www.biomathforum.org/biomath/index.php/biomath ORIGINAL ARTICLE Optimal control of the treatment frequency in a stochastic model of Tuberculosis Bongor Danhree∗, Emvudu Yves∗ and Koı̈na Rodoumta ‡ ∗Department of Mathematics, Faculty of Science, University of Yaounde 1, Cameroon sbongordanhree@yahoo.com, yemvudu@yahoo.fr ‡Department of Mathematics, Faculty of Exact and Applied Sciences University of Djamena, Chad koinarodoumta@yahoo.fr Received: 10 March 2016, accepted: 7 May 2017, published: 6 June 2017 Abstract—This paper presents a stochastic model of the Tuberculosis(TB) infection with treatment in a population composed of four individuals compart- ments: susceptible individuals, latent infected indi- viduals, active infected individuals and recovered individuals after the therapy. A preliminary survey of the model is performed on the stability before approaching the crucial left of the topic. The aim in this paper is to control the treatment frequency in a stochastic model of the TB infection while minimiz- ing the cost of the measures. Then, we formulate an optimal control problem that consists in minimizing the relative cost of the dynamics of TB-model in order to reduce the prevalence and the mortality due to this infection. The optimal problem is solved by applying the Projection Stochastic Gradient Method in order to find the optimal numerical solution. Finally, we provide some numerical simulations of the controlled model. Keywords-Stochastic Model of TB; local and Global Stability; Optimal Control; Functional Cost; Projection Stochastic Gradient. I. INTRODUCTION The tuberculosis (TB) continues to make a lot of victims in our societies despite of the exist- ing treatment: the Bacillus Calmette- Guerin. The vaccine anti tubercular is used for preventive treat- ment for children. Nevertheless, other medicines exist as Rifampicin, Isoniazid, Pyrazinamide... for the curative treatment of the patients [23]. The expenses are enormous when the treatment is long. The tuberculosis is one of the causes of elevated mortality in humane communities irrespective of the enormous financial resources made by world- wide governments for the treatment of this disease in the purpose of its eradication. So there is the necessity to integrate to a set of the available con- trol an optimal measure that consists on respecting the dose of the treatment to short length in order to reduce this infection. In this paper, we consider a stochastic model of the Tuberculosis (TB) infection in presence of treatment in a population composed of four com- partments of individuals: susceptible individuals, latent infected individual, active infected individ- uals and recovered individuals after the therapy. The mathematical model of TB infection include in addition to the deterministic term, a stochastic term that translates the random noise. The random nature of this model is due to the fact that the Citation: Bongor Danhree, Emvudu Yves, Koı̈na Rodoumta, Optimal control of the treatment frequency in a stochastic model of Tuberculosis, Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis contraction of the Mycobacterium Tuberculosis, the vector agent