛾₁𝜇+𝑑₂𝜇+𝑘₂1−𝑢₂(𝑡)1−𝑟₁+𝛾₂𝜇+𝑑₁𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁𝑘₁1−𝑟₁𝜇+𝑑₁+𝛾₁𝑟₂𝜇.(17)

Proof. The system (2) has an evident equilibrium (𝑆₀,0,0,0), where there is no disease. This equilibrium is the disease-free equilibrium (DFE). We calculate the basic reproduction ratio, 𝑅₀(𝑢), using the Van Den Driesseche and Watmough next generation approach [9] and the techniques reported in [10, 11]. In order to compute the basic reproduction ratio, it is important to distinguish new infections from all other class transitions in the population. The infected classes are 𝐼, 𝐸, and 𝐿. We can write system (2) as 𝑥̇=ℱ(𝑥)−𝒱(𝑥),(18) where 𝑥=(𝐸,𝐼,𝐿,𝑆), ℱ is the rate of new infections in each class, 𝒱⁺ is the rate of transfer into each class by all other means, and 𝒱⁻(𝑥) is the rate transfer out of each class. Hence, ℱ(𝑥)=𝒱(𝛽𝑥𝑝)₂=(𝐼⎛⎜⎜⎜⎝+𝛿𝐿)𝑆,𝛽𝑎𝑝₁₁₁(𝐼+𝛿𝐿)𝑆,𝛽𝑝₃(𝐼+𝛿𝐿)𝑆,0^𝑇,𝑎₂₂𝐸−𝑟₂𝐼−𝛾₂𝐿𝑎₃₃𝐼−𝑘₁1−𝑟₁𝐸−𝛾₁𝐿𝐿−𝜙1−𝑟₂01−𝑢₁𝐼−𝑎₃₁𝐸⎞⎟⎟⎟⎠.(19) The Jacobian matrices of ℱ and 𝒱 at the disease-free equilibrium DFE can be partitioned as 𝐷ℱ(DFE)=𝐹000,𝐷𝒱(DFE)=𝑉000,(20) where 𝐹 and 𝑉 correspond to the derivatives of 𝐷ℱ and 𝐷𝒱 with respect to the infected classes: 𝐹=0𝛽𝑝²𝑆₀𝛿𝛽𝑝₂𝑆₀0𝛽𝑝₁𝑆₀𝛿𝛽𝑝₁𝑆₀𝑉=0𝑎𝛽₁₁𝑝₃𝑆₀𝛿𝛽𝑝₃𝑆₀,−𝑟₂−𝛾₂−𝑘₁1−𝑟₁𝑎₂₂−𝛾₁−𝑎₃₁−𝑎₃₂𝑎₃₃.(21) The basic reproduction ratio is defined, following Van den Driessche and Watmough [9], as the spectral radius of the next generation matrix, 𝐹𝑉⁻¹.

From 𝑅₀(𝑢), we deduce 𝑅₀(0) (basic reproduction ratio of the system without control) by taking 𝑢≡(0,0). We are going to compare 𝑅₀(𝑢) and 𝑅₀(0).

Note
We have 𝑅𝑅_0,1(𝑢)=𝑅_0,1(0)−𝜔₁(𝑢),_0,2𝑅(𝑢)=𝑅_0,2(0)−𝜔₂(𝑢),_0,3(𝑢)=𝑅_0,3(0)−𝜔₃(𝑢),(22) where 𝜔₁, 𝜔₂, and 𝜔₃ are nonnegative functions defined by 𝜔𝜔₁₂(𝑢)=1−𝑟₁𝛾₁+𝑝₁𝜇+𝑑₂𝑘₂𝑢₂,(𝑢)=×𝜙11−−𝑟𝑟₂₁𝑘₂𝑢₂+𝑢₁𝑘₁+𝑘₂−𝑘₂𝑢₂+𝜙𝜇1−𝑟₂𝑝₁+𝑝₂𝑢₁𝜔₃+𝑘₂1−𝑟₁𝑟₂+𝜇+𝑑₁𝑝₃+𝑝₂𝑢₂,(𝑢)=×𝜇𝑘+₂𝑑₂𝜙1−𝑟₂+𝑘₂𝑢₂1−𝑟₁+𝑢₁𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂1−𝑟₁+𝛾₂1−𝑟₁𝑢₂𝛾₁𝜇+𝑑₁+𝑟₂𝜇+𝑑₂𝜇𝜙1−𝑟₂𝑢₁.(23)

