Computational and Mathematical Methods in Medicine

Computational and Mathematical Methods in Medicine / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 398476 | https://doi.org/10.1155/2011/398476

Yves Emvudu, Ramsès Demasse, Dany Djeudeu, "Optimal Control of the Lost to Follow Up in a Tuberculosis Model", Computational and Mathematical Methods in Medicine, vol. 2011, Article ID 398476, 12 pages, 2011. https://doi.org/10.1155/2011/398476

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

Academic Editor: Thierry Busso
Received28 Apr 2011
Revised23 Jun 2011
Accepted22 Jul 2011
Published11 Oct 2011

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 [35]. 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=1𝑝1𝑝3 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 1𝑟2 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 (1𝑟1)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.

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: ̇̇𝑆=Λ𝜇𝑆𝛽(𝐼+𝛿𝐿)𝑆,𝐸=𝛽𝑝2(𝐼+𝛿𝐿)𝑆+𝛾2𝐿+𝑟2𝑘𝐼𝜇+1+𝑘21𝑟1̇𝐸,𝐼=𝛽𝑝1(𝐼+𝛿𝐿)𝑆+𝑘11𝑟1𝐸+𝛾1𝐿𝑟2+𝜇+𝑑1+Φ1𝑟2̇𝐼,𝐿=𝛽𝑝3(𝐼+𝛿𝐿)𝑆+𝑘21𝑟1𝐸+Φ1𝑟2𝐼𝛾1+𝛾2+𝜇+𝑑2𝐿.(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 𝑡𝑓. 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 𝑢𝑖(𝑡); 𝑖=1,2, we obtain the following compartmental model.

Figure 2 leads us to the following differential system:̇̇𝑆=Λ𝜇𝑆𝛽(𝐼+𝛿𝐿)𝑆,𝐸=𝛽𝑝2(𝐼+𝛿𝐿)𝑆+𝛾2𝐿+𝑟2𝐼𝑘𝜇+1+𝑘21𝑢2(𝑡)1𝑟1̇𝐸,𝐼=𝛽𝑝1(𝐼+𝛿𝐿)𝑆+𝑘11𝑟1+𝛾1𝐿𝑟2+𝜇+𝑑1+Φ1𝑢1(𝑡)1𝑟2̇𝐼,𝐿=𝛽𝑝3(𝐼+𝛿𝐿)𝑆+𝑘21𝑢2(𝑡)1𝑟1𝐸+Φ1𝑢1(𝑡)1𝑟2𝛾𝐼1+𝛾2+𝜇+𝑑2𝐿.(2) With initial conditions (𝑆(0);𝐸(0);𝐼(0);𝐿(0))4+.

We seṫ̇̇̇𝐿𝑔𝑆;𝐸;𝐼;=1(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 𝑢𝑖(𝑡); 𝑖=1,2 are assumed to be integrable in the sense of Lebesgue, bounded with (0𝑢𝑖(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:̇̇𝑆=𝜑(𝑆)𝑆𝜂,𝑌,𝑌=𝑆𝜂,𝑌𝐵+𝐴(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)=Λ𝜇𝑆 is a function that depends on 𝑆+, 𝜂=(0,𝛽,𝛽𝛿)𝑇, 𝐵=(𝑝2,𝑝1,1𝑝1𝑝2), , is the usual scalar product in 3, and 𝐴 is a Metzler [8] 3×3 nonconstant matrix defined as𝐴(𝑡)=𝑎11(𝑡)𝑟2𝛾2𝑘11𝑟1𝑎22(𝑡)𝛾1𝑎31(𝑡)𝑎32(𝑡)𝑎33(𝑡)(5) with𝑎11𝑘(𝑡)=𝜇+1+𝑘2𝑘2𝑢2(𝑡)1𝑟1,𝑎22(𝑡)=𝑟2+𝜇+𝑑1+𝜙1𝑢1(𝑡)1𝑟2,𝑎31𝑘(𝑡)=2𝑘2𝑢2(𝑡)1𝑟1,𝑎32(𝑡)=𝜙1𝑢1(𝑡)1𝑟2,𝑎33(𝑡)=𝛾1+𝛾2+𝜇+𝑑2.(6)

Remark 2. The dynamic of the susceptibles is asymptotically stable. In other words, for the system ̇𝑆=𝜑(𝑆),(7) there exists a unique equilibrium 𝑆0=Λ/𝜇 such that 𝜑(𝑆)>0for0<𝑆<𝑆0,𝜑(𝑆)<0for𝑆0<𝑆.(8)

3.1. Positive Invariance of the Nonnegative Orthant

We have the following result.