of the TB infection and his trans- mission within the population are done in an ran- dom manner according to the variable efficiency of control of the immune system of the individuals. The infection of TB contracts itself mainly by the inhalation of the bacteria distributed by the cough or the sneeze of a sick individual. The vector agent of this infection accommodates itself to the level of the lungs of an individual exposition susceptible of contamination and the immune system of this one controls and maintains the infection in the latent state; otherwise there is the risk that this infec- tion develops itself toward the active state. While supposing that only the infected individuals of active TB transmit the infection, they must observe some hygienic rules, they must adopt a positive behavior with respect to the susceptible individuals (who must also take precaution), to follow the treatment up to finish as early as possible (in less than one year), constitute measures of adequate control. A preliminary survey of the model is performed before introducing a function of control representing the necessary dose of medicines in order to control the frequency of the treatment and to reduce considerably and quickly the prevalence of the disease. The main objective is the control of the treatment frequency in the stochastic model of the TB infection. So we formulate an optimal control problem that consists in minimizing the relative cost of the dynamics of the model in order to reduce the prevalence and mortality due to this infection. To solve this optimal control problem, we are going to apply the Stochastic Gradient Method with Projection in order to find the optimal numeric solution. Finally, thanks to the numerical simulation tool, we simulate this model without or with control as well as the optimal solution and the associated cost function in order to characterize an optimal decision. In epidemiology and others domain as biology, demography, economy..., many stochastic models deriving from their deterministic formulation. The reference of the literature for a variety of well- known stochastic models deriving from their de- terministic counterparts include the books [1], [5], [6], [7], and [22]. Our contribution is first in Sub Section II.A, the formulation of a stochastic model of TB with treatment from a deterministic model of TB-only (Sharomi [18]) which is formulated along the lines of the model in Feng and al. [26]. Secondly in Sub Section II.B, we change this stochastic model by perturbations or by an affine change of variables affine to lead the survey of the stability of the random equilibrium because the used transformation keeps the law of probability of an random variable [12]. Finally in Sub Section III.A, we control the treatment frequency in this stochastic model in order to reduce mortality due to the infection. The continuation of the paper is like follows: we recall the results that concern the projection method in Sub Section III.B. The gradient projection method is applied to the model in Sub Section III.C, and the numerical simulations are plotted in Sub Section III.D. II. STOCHASTIC MODEL OF TB