Remark 5. Note that 𝑅₀(𝑢)−𝑅₀(0)=𝜔³(𝑢)𝑅_0,3(𝑢)𝑅⁰(0)−𝜔¹(𝑢)+𝛿𝜔₂(𝑢)𝜔₃(𝑢).(24) We can remark that in some conditions, depending only on system parameters, we can have 𝑅₀(𝑢)≤𝑅₀(0).(25)

Remark 6. Let us examine sensitivity of the basic reproduction ratio without control 𝑅₀(0) with respect to 𝛽. It is easy to prove that 𝜕𝑅₀(0)𝜕𝛽=𝑅𝑆⁰_0,3(0)𝑅^0,1(0)+𝛿𝑅_0,2(0)>0.(26) Thus, 𝑅₀(0) increases with 𝛽.

3.4. Equilibria

The equilibrium (𝑆,𝑌) on system (2) can be obtained by setting the right-hand side of all the equations in model (4) equal to zero, that is,𝑆𝜑⟨𝜂(𝑆,𝑌)⟩−𝐵𝑆+⟨𝜂𝐴,𝑌(𝑡⟩)𝑌==0,0.(27) From the second equation of (27), one has 𝑌=𝑆(−𝐴⁻¹(𝑡))⟨𝜂,𝑌⟩𝐵. And replacing in ⟨𝜂,𝑌⟩ yields⟨𝜂,𝑌⟩=𝑆𝜂,−𝐴⁻¹(𝑡)𝐵⟨𝜂,𝑌⟩.(28) The case ⟨𝜂,𝑌⟩=0 implies 𝜑(𝑆)=0 and 𝐴(𝑡)𝑌=0. Since 𝐴 is nonsingular, this gives the disease-free equilibrium 𝑃⁰=(𝑆₀,0,0,0).

The case ⟨𝜂,𝑌⟩≠0 implies 𝑆^∗=𝑆₀/𝑅₀(𝑢). From (28), we have 𝑌^∗=(𝐸^∗,𝐼^∗,𝐿^∗)^𝑇=(−𝐴⁻¹(𝑡))𝐵𝜑(𝑆^∗).

After calculations, we obtained that, with 𝑅₀(𝑢)>1, the model (4) has a unique endemic equilibrium 𝑃^∗(𝑢)=(𝑆^∗(𝑢),𝐸^∗(𝑢),𝐼^∗(𝑢),𝐿^∗(𝑢)) given by𝑆^∗(𝑢)=𝑆⁰𝐸𝑅₀(𝑢)_∗,(𝑢)=𝑄¹(𝑢)Λ𝐼𝑅³⁰(𝑢)1−𝑅₀1(𝑢)_∗,(𝑢)=𝑄²(𝑢)Λ𝐿𝑅³⁰(𝑢)1−𝑅₀1(𝑢)_∗,(𝑢)=𝑄³(𝑢)Λ𝑅³⁰(𝑢)1−𝑅₀1(𝑢),(29) where𝑄₁(𝑢)=+𝑝₁𝑝𝑟₂₃𝛾₁+𝛾₂+𝜇+𝑑₂+𝑟₂+𝜇+𝑑₁𝑝₂𝛾₁+𝑝₃𝛾₃+𝑝₂𝑟₂𝛾₁+𝛾₂𝜙1−𝑢₁(𝑡)1−𝑟₂𝑝₁+𝑝₃𝑄𝛾₂+𝜇+𝑑₂𝑟₂+𝜇+𝑑₁+𝜙1−𝑟₂₂,(𝑢)=+𝑝₂𝑝𝑘₁₂1−𝑟₁𝛾₂+𝜇+𝑑₂+𝑝₁𝛾₂𝜇+𝑘₁1−𝑟₁+𝑝₁𝛾₁𝑘₁+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁+𝑝₃𝛾₂𝑘₁1−𝑟₁+𝑝₃𝛾₁+𝜇+𝑑₂𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂1−𝑟₁𝑄𝛾₁𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁₃,(𝑢)=×𝑝₃𝑟𝑘₂₁𝜇+𝜇+𝑑₁𝜇+𝑘₁1−𝑟₁+𝜙1−𝑢₁(𝑡)+𝜙1−+𝑢𝑘₂₁−𝑘₂𝑢₂(𝑡)1−𝑟₁×1−𝑟₁(𝑡)𝜇𝑝₁+𝑝₃+𝑘₂1−𝑢₂(𝑡)𝑟₂+𝜇+𝑑₁𝑝₂+𝑝₃.(30)

Lemma 7. When 𝑅₀(𝑢)>1, model (2) has a unique endemic equilibrium defined as in (29).