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

Proof. The system (4) can be written as ̇̇𝑆=𝜑(𝑆)𝑆𝜂,𝑌,𝑌=𝑆𝐵𝜂𝑇+𝐴(𝑡)𝑌.(9) The fist equation of system (9) implies that 𝐾𝑆(𝑡)=𝐾𝑆0𝑒𝐾(𝑡𝑡0)+Λ1𝑒𝐾(𝑡𝑡0)(10) for 𝑡𝑡0.
With 𝐾=𝜇+𝛽(𝐼+𝛿𝐿). For 𝐼0, 𝐿0, and 𝑆00, it comes that 𝑆(𝑡)0 for all 𝑡𝑡0. As a consequence, + is invariant for the system ̇𝑆=𝜑(𝑆)𝑆𝜂,𝑌. For 𝑆0, the matrix (𝑆𝐵𝜂𝑇+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 4+ for the system (4).

3.2. Boundedness of Trajectories

Adding all equations of model (2), one haṡ𝑁(𝑡)=Λ𝜇(𝑆+𝐸+𝐼+𝐿)𝑑1𝐼𝑑2𝐿.(11) Thus, one can deduce thaṫ𝑁(𝑡)Λ𝜇𝑁(𝑡).(12) After integration, using the constant variation formula, we haveΛ𝑁(𝑡)𝜇+𝑒𝜇𝑡𝑁(0).(13) It then follows thatlim𝑡+𝑁(𝑡)𝑆0.(14) It is straightforward to prove that for 𝜖>0 the simplexΩ𝜖=(𝑆,𝐸,𝐼,𝐿)4+Λ;𝑁(𝑡)𝜇+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)=𝛽𝑆0𝑅0,3𝑅(𝑢)0,1(𝑢)+𝛿𝑅0,2(𝑢),(16) where 𝑅0,1(𝑢)=𝑝2𝑘11𝑟1𝛾2+𝜇+𝑑2+𝑝1𝛾2×𝜇+𝑘11𝑟1+𝑝2𝛾1×𝜇+𝑘1+𝑘2𝑘2𝑢2(𝑡)1𝑟1+𝑝1𝛾1+𝜇+𝑑2×𝑘𝜇+1+𝑘2𝑘2𝑢2(𝑡)1𝑟1+𝑝3𝛾2𝑘11𝑟1+𝑝3𝛾1×𝑘𝜇+1+𝑘2𝑘2𝑢2(𝑡)1𝑟1,𝑅0,2(𝑢)=𝑝3𝑟2𝜇+𝜇+𝑑1𝜇+𝑘11𝑟1+Φ1𝑢1(𝑡)1𝑟2×𝑘1+𝑘2𝑘2𝑢2(𝑡)1𝑟1+𝜇Φ1𝑢1(𝑡)1𝑟2𝑝1+𝑝2+𝑘21𝑢2(𝑡)1𝑟1×𝑟2+𝜇+𝑑1𝑝3+𝑝2,𝑅0,3(𝑢)=𝜇+𝑑2𝜇+𝑑1+Φ1𝑢1(𝑡)1𝑟2×𝑘𝜇+1+𝑘2𝑘2𝑢2(𝑡)1𝑟1+𝛾2𝜇𝑟2+𝜇+𝑑1+Φ1𝑢1(𝑡)1𝑟2+𝑟2𝜇+𝑑2𝜇+𝑘21𝑢2(𝑡)1𝑟1+𝛾1𝜇+𝑑1𝑘𝜇+1+𝑘2𝑘2𝑢2(𝑡)1𝑟1+𝛾2𝑘11𝑟1𝜇+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 ̇𝑥=(𝑥)𝒱(𝑥),(18) where 𝑥=(𝐸,𝐼,𝐿,𝑆), 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, (𝑥)=𝛽𝑝2(𝐼+𝛿𝐿)𝑆,𝛽𝑝1(𝐼+𝛿𝐿)𝑆,𝛽𝑝3(𝐼+𝛿𝐿)𝑆,0𝑇,𝑎𝒱(𝑥)=11𝐸𝑟2𝐼𝛾2𝐿𝑎22𝐼𝑘11𝑟1𝐸𝛾1𝐿𝑎33𝐿𝜙1𝑟21𝑢1𝐼𝑎31𝐸0.(19) The Jacobian matrices of and 𝒱 at the disease-free equilibrium DFE can be partitioned as 𝐷(DFE)=𝐹000,𝐷𝒱(DFE)=𝑉000,(20) where 𝐹 and 𝑉 correspond to the derivatives of 𝐷 and 𝐷𝒱 with respect to the infected classes: 𝐹=0𝛽𝑝2𝑆0𝛿𝛽𝑝2𝑆00𝛽𝑝1𝑆0𝛿𝛽𝑝1𝑆00𝛽𝑝3𝑆0𝛿𝛽𝑝3𝑆0,𝑎𝑉=11𝑟2𝛾2𝑘11𝑟1𝑎22𝛾1𝑎31𝑎32𝑎33.(21) The basic reproduction ratio is defined, following Van den Driessche and Watmough [9], as the spectral radius of the next generation matrix, 𝐹𝑉1.