WITHOUT CONTROL We start this section by the description of the variables and parameters of the model (see Table I) then follows it by the presentation of the model. A. Diagram and Mathematical Stochastic Model of TB Fig. 1. Diagram of the stochastic model of TB with treatment Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis TABLE I RANDOM VARIABLES AND PARAMETERS DESCRIPTION Variable Description St = S(t) Susceptible individuals number Lt = L(t) Number of the TB- infected individuals in the latent state Tt = T(t) Number of the TB-infected individuals in the active state Rt = R(t) Number of recovered individuals λI Force of TB infection in presence of the treatment λr Force of exogenous infection again Parameters Description Λ Recruitment rate of susceptible individuals µ Naturel mortality rate σ Progression rate of TB-infected individuals from latent state to active state ρ Infection rate of recovered individuals n Proportion of susceptible individuals that enter in (T) by infection βT Number of effective contact of susceptible with TB vector δT Mortality rate caused by TB ηr Proportion of infected individuals by exogenous infection again ηT Proportion of recovered individuals by par the active TB-infected τ Treatment rate of TB The diagram of the stochastic model of the TB infection is given by Fig 1. The mathematical stochastic model of TB infec- tion in the presence of treatment is written under the compact form by the following equation (1) (its formulation uses [1], [5], [6], [7], and [22]) dXt = f(t,Xt)dt + G(t,Xt)dWt, (1) where Xt = (St,Lt,Tt,Rt)T is a 4-dimensional random vector of the states St,Lt,Tt,Rt; Wt = (W j t ) T j=1,...,m=10, is a 10-dimensional Brownian motion process and is defined on a space of (Ω,F,{Ft}t≥0,P); f(t,Xt) = (fi(t,Xt)) T i=1,...,d=4 is a vectorial function of evolution with components fi = fi(t,Xt) defined by  f1 = Λ − (µ + λI)St, f2 =nλISt−(µ+σ+λr)Lt+ρRt, f3 = (1−n)λISt+(σ+λr) Lt−(µ + δT +τ)Tt, f4 = τTt − (µ + ρ)Rt, (2) G = G(t,Xt) = (Gij)i=1,...,d=4;j=1,...,m=10 below is a (4 × 10)−dimensional matrix such that G = ( M1 O2×3 O2×3 M2 ) , (3) where O2×3 = ( 0 0 0 0 0 0 ) , M1 = ( G11 G12 G13 G14 0 0 0 0 0 G23 0 G25 G26 G27 ) , M2 = ( G34 0 G36 0 G38 G39 0 0 0 0 G47 0 G49 G410 ) , with G11 = √ Λ, G12 = − √ µSt, G13 = −G23 = − √ nλISt, G14 = −G34 = − √ (1 −n)λISt, G25 = − √ µLt, G26 = −G36 = − √ (σ + λr)Lt, G27 = −G47 = √ ρRt, G38 =− √ (µ+δT )Tt, G39 =−G49 =− √ τTt, G410 = − √ µRt. (4) The TB force of infection λI is defined by: λI = βT Tt + ηTRt N (5) with N = St + Lt + Tt + Rt The force of exogenous infection λr is defined by: λr = βT ηrTt N . (6) Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis B. Analysis of the solution of model and Stability In this part, we are going to show the existence and the uniqueness of the global solution positive of the model (1). We also address the existence and stability characterization of the Disease Free-Equilibrium (DFE) and of the endemic equilibrium point. 