Remark 8. It is showed in [7] that(i)if 𝑅₀(0)≤1, the disease-free equilibrium 𝑃⁰ is globally asymptotically stable on the nonnegative orthant ℝ⁺⁴. This means that, the disease naturally dies out in the host population; (ii)If 𝑅₀(0)>1, then the positive endemic equilibrium state 𝑃^∗(0) of model (2) is globally asymptotically stable on the set Ω_𝜖 when𝜇+𝛼₄𝑘+₁𝛿𝛼₂𝛾₁𝑄₃(0)=𝛼₃+𝛿𝛼₁𝑄₂(0)𝑝₃𝛽𝑆^∗(0),=𝑘+𝑘₂1−𝑟₁𝑘₂1−𝑟₁𝑄₁(0)₁1−𝑟₁𝛼⁴+𝛿𝛼₂𝛼₃+𝛿𝛼₁+𝑘₂1−𝑟₁𝑄₃(0)𝑝₂𝛽𝛿𝑆^∗(0),(31) with 𝛼₁=+𝜇𝑟₂+𝑘₁+𝑘₂1−𝑟₁𝜇+𝑑₁+𝜙1−𝑟₂𝛼𝜇+𝑘₂1−𝑟₁₂,𝛼₃=𝜇+𝑘₁+𝑘₂1−𝑟₁𝜙1−𝑟₂+𝑘₂1−𝑟₁𝑟₂,𝛼₄=𝜇+𝑘₁+𝑘₂1−𝑟₁𝛾₁+𝛾₂𝑘₁1−𝑟₁,=+𝜇𝛾₂+𝑘₁+𝑘₂1−𝑟₁𝛾₁+𝜇+𝑑₂𝜇+𝑘₁1−𝑟₁.(32)

4. Optimal Control

4.1. Definition of the Cost Function

Let 𝐵_𝑖, 𝑖=1,2, be the cost associated to the control 𝑢_𝑖(𝑡), 𝑖=1,2. (𝐵_𝑖 represents the necessary means to realize the control defined by 𝑢_𝑖). Our cost function is hence𝐽𝑢₁,𝑢₂=^𝑡𝑓₀𝐿(𝑡)+²_𝑖=1𝐵_𝑖2𝑢^2𝑖(𝑡)𝑑𝑡.(33) The cost function is defined having in mind that, we are going to penalize the number of lost sight person. This justifies the presence of the term 𝐿.

The problem now is to find 𝑢^∗=(𝑢^∗1,𝑢^∗2) satisfying𝐽𝑢^∗1,𝑢^∗2=min_Ω𝐽𝑢₁,𝑢₂,(34) where Ω={(𝑢₁,𝑢₂)∈𝐿¹(𝑜,𝑡_𝑓);𝑎_𝑖≤𝑢_𝑖≤𝑏_𝑖,𝑖=1,2} and 𝑎_𝑖, 𝑏_𝑖 are nonnegative constants such that 𝑎_𝑖,𝑏_𝑖∈[0,1].

4.2. Resolution of the Optimal Problem

Using the Pontryagin's maximum principle [12], problems (2)–(34) are reduced to minimize the function 𝐻 defined by𝐻(𝑢,𝑆,𝐸,𝐼,𝐿)=𝐿(𝑡)+12²_𝑗=1𝐵_𝑗𝑢^2𝑗(𝑡)+⁴_𝑖=1𝜆_𝑖𝑔_𝑖,(35) where the functions (𝑔_𝑖, 𝑖=1,2,3,4) are defined by (2).