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 𝑅0,1(𝑢)=𝑅0,1(0)𝜔1𝑅(𝑢),0,2(𝑢)=𝑅0,2(0)𝜔2(𝑅𝑢),0,3(𝑢)=𝑅0,3(0)𝜔3(𝑢),(22) where 𝜔1, 𝜔2, and 𝜔3 are nonnegative functions defined by 𝜔1(𝑢)=1𝑟1𝛾1+𝑝1𝜇+𝑑2𝑘2𝑢2,𝜔2(𝑢)=𝜙1𝑟2𝑘2𝑢2+𝑢1𝑘1+𝑘2𝑘2𝑢2×1𝑟1+𝜙𝜇1𝑟2𝑝1+𝑝2𝑢1+𝑘21𝑟1𝑟2+𝜇+𝑑1𝑝3+𝑝2𝑢2,𝜔3(𝑢)=𝜇+𝑑2𝜙1𝑟2×𝑘2𝑢21𝑟1+𝑢1𝑘𝜇+1+𝑘2𝑘2𝑢21𝑟1+𝑘21𝑟1𝑢2𝛾1𝜇+𝑑1+𝑟2𝜇+𝑑2+𝛾2𝜇𝜙1𝑟2𝑢1.(23)

Remark 5. Note that 𝑅0(𝑢)𝑅0𝜔(0)=3(𝑢)𝑅0,3𝑅(𝑢)0𝜔(0)1(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)=𝑆𝜕𝛽0𝑅0,3𝑅(0)0,1(0)+𝛿𝑅0,2(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,𝜑(𝑆)𝑆𝜂,𝑌=0,𝑆𝜂,𝑌𝐵+𝐴(𝑡)𝑌=0.(27) From the second equation of (27), one has 𝑌=𝑆(𝐴1(𝑡))𝜂,𝑌𝐵. And replacing in 𝜂,𝑌 yields𝜂,𝑌=𝑆𝜂,𝐴1𝐵(𝑡)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,𝑌0 implies 𝑆=𝑆0/𝑅0(𝑢). From (28), we have 𝑌=(𝐸,𝐼,𝐿)𝑇=(𝐴1(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 by𝑆𝑆(𝑢)=0𝑅0,𝐸(𝑢)𝑄(𝑢)=1(𝑢)Λ𝑅30(1𝑢)1𝑅0,𝐼(𝑢)𝑄(𝑢)=2(𝑢)Λ𝑅30(1𝑢)1𝑅0,𝐿(𝑢)𝑄(𝑢)=3(𝑢)Λ𝑅30(1𝑢)1𝑅0,(𝑢)(29) where𝑄1(𝑢)=𝑝1𝑟2𝛾1+𝛾2+𝜇+𝑑2+𝑟2+𝜇+𝑑1𝑝2𝛾1+𝑝3𝛾3+𝑝3𝑟2𝛾1+𝛾2𝜙1𝑢1(𝑡)1𝑟2𝑝1+𝑝3+𝑝2𝛾2+𝜇+𝑑2𝑟2+𝜇+𝑑1+𝜙1𝑟2,𝑄2(𝑢)=𝑝2𝑘11𝑟1𝛾2+𝜇+𝑑2+𝑝1𝛾2𝜇+𝑘11𝑟1+𝑝2𝛾1𝑘1+𝑘2𝑘2𝑢2(𝑡)1𝑟1+𝑝3𝛾2𝑘11𝑟1+𝑝1𝛾1+𝜇+𝑑2𝑘𝜇+1+𝑘2𝑘2𝑢21𝑟1+𝑝3𝛾1𝑘𝜇+1+𝑘2𝑘2𝑢2(𝑡)1𝑟1,𝑄3(𝑢)=𝑝3𝑟2𝜇+𝜇+𝑑1𝜇+𝑘11𝑟1+𝜙1𝑢1×𝑘(𝑡)1+𝑘2𝑘2𝑢2(𝑡)1𝑟1+𝜙1𝑢1𝜇𝑝(𝑡)1+𝑝3+𝑘21𝑢2×(𝑡)1𝑟1𝑟2+𝜇+𝑑1𝑝2+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 +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 Ω𝜖 when𝛼4+𝛿𝛼2𝛾1𝑄3𝛼(0)=3+𝛿𝛼1𝑄2(0)𝑝3𝛽𝑆𝑘(0),𝜇+1+𝑘21𝑟1𝑘21𝑟1𝑄1(=𝑘0)11𝑟1𝛼4+𝛿𝛼2𝛼3+𝛿𝛼1+𝑘21𝑟1𝑄3(0)𝑝2𝛽𝛿𝑆(0),(31) with 𝛼1=𝑘𝜇+1+𝑘21𝑟1𝜇+𝑑1+𝜙1𝑟2+𝑟2𝜇+𝑘21𝑟1,𝛼2=𝑘𝜇+1+𝑘21𝑟1𝜙1𝑟2+𝑘21𝑟1𝑟2,𝛼3=𝑘𝜇+1+𝑘21𝑟1𝛾1+𝛾2𝑘11𝑟1,𝛼4=𝑘𝜇+1+𝑘21𝑟1𝛾1+𝜇+𝑑2+𝛾2𝜇+𝑘11𝑟1.(32)