1) Existence and Uniqueness of solution: Consider a region Ω ⊂ R4+ defined by Ω ={(St,Lt,Tt,Rt)∈R4+; St+Lt+Tt+ Rt≤ Λ µ }. Then, we has the following result: Theorem 1. Let (S0,L0,T0,R0) ∈ Ω an ini- tial condition. Then there is a unique solu- tion of the stochastic model (1) denoted Xt = (St,Lt,Tt,Rt) T such that P{Xt = (St,Lt,Tt,Rt)T ∈ Ω} = 1 ∀ t ≥ 0. Proof: See Appendix A. 2) Stochastic Stability of the random DFE: Let us recall the following that will a very helpful in the sequel Lemma 1. Let p ≥ 2, x,y ∈ R+ and ε > 0 sufficiently small xyp−1 ≤ ε1−p p xp + (p− 1)ε p yp x2yp−2 ≤ 2ε 2−p 2 p xp + (p− 2)ε p yp Proof: The inequalities above can be demon- strated with the help of the inequalities of Young: for p,q > 0 and 1 p + 1 q = 1, xy ≤ xp p + yp q . Proposition 1. The stochastic model (1) admits a random equilibrium point without TB (Disease- Free random Equilibrium) [ X0 = ( Λ µ , 0, 0, 0 )] that is exponentially p-stable if p ≥ 2 and globally asymptotically stable. Proof: By translation, we can always bring back a random equilibrium point Xe to Xe = 0 like in [25]. The existence of X0, disease-free random equi- librium point is proved by the following change variable for the stochastic model (1) S̃t = Λ µ −St. (7) As a consequence, the stochastic model (1) reads as dX̃t = f̃(t,X̃t)dt + G̃(t,X̃t)dW̃t, (8) wherein X̃t = (S̃t,Lt,Tt,Rt) T , W̃ = (Wi) T , i = 2, ..., 10., f̃(t,Xt) = (f̃i(t,Xt)) T i=1,...,4 = (f̃i) T i=1,...,4 such that  f̃1 = λ̃I ( Λ µ − S̃t ) −µS̃t, f̃2 =nλ̃I ( Λ µ −S̃t ) −(µ+σ+λ̃r)Lt+ρRt, f̃3 = (1 −n)λ̃I ( Λ µ − S̃t ) + (σ + λ̃r)Lt − (µ + δT + τ)Tt, f̃4 = τTt − (µ + ρ)Rt. (9) The noise G̃ = G̃(t,X̃t) is a matrix (4 × 9) given by G̃ = ( M̃1 O2×3 O2×2 M̃2 ) , (10) where O2×2 = ( 0 0 0 0 ) , M̃1 = ( G̃12 G̃13 0 0 0 0 0 G̃23 0 G25 G26 G27 ) , M̃2 = ( G̃34 0 G36 0 G38 G39 0 0 0 0 G47 0 G49 G410 ) , Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis with G̃12 = − √ µS̃ t , G̃34 = (1 −n) √ λ̃I ( Λ µ − S̃t ) , G̃23 = n √ λ̃I ( Λ µ − S̃t ) , G̃13 = √ λ̃I ( Λ µ − S̃t ) , λ̃I = βT Tt + ηT Rt( Λ µ − S̃t ) + Lt + Tt + Rt , λ̃r = βT ηrTt( Λ µ − S̃t ) + Lt + Tt + Rt . The existence of a disease-free random equi- librium of the model (8) gives the existence of disease-free random equilibrium of (1). In fact, Denote by X̃(0) ≡ 0 ∈ R4. The equalities f̃(t, 0) = 0 and G̃(t, 0) = 0 are verified for t ≥ 0. So X̃(0) a disease-free random equilibrium of the model (8). Therefore, we have S̃t = 0, Lt = 0, Tt = 0, Rt = 0, that gives St = Λ µ , Lt = 0, Tt = 0, Rt = 0, i.e., X0 = ( Λ µ , 0, 0, 0 ) is a disease-free random equilibrium of the model (1). Now, consider a Lyapunov function: V = 1 2p ( K ( Λ µ −S̃t )p +K1L p t +K2T p t +K3R p t ) (11) with K > 0,K1 > 0,K2 > 0,K3 > 0,p ≥ 2. Let us note by A a differential operator asso- ciated to the stochastic model (1), operating on a function V = V (t,x) ∈C1,2(R×Rd) by AV = ∂V ∂t +f(t,x) ∂V ∂x + 1 2 tr[GT (t,x) ∂2V ∂x2 G(t,x)]. Then AV =−[K1(µ+σ+λr)L p t +K2(µ+δT +τ)T p t +K3(µ+ρ)R p t ] +K1nλIStL p−1 t +K1ρRtL p−1 t + K2(1 −n)λIStT p−1 t + K2(σ + λr)LtT p−1 t +K3τTtR p−1 t + 1 4 (p−1)[KG211 ( Λ µ − S̃t )p−2 +KG212 ( Λ µ − S̃t )p−2 +K 1 n G223 ( Λ µ − S̃t )p−2 + K1G 2 23L p−2 t + K1G 2 47L p−2 t + K2G 2 34T p−2 t + K2G 2 36T p−2 t + K3G 2 49R p−2 t + K1G 2 25L p−2 t + K1G 2 36L p−2 t + K2G 2 38T p−2 t + K1G 2 49T p−2 t + K3G 2 47R p−2 t + K3G 2 410R p−2 t ] The application of the lemma 1 and the theorem given by Afanas’ev in [24], allows us to obtain finally AV ≤−[K1(µ + σ + λr)L p t + K2(µ + δT + τ)T p t +K3(µ + ρ)R p t ] AV ≤ 0 (necessary to demonstrate). Therefore, X0 = ( Λ µ , 0, 0, 0 ) is exponentially p−stable (p ≥ 2). For, p = 2, we say that X0 is exponentially 2- stable or stable in mean square [24]. In the sense of Lyapunov, X0 is globally asymptotically stable. It marks the end of the proof. 