The necessary conditions for the existence of the solution for problem (34) are𝜕𝜆₁𝜕𝜕𝜆𝑡₂=−𝜕𝜕𝐻𝑆,𝜕𝜕𝜆𝑡₃=−𝜕𝜕𝐻𝐸,𝜕𝜕𝜆𝑡₄=−𝜕𝜕𝐻𝐼,𝜕𝐻𝜕𝑡=−𝜕𝜕𝐻𝐿,(36)𝜕𝑢_𝑖=0(𝑖=1,2).(37)

System (36) leads to the adjoint system:𝜆̇(𝑡)=(0,0,0,−1)^𝑇+Γ(𝑡)𝜆(𝑡),(38) with 𝜆(𝑡)=(𝜆_𝑖(𝑡))_{𝑖∈{1,2,3,4}}, and Γ(𝑡)=(Γ_𝑖𝑗)_{1≤𝑖,𝑗≤4} is a nonconstant 4×4 matrix defined asΓΓ₁₁=𝜇𝑆+𝛽(𝐼+𝛿𝐿),₁₂Γ=−𝛽𝑝₂(𝐼+𝛿𝐿),₁₃Γ=−𝛽𝑝₁(𝐼+𝛿𝐿),₁₄Γ=−𝛽𝑝₃(𝐼+𝛿𝐿),₂₁Γ=0,₂₂Γ₂₃=𝜇+𝑘₁+𝑘₂−𝑘₂𝑢₂(𝑡)1−𝑟₁,Γ₂₄=−𝑘₁1−𝑟₁,Γ₃₁=−𝑘₂1−𝑢₂1−𝑟₁,Γ=𝛽𝑆,₃₂Γ₃₃=−𝛽𝑝₂𝑆+𝑟₂,Γ=𝑟₂+𝜇+𝑑₁+Φ1−𝑢₁(𝑡)1−𝑟₂−𝛽𝑝₁𝑆,₃₄Γ=−Φ1−𝑢₁(𝑡)1−𝑟₂−𝛽𝑝₃𝑆,₄₁Γ=𝛽𝛿𝑆,₄₂Γ₄₃=−𝛽𝑝₂𝛿𝑆+𝛾₂,Γ=−𝛾₁+𝛽𝑝₁𝛿𝑆₄₄,=𝛾₁+𝛾₂+𝜇+𝑑₂−𝛽𝑝₃𝛿𝑆,(39) with transversality conditions 𝜆_𝑖𝑡_𝑓=0;𝑖∈{1,2,3,4}.(40)

Remark 9. The transversality conditions are due to the fact that after the period of control (𝑡_𝑓), there is no more information given by the adjoint system.

Proposition 10. System (37) leads to 𝑢^∗1(𝑡)=minmax𝑎₁;𝜙1−𝑟²𝐼𝐵₁𝜆₄−𝜆₃;𝑏₁𝑢_∗2,(𝑡)=minmax𝑎₂;𝑘²1−𝑟₁𝐸𝐵₂𝜆₄−𝜆₂;𝑏₂.(41)

Proof. The existence of an optimal control pair is due to the convexity of integrand of 𝐽 with respect to (𝑢₁,𝑢₂), a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables [12]. By considering the optimality conditions (37), and solving for 𝑢^∗1, 𝑢^∗2, subject to the constraints, the characterizations (41) are derived. To illustrate the characterization of 𝑢^∗1, we have 𝜕𝐻𝜕𝑢₁=𝐵₁𝑢₁+𝜙1−𝑟₂𝐼𝜆₃−𝜆₄=0,(42) at 𝑢^∗1 on the set {𝑡/𝑎₁<𝑢^∗1(𝑡)<𝑏₁}. On this set, 𝑢^∗1(𝑡)=𝜙1−𝑟²𝐵₁𝐼𝜆₄−𝜆₃.(43) Taking into account the bounds on 𝑢^∗1, we obtain the characterization of 𝑢^∗1 in (41).

4.3. Determination of the Control Function

In this section, we are going to show step by step, how to determine the optimal functions numerically.

Remark 11. The main difficulty here for the optimal control is that we have initial conditions for system (2) and final conditions for the adjoint system (transversality conditions).

To overcome this difficulty, we proceed as follows.