4. Optimal Control

4.1. Definition of the Cost Function

Let 𝐵𝑖, 𝑖=1,2, be the cost associated to the control 𝑢𝑖(𝑡), 𝑖=1,2. (𝐵𝑖 represents the necessary means to realize the control defined by 𝑢𝑖). Our cost function is hence𝐽𝑢1,𝑢2=𝑡𝑓0𝐿(𝑡)+2𝑖=1𝐵𝑖2𝑢2𝑖(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=(𝑢1,𝑢2) satisfying𝐽𝑢1,𝑢2=minΩ𝐽𝑢1,𝑢2,(34) where Ω={(𝑢1,𝑢2)𝐿1(𝑜,𝑡𝑓);𝑎𝑖𝑢𝑖𝑏𝑖,𝑖=1,2} and 𝑎𝑖, 𝑏𝑖 are nonnegative constants such that 𝑎𝑖,𝑏𝑖[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 by1𝐻(𝑢,𝑆,𝐸,𝐼,𝐿)=𝐿(𝑡)+22𝑗=1𝐵𝑗𝑢2𝑗(𝑡)+4𝑖=1𝜆𝑖𝑔𝑖,(35) where the functions (𝑔𝑖, 𝑖=1,2,3,4) are defined by (2).

The necessary conditions for the existence of the solution for problem (34) are𝜕𝜆1𝜕𝑡=𝜕𝐻,𝜕𝑆𝜕𝜆2𝜕𝑡=𝜕𝐻,𝜕𝐸𝜕𝜆3𝜕𝑡=𝜕𝐻,𝜕𝐼𝜕𝜆4𝜕𝑡=𝜕𝐻,𝜕𝐿(36)𝜕𝐻𝜕𝑢𝑖=0(𝑖=1,2).(37)

System (36) leads to the adjoint system:̇𝜆(𝑡)=(0,0,0,1)𝑇+Γ(𝑡)𝜆(𝑡),(38) with 𝜆(𝑡)=(𝜆𝑖(𝑡))𝑖{1,2,3,4}, and Γ(𝑡)=(Γ𝑖𝑗)1𝑖,𝑗4 is a nonconstant 4×4 matrix defined asΓ11Γ=𝜇𝑆+𝛽(𝐼+𝛿𝐿),12=𝛽𝑝2Γ(𝐼+𝛿𝐿),13=𝛽𝑝1Γ(𝐼+𝛿𝐿),14=𝛽𝑝3Γ(𝐼+𝛿𝐿),21Γ=0,22𝑘=𝜇+1+𝑘2𝑘2𝑢2(𝑡)1𝑟1,Γ23=𝑘11𝑟1,Γ24=𝑘21𝑢21𝑟1,Γ31Γ=𝛽𝑆,32=𝛽𝑝2𝑆+𝑟2,Γ33=𝑟2+𝜇+𝑑1+Φ1𝑢1(𝑡)1𝑟2𝛽𝑝1Γ𝑆,34=Φ1𝑢1(𝑡)1𝑟2𝛽𝑝3Γ𝑆,41Γ=𝛽𝛿𝑆,42=𝛽𝑝2𝛿𝑆+𝛾2,Γ43𝛾=1+𝛽𝑝1,Γ𝛿𝑆44=𝛾1+𝛾2+𝜇+𝑑2𝛽𝑝3𝛿𝑆,(39) with transversality conditions 𝜆𝑖𝑡𝑓=0;𝑖{1,2,3,4}.(40)