3) Stability of the endemic random equilibrium: Preliminary: Suppose that the infection of TB evolves of manner linear i.e. without random noise G(Xt, t) ≡ 0. Then the model (1) become dXt = f(Xt, t)dt which admits a basic reproduc- tion number Rτ0 given by: Rτ0 = βT (µ + ρ + τηT )[(1 −n)µ + σ] (µ + σ)[(µ + ρ)(µ + δT ) + µτ] + µρτ . (12) If Rτ0 > 1, then the model dXt = f(Xt, t)dt admits a unique endemic equilibrium point bio- logically meaningful, X∗ that is locally asymp- totically stable [18]. The existence of a random endemic equilibrium [X∗ = (S∗,L∗,T∗,R∗)] is guaranteed by the condition Rτ0 > 1 almost surely (see [8]). At this random endemic equilibrium Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis [X∗ = (S∗,L∗,T∗,R∗)T ] we have [λI = λ∗I] and [λr = λ ∗ r] such that λ∗I = µ + ρ + τηT ηr(µ + ρ) λ∗r. (13) and a2(λ ∗ I) 2 + a1λ ∗ I + a0 (14) where a0 = (1−Rτ0 )(µ+ρ+τηT ){(µ+σ)[(µ+ρ)(µ+δT ) +µτ] +µρτ}, a1 = (µ + ρ + τηT ){(µ + ρ)(µ + σ) + n[δT (µ + ρ) +µτ]+µτ}+ηr(µ+ρ)[(µ+ρ)(µ+δT )+µτ], a2 = ηr(µ+ρ). If Rτ0 = 1 ie. a0 = 0, the equation (14) admits a hopeless solution corresponding to X0 the unique equilibrium without TB and it admits another solution to the real negative part corresponding to the endemic equilibrium which is biologically not pertinent. If Rτ0 < 1 ie. a0 > 0, then a2a0 < 0 and if the discriminant of (14) is positive i.e. a21−4a2a0 > 0. It follows itself that the equation (14) admits two solutions to part real negatives that corresponding to two equilibriums no pertinent. If Rτ0 > 1 ie. a0 < 0 then according to the Descartes rule of sign, the equation (14) admits one positive solution λ∗I = −a1 + √ a21 − 2a2a0 2a2 corresponding to an endemic equilibrium X∗. Now, suppose that the random noise of the dynamic system of TB has a nature to perturb the states variables St, Lt, Tt, and Rt of the stochastic term G(t,Xt) around of S∗, L∗, T∗, and R∗ respectively (see also [25]). Then the model (1) becomes dXt = f(t,Xt)dt + G(t,Xt −X∗)dWt, (15) that can be centered to X∗ by the change variables Y1 =St−S∗, Y2 =Lt−L∗, Y3 =Tt−T∗, Y4 =Rt−R∗ (16) The linearized system of (15) around X∗ = (S∗,L∗,T∗,R∗)T as in [4] takes the form dYt = f y(Yt)dt + G y(Yt)dξt, (17) where fy = fy(Yt) = Jf (X∗).Yt with Jf (X∗) the jacobian matrix of f at X∗; Yt = Y = (Y1,Y2,Y3,Y4) T ; ξt = (W it )i=2,...10; fy =   −∂11 ∂12 ∂13 ∂14 ∂21 −∂22 ∂23 ∂24 ∂31 ∂32 −∂33 ∂44 0 0 τ −(µ+ρ)     Y1 Y2 Y3 Y4   wherein −∂11 = µ+λ∗I ( 1 − S∗ N∗ ) , ∂12 = −λ∗I S∗ N∗ , ∂13 = (λ ∗ I −βT ) S∗ N∗ ∂14 = (λ ∗ I −βTηT ) S∗ N∗ , ∂21 = nλ ∗ I ( 1 − S∗ N∗ ) + λ∗r L∗ N∗ , −∂22 = nλ∗I S∗ N∗ +λ∗r L∗ N∗ +µ+ρ, ∂23 = −n(λ∗I −βT ) S∗ N∗ + (λ∗r −βTηr) L∗ N∗ , ∂24 = −n(λ∗I −βTηT ) S∗ N∗ + λ∗r L∗ N∗ + ρ, ∂31 = (1 −n)λ∗I ( 1 − S∗ N∗ ) −λ∗r L∗ N∗ , ∂32 = −(1−n)λ∗I S∗ N∗ +λ∗r ( 1 − L∗ N∗ ) +σ, ∂33 = (n−1)(λ∗I−βT ) S∗ N∗ +(λ∗r−βTηr) L∗ N∗ +µ+δT+τ, ∂34 = −(1−n)(λ∗I −βTηT ) S∗ N∗ −λ∗r L∗ N∗ ; and Gy(Yt) =  G y 12 G y 13 G y 14 0 0 0 0 0 0 0 G y 23 0 G y 25 G y 26 G y 27 0 0 0 0 0 G y 34 0 G y 36 0 G y 38 G y 39 0 0 0 0 0 0 G y 47 0 G y 49 G y 410  , Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis with G y 12 = − √ µY1, G y 13 = −G y 23 = − √ nλIY1, G y 14 = −G y 34 = − √ (1 −n)λIY1, G y 25 = − √ µY2, G y 26 = −G y 36 = − √ (σ + λr)Y3, G y 27 = −G y 47 = √ ρY4, G y 38 = − √ (µ + δT )Y3, G y 39 = −G y 49 = − √ τY3, G y 410 = − √ µY4. (18) . Theorem 2. The stochastic model (1) ad- mits a random endemic equilibrium [X∗ = (S∗,L∗,T∗,R∗)] exponentially 2-stable and glob- ally stable if the following conditions (i.), (ii.) are satisfied: (i.) : Rτ0 > 1 (ii.) :   ∂11 > 1 2 ( ω1 + κ1βT (1 + ηT ) Λ µ ) ∂22 > 1 2 (ω2 + µ) ∂33 > 1 2 ( ω3 + βT Λ µ ( c2 c3 + 1 ) + κ2 ) µ + ρ > 1 2 (ω4 + κ3). where, for all real constants ci > 0, i = 1, ..., 4, we have κ1 = 1+n c2 c1 +(1−n) c3 c1 ; κ3 =ρ ( c2 c4 +1 ) +µ; κ2 = σ ( c2 c3 + 1 ) +τ ( c4 c3 + 1 ) +µ+δT ; and ωi > 0, i = 1, ..., 4 such that ω1 = 2λ ∗ I S∗ N∗ + c2 c1 ( nλ∗I ( 1− S∗ N∗ ) +λ∗r L∗ N∗ ) + c3 c1 (1 −n) λ∗I ( 1 − S∗ N∗ ) , ω2 = nλ ∗ I ( 1 − S∗ N∗ ) + nβT (1 + ηT ) S∗ N∗ + 3λ∗r L∗ N∗ + ρ + c3 c2 (λ∗r + σ) ω3 = c1 c3 λ∗I S∗ N∗ + c2 c3 + ( nβT S∗ N∗ + λ∗r ) + λ∗r + (1−n) ( λ∗I ( 1− S∗ N∗ ) +βTηT S∗ N∗ ) +σ ω4 = c1 c4 λ∗I S∗ N∗ + c2 c4 ( nβTηT S∗ N∗ +λ∗r L∗ N∗ +ρ ) + c3 c4 (1 −n)βTηT S∗ N∗ . Proof: The trivial solution Yt = 0 of the linearized system (17) corresponds to the equilib- rium X∗ that the existence is guaranteed by the condition (i). Consider now the Lyapunov function defined by: V y =V y(Y ) = 1 2 4∑ i=1 ciY 2 i , ci>0, i= 1, ..., 4. (19) Then AV y =−c1∂11Y 21 −c2∂22Y 2 2 −c3∂33Y 2 3 −c4(µ+ρ)Y 2 4 + 3∑ i,j=1 4∑ i 6=j ci∂ijYiYj+ 1 2 4∑ i,j=1 tr(GyG yT ij ∂2V y(Y ) ∂Yi∂Yj ) AV y = −c1 ( µ + λ∗I(1 − S∗ N∗ ) ) Y 21 −c2(nλ ∗ I S∗ N∗ +λ∗r L∗ N∗ +µ+ρ)Y 22 −c3[(1−n) (λ ∗ I −βT ) S∗ N∗ +(λ∗r−βTηr) L∗ N∗ +µ+δT +τ]Y 2 3 −c4(µ+ρ)Y 2 4 + 3∑ i,j=1 4∑ i 6=j ci∂ijYiYj+ 1 2 4∑ i,j=1 tr(GyGyT )ij ∂2V y(Y ) ∂Yi∂Yj . To increase the last two terms of AV y(Y ) that we pose: sum1 = 3∑ i,j=1 4∑ i 6=j ci∂ijYiYj, sum2 = 1 2 4∑ i,j=1 tr(GyGyT )ij ∂2V y(Y ) ∂Yi∂Yj . sum1 = 3∑ i,j=1 4∑ i 6=j,∂ij>0 ci∂ijYiYj+ 3∑ i,j=1 4∑ i 6=j,∂ij<0 ci∂ijYiYj Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis ≤ 1 2 3∑ i,j=1 4∑ i 6=j,∂ij>0 ci∂ij(Y 2 i + Y 2 j ) + 3∑ i,j=1 4∑ ,i6=j,∂ij<0 ci∂ijYiYj sum1 ≤ 1 2 3∑ i,j=1 4∑ ,i6=j,∂ij>0 ci∂ij(Y 2 i +Y 2 j ) sum1 ≤ 1 2 {[2λ∗I S∗ N∗ + c2 c1 ( nλ∗I ( 1 − S∗ N∗ ) + λ∗r L∗ N∗ ) + c3 c1 (1 −n)λ∗I ( 1 − S∗ N∗ ) ]c1Y 2 1 + [nλ ∗ I ( 1 − S∗ N∗ ) +nβT (1 + ηT ) S∗ N∗ + 3λ∗r L∗ N∗ + ρ + c3 c2 (λ∗r + σ)]c2Y 2 2 + [ c1 c3 λ∗I S∗ N∗ + c2 c3 ( nβT S∗ N∗ + λ∗r ) +(1−n)(λ∗I ( 1 − S∗ N∗ ) +βT ηT S∗ N∗ )+λ∗r +σ]c3Y 2 3 +[ c1 c4 λ∗I S∗ N∗ + c2 c4 (nβT ηT S∗ N∗ +λ∗r L∗ N∗ +ρ) + c3 c4 (1−n)×βT ηT S∗ N∗ ]c4Y 2 4 }. From where sum1 ≤ 1 2 {ω1c1Y 21 + ω2c2Y 2 2 + ω3c3Y 2 3 + ω4c4Y 2 4 } and sum2 = 1 2 {c1(G212 + G 2 13 + G 2 14) + c2(G 2 23 + G 2 25 + G226 + G 2 27) + c3(G 2 34 + G 2 36 + G 2 38 + G 2 39) + c4(G 2 47 + G 2 49 + G 2 410)} + 