Step 1. We choose a control function 𝑢(𝑡)≡𝑢^𝑐(𝑡) in the set Ω. However, this choice is not a random process; it depends on the strategy we need to adopt. For example, in this paper, we adopt a strategy which is very strict at the beginning of the control. We choose 𝑢^𝑐1(𝑡)=𝑏₁,𝑢^𝑐2(𝑡)=𝑏₂,∀𝑡∈0,𝑡_𝑓.(44)

Step 2. Then, with this choice of the control function 𝑢^𝑐(𝑡), one determines the solution (𝑆(𝑡), 𝐸(𝑡), 𝐼(𝑡), 𝐿(𝑡)) of the Cauchy problem associated to system (2).

Step 3. The knowledge of 𝑢(𝑡)≡𝑢^𝑐(𝑡) and (𝑆(𝑡), 𝐸(𝑡), 𝐼(𝑡), 𝐿(𝑡)) allows us to determine the solution 𝜆(𝑡) of the Cauchy problem associated to the adjoint system with transversality conditions. This takes us to the control functions defined in (41) by 𝑢^∗=(𝑢^∗1,𝑢^∗2).

Step 4. For one thing we have the chosen control function 𝑢^𝑐, for another thing we have the control function 𝑢^∗. We take a convex combination of those functions as follows: 𝑢(𝑡)=1−𝑡𝑡_𝑓𝑢^𝑐(𝑡)+𝑡𝑡_𝑓𝑢^∗(𝑡)(45) for 𝑡∈[0,𝑡_𝑓].

Step 5. This process is repeated (Steps 2, 3, and 4), and iterations are stopped when the values at the unknown iteration are very closed to the ones at the present iteration.

5. Numerical Simulations

We are going to provide numerical simulations to illustrate our studies.

We assumed that 𝛽 is variable because it strongly influences the basic reproduction ratio (Remark 6). This is illustrated by Figure 3.

Figure 3 

Variation of the basic reproduction ratio without control as a function of 𝛽, with 𝜙=0.0022 and 𝑘2=0.0006.

We also assume that the parameters 𝜙 and 𝑘₂, which denote the rate of progression from infectious to lost to follow up and the rate of progression from latently infected to lost follow up, respectively, are variable just to highlight the fact that the optimal control depends on that parameters.

For numerical simulations the values of the above parameters are 𝛽∈{0.002;0.003;0.02}, 𝜙∈{0.0022;0.1;0.5}, and 𝑘₂∈{0.0006;0.006}. The values of the other parameters are given in Table 1.

Table 1 

Table of parameter values [14–16].

We solve the state equation (2) with the chosen functions 𝑢_𝑖=𝑢^𝑐𝑖 (𝑖=1,2) using the Runge-Kutta forward scheme of order 4. Then, we solve the adjoint system using the backward Runge-Kutta scheme of order 4.

We deduce 𝑢^∗𝑖 (𝑖=1,2) from system (41).

For those simulations, we take 𝑡_𝑓=5 years as control period. We also assume that the total population number is 𝑁=500 individuals subdivided as follows: 𝑆(0)=50, 𝐸(0)=100, 𝐼(0)=150, and 𝐿(0)=200.

In Figure 4, 𝛽=0.002 is chosen to assure that the reproduction ratio 𝑅₀ without control is less than 1. The values of 𝜙 and 𝑘₂ are chosen here small enough to show that the control would not really be necessary (Figure 4(a)). Figure 4(b): the average basic reproduction ratio is about 0.4020 without control and about 0.3974 with it. This is due to the fact that our control is not rigorous enough. Figure 4(c): the average number during 𝑡_𝑓=5 years of lost to follow up is about 86.4411 individuals without control. This value is approximately the same with control 86.3869. This is because the rate at which infectious becomes lost to follow up 𝜙=0.0022 and the rate at which latently infected becomes lost to follow up 𝐾₂=0.0006 are very small.

(a) Control functions

(b) Basic reproduction ratio

(c) Evolution of the lost to follow up

Figure 4 

The influence of the control with 𝑆(0)=50, 𝐸(0)=100, 𝐼(0)=150, 𝐿(0)=200, 𝛽=0.002, 𝜙=0.0022, and 𝑘2=0.0006. All the other parameter values are as in Table 1.