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

Proposition 10. System (37) leads to 𝑢1𝑎(𝑡)=minmax1;𝜙1𝑟2𝐼𝐵1𝜆4𝜆3;𝑏1,𝑢2𝑎(𝑡)=minmax2;𝑘21𝑟1𝐸𝐵2𝜆4𝜆2;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 𝑢1, 𝑢2, subject to the constraints, the characterizations (41) are derived. To illustrate the characterization of 𝑢1, we have 𝜕𝐻𝜕𝑢1=𝐵1𝑢1+𝜙1𝑟2𝐼𝜆3𝜆4=0,(42) at 𝑢1 on the set {𝑡/𝑎1<𝑢1(𝑡)<𝑏1}. On this set, 𝑢1𝜙(𝑡)=1𝑟2𝐵1𝐼𝜆4𝜆3.(43) Taking into account the bounds on 𝑢1, we obtain the characterization of 𝑢1 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)𝑢𝑐(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 𝑢𝑐1(𝑡)=𝑏1,𝑢𝑐2(𝑡)=𝑏2,𝑡0,𝑡𝑓.(44)

Step 2. Then, with this choice of the control function 𝑢𝑐(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)𝑢𝑐(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=(𝑢1,𝑢2).

Step 4. For one thing we have the chosen control function 𝑢𝑐, for another thing we have the control function 𝑢. We take a convex combination of those functions as follows: 𝑡𝑢(𝑡)=1𝑡𝑓𝑢𝑐𝑡(𝑡)+𝑡𝑓𝑢(𝑡)(45) for 𝑡[0,𝑡𝑓].

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.

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.


ParametersDescriptionEstimated valuesSource

Λ Recruitment rate of susceptible individuals5  ( y r ) 1 Assumed
𝛽 Transmission ratevariable Assumed
𝜇 Natural death rate0.019896  ( y r ) 1 [14]
𝑑 1 TB-induced mortality for the follow up0.02272  ( y r ) 1 [15]
𝑑 2 TB-induced mortality for the lost to follow up0.20  ( y r ) 1 [15]
𝛿 Fraction of lost to follow up that are still infectious1  ( y r ) 1 Assumed
𝜙 Rate at which infectious become lost to follow upVariableAssumed
𝑝 1 Fast route to infectious class0.3  ( y r ) 1 [15]
𝑝 3 Fast route to lost to follow up class0.1  ( y r ) 1 Assumed
𝑟 1 Chemoprophylaxis of latently infected individuals0.001  ( y r ) 1 [15]
𝑟 2 Recovery rate of the infectious0.7311  ( y r ) 1 [15]
𝛾 1 Rate at which the lost to follow up return to the hospital0.2  ( y r ) 1 Assumed
𝛾 2 Recovering rate for the lost to follow up0.001  ( y r ) 1 Assumed
𝑘 1 Rate of progression from infected latently to infectious0.0005  ( y r ) 1 [16]
𝑘 2 Rate of progression from infected latently to lost to follow upVariableAssumed

We solve the state equation (2) with the chosen functions 𝑢𝑖=𝑢𝑐𝑖 (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 𝑢𝑖 (𝑖=1,2) from system (41).

For those simulations, we take 𝑡𝑓=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 𝑡𝑓=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.

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 𝑡𝑓=5 years of lost to follow up is about 89.4644 individuals without control and about 86.7582 with it. In a period of 𝑡𝑓=5 years of control, we succeed in keeping about 3 infectious individuals in a care center.

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 𝑡𝑓=5 years of lost to follow up is about 100.4334 individuals without control and about 87.5361 with it. In a period of 𝑡𝑓=5 years of control, we succeed in keeping about 13 infectious individuals in a care center.

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 𝑡𝑓=5 years of lost to follow up is about 102.3067 individuals without control and about 87.4159 with it. In a period of 𝑡𝑓=5 years of control, we succeed in keeping about 15 infectious individuals in a care center.

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.

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Copyright © 2011 Yves Emvudu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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