1 2 {(κ1λI + c1ρ)Y1 + c2µY2 + (λr(c2 + c3) + κ2)Y3 + κ3Y4} sum2 ≤ 1 2 { ( κ1βT Λ µ (1 + ηT ) + ρ ) c1Y 2 1 + c2µY 2 2 + ( βT Λ µ ( c2 c3 + 1 ) + κ2 ) c3Y 2 3 + κ3Y 2 4 }. Hence AV y ≤− ( ∂11 − 1 2 ( ω1 + κ1βT (1 + ηT ) Λ µ )) c1Y 2 1 − ( ∂22 − 1 2 (ω2 + µ) ) c2Y 2 2 −[∂33− 1 2 (βT Λ µ ( c2 c3 + 1 ) +ω3 + κ2)]c3Y 2 3 − ( µ + ρ− 1 2 (ω4 + κ3) ) c4Y 2 4 . According to the condition (ii.), we has AV y ≤ 0 marking the end of this proof. The random endemic equilibrium [X∗ = (S∗,L∗,T∗,R∗)T ] of the model (1) exists when- ever Rτ0 > 1 and condition (i.) is fulfilled. It is exponentially 2-stable and globally asymptotically stable in sense of Lyapunov if the supplementary condition (ii.) is satisfied. We study in the following section, the optimal control of the treatment frequency in a stochastic model of TB. The condition Rτ0 < 1 is needed for the effective stability of TB in a population because the biological pertinence of the endemic equilibrium exists whenever Rτ0 > 1 almost surely. The control permits then to adjust this endemic situation unstable. III. OPTIMAL CONTROL OF THE TREATMENT FREQUENCY IN THE TB MODEL A. Optimal control problem Let (Ω,F,{Ft}t≥0,P) a complete filtered prob- ability space {Ft}t≥0 produced by a standard 10- dimensional Brownian Motion {Wt}t≥0. Let T > 0 a fixed real number named the horizon of the finite time. Let’s note by L2(Ω,FT ,R) the space of random variables. FT -measurable to real values and integrable square and by L2F(0;T ,R) a space of process Ft- adapted to real values and integrable square such that E[ ∫ T 0 |Xt|2dt] < +∞. Let K ∈ Uad a compact convex sub set of L2(0,T ). Consider an optimal control problem that consists in minimizing the cost J(., .), the objective function defined for the time t ∈ [0,T ], the state X ∈ R4 and function of control u ∈Uad by: J(X,u) = ∫ T 0 E[ϕ(Xt,ut)]dt + ∫ T 0 h(ut)dt, (20) relative to the state Xt ∈ R4 of the TB model governed in general by:{ dXt =f(t,Xt,ut)dt+G(t,Xt,ut)dWt, t∈[0,T ] X0 = X(0) ∈ R4 (21) Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis and in particular by:{ dXt =f(t,Xt,ut)dt+G(t,Xt)dWt, t∈[0,T ] X0 = X(0) ∈ R4 (22) where u = ut : τ 7−→ u(t)τ, for all rate τ of (2). This part deals with the study of the particu- lar case where the control doesn’t appear in the stochastic term. The control is said optimal when this dose reached its value optimal positive i.e. u = uop > 0. If this optimal value is not reached, i.e. u ∈ [−1; 0[∪]0; uop[, then the control is said less efficient; it is said without effect when u = 0 and finally the control is said efficient when the optimal value is passed) i.e. u ∈]uop, 1]. The aim is therefore to control the frequencies of the TB treatment in order to reduce number of new cases. The problem of the optimal control is translated to: Find an admissible control optimal u = u∗ such that J(X,u∗) = min u∈K⊂Uad J(X,u) (23) ie. J(X,u∗) ≤J(X,u) ∀ u ∈ K ⊂Uad Set F(u) = J(X,u), then the optimal control problem (23) becomes a optimization problem F(u∗) = min u∈K⊂Uad F(u), (24) wherein F(u) is a functional convex. B. Gradient Projection Method We want to solve (24) by the projection stochas- tic gradient method. For this purpose, let us recall the results that concern the projection method on a convex closed K an the stochastic algorithm: Proposition 2. Let, H a Hilbert space, provided with a norm ‖.