In Figure 5, 𝛽=0.003 is chosen to assure that the reproduction ratio 𝑅₀ without control is less than 1. The value of 𝑘₂ is chosen here small enough to show that the associated control function 𝑢₂ would not really be necessary. Unlike the value of 𝜙 which the associated control function 𝑢₁ is strict (Figure 5(a)). Figure 5(b): the average basic reproduction ratio is about 0.6482 without control and about 0.6033 with it. Figure 5(c): the average number during 𝑡_𝑓=5 years of lost to follow up is about 89.4644 individuals without control and about 86.7582 with it. In a period of 𝑡_𝑓=5 years of control, we succeed in keeping about 3 infectious individuals in a care center.

(a) Control functions

(b) Basic reproduction ratio

(c) Evolution of the lost to follow up

Figure 5 

The influence of the control with 𝑆(0)=50, 𝐸(0)=100, 𝐼(0)=150, 𝐿(0)=200, 𝛽=0.003, 𝜙=0.1, and 𝑘2=0.0006. All the other parameter values are as in Table 1.

In Figure 6, 𝛽=0.02 is chosen to assure that the reproduction ratio 𝑅₀ without control is greater than 1. The value of 𝑘₂ is chosen here small enough to show that the associated control function 𝑢₂ would not really be necessary. Unlike the value of 𝜙 which the associated control function 𝑢₁ is very strict during the whole control period (Figure 6(a)). Figure 6(b): the average basic reproduction ratio is about 5.4460 without control and about 4.0424 with it. Figure 6(c): the average number during 𝑡_𝑓=5 years of lost to follow up is about 100.4334 individuals without control and about 87.5361 with it. In a period of 𝑡_𝑓=5 years of control, we succeed in keeping about 13 infectious individuals in a care center.

(a) Control functions

(b) Basic reproduction ratio

(c) Evolution of the lost to follow up

Figure 6 

The influence of the control with 𝑆(0)=50, 𝐸(0)=100, 𝐼(0)=150, 𝐿(0)=200, 𝛽=0.02, 𝜙=0.5, and 𝑘2=0.0006. All the other parameter values are as in Table 1.

In Figure 7, 𝛽=0.02 is chosen to assure that the reproduction ratio 𝑅₀ without control is greater than 1. The values of 𝑘₂ and 𝜙 are chosen in order to make both control functions 𝑢₁ and 𝑢₂ strict (Figure 7(a)). Figure 7(b): the average basic reproduction ratio is about 8.4875 without control and about 3.9675 with it. The basic reproduction ratio without control is about twice as large as the one with control. Figure 7(c): the average number during 𝑡_𝑓=5 years of lost to follow up is about 102.3067 individuals without control and about 87.4159 with it. In a period of 𝑡_𝑓=5 years of control, we succeed in keeping about 15 infectious individuals in a care center.

(a) Control functions

(b) Basic reproduction ratio

(c) Evolution of the lost to follow up

Figure 7 

The influence of the control with 𝑆(0)=50, 𝐸(0)=100, 𝐼(0)=150, 𝐿(0)=200, 𝛽=0.02, 𝜙=0.5, and 𝑘2=0.006. All the other parameter values are as in Table 1.

6. Summary and Discussion

This has considered the problem of optimal control of the transmission dynamics of TB. A model considering a new class has been investigated and analyzed. An optimal control strategy has been presented, and the results show how important it is to control the lost to follow up class, which is very crucial to the study of the disease. Numerical simulations have been given to illustrate the effectiveness and efficiency of the proposed control scheme. In Africa, it is very important to keep infectious individuals in a care center in order to complete their treatment and avoid the quick transmission of the disease. Our control strategy helps to do so, though other control strategies could be investigated.

For discussion, it should be noted that the model investigated here is based on some restrictive assumptions as an epidemic model. We have assumed that(1)any recruitment, is into the susceptible class and occur at a constant rate Λ, (2)we have not taken into account the class of recovered individuals.

The first assumption is met for the dynamical study of a host population evolving in a restrictive domain. To overpass this assumption, we could introduce the diffusion phenomenon in the model.

The second is due to the fact that the complete recovering from TB is just apparent in general [13]. In other words, some infectious individuals apparently recover but actually harbor TB bacteria, which are in an inactive state. Thus, those TB bacteria are undetectable by the antibodies or other molecules aiming to fight the disease.

Acknowledgment

The authors would like to thank the reviewers and the handling editor very much for their valuable comments and suggestions.