‖ induced by the scalar product (·|·) and let K ⊂ H a nonempty convex closed set. Then for all u ∈ H, 1) an unique ũ ∈ K exists such that ‖u− ũ‖ = min v∈K ‖u−v‖ for all v ∈ K, where ũ = PK(u) is the orthogonal projec- tion of u on K. 2) ũ is charcterized by ũ = PK(u) ⇐⇒ (ũ−u | v − ũ) ≥ 0 Proof: 1) The existence of ũ ∈ K holds true because K is closed. Let’s suppose that the dimension of H is finite. Let us consider K∩B(u;‖u− v‖) the intersection of K with a ball B. On this compact, the function v 7−→‖u−v‖ is continuous. Of all minimizing sequence we can extract a convergent sequence, its limit is ũ. The uniqueness comes from the convexity of K and Pythagoras’ theorem. 2) For the characterization of ũ; suppose ũ = PK(u) then we has for all v ∈ K ‖u−ũ‖ = min v∈K ‖u−v‖ =⇒‖u−ũ‖≤‖u−v‖ Let v ∈ K, pose vε = ũ + ε(v − ũ) ε ∈ ]0; 1[ vε ∈ K which implies that ‖u−ũ‖2 ≤‖u−vε‖2 = ‖u−ũ‖2+ε2‖v−ũ‖2 +2ε(ũ−u | v − ũ) ‖u− ũ‖2 ≤‖u− ũ‖2 + ε2‖v − ũ‖2 +2ε(ũ−u | v − ũ) Dividing by ε then we obtain 0 ≤ ε‖v − ũ‖2 + 2(ũ−u | v − ũ) =⇒ (ũ−u | v − ũ) ≥ 0 Reciprocally, let’s suppose that (ũ−u | v − ũ) ≥ 0 0 ≥ (u−ũ | v−ũ) = (u−ũ | v−u+u−ũ) 0 ≥‖u− ũ‖2 + (u− ũ | v −u) Applying the inequality of Cauchy-Schwarz, we have 0 ≥‖u− ũ‖2 −‖u− ũ‖‖v −u‖ =⇒‖v −u‖≥‖ũ−u‖ Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis Proposition 3. The algorithm of the stochastic gradient consists in making evolve the variable u of the optimization problem (24) according to the formula of the following recurrence convergent sequence (un)n≥0 of limit u: un+1 = un + qn(−∇F(un)) where qn > 0 with lim n−→+∞ qn = q and ∇ denotes the gradient.  1. Initialization : u0 ∈ H 2. u = un is for n ≥ 0 a) Calculate ωn = −∇F(un) Choisir qn ≥ 0 such that F(un −qnωn) ≤ F(un −qωn) ∀q > 0 un+1 = un + qnωn b) Calculate vn+1 = PK(vn+1) c) Test the covergence of the iteration εn = ‖un+1 −un‖ : − if εn < ε Stop − otherwise : u = un+1 and repeat iteration. (25) C. Projection gradient method applied to the stochastic model of TB with control Proposition 4. Consider H = U is a Hilbert space and Uad ⊂ U a closed convex subset. Let PK the projection operator on K defined in U by PK(ω) = PKω ∈ K; ∀ ω ∈U, then problem (24) admits an unique solution u or an optimal control such that u = u(·) = PK[u−q(· |F ′(u))] Proof: H = U is a Hilbert space and Uad ⊂U a closed convex subset. The necessary and suffi- cient condition of the optimality problem (24) is given by (F ′(u) | v −u) ≥ 0 ∀ v ∈ K. Let PK the projection operator on K defined in U by PK(ω) = PKω ∈ K; ∀ ω ∈U, such that we have (PKω−ω |PKω−ω) = min u∈K⊂Uad (u−ω |u−ω) ∀ω∈U. It is equivalent to (PKω−ω | v−PKω) ≥ 0 ∀v∈K ⇐⇒ ω =PKω. It follows that the solution u of (24) is given by u = u(·) = PK[u−q(· |F ′(u))]. Indeed, the optimality condition gives (F ′(u) | v −u) ≥ 0 ∀ v ∈ K, then for q > 0 we have q(· | F ′(h)) | v−u)≥0 =⇒ (q(· | F ′(h)) | v−u)≥0 =⇒ (u−u + q(· | F ′(h)) | v −u) ≥ 0. With ω = u − q(· | F ′(h)), the last implication gives (u−ω |v−u) ≥ 0 ⇐⇒ u = PKω ⇐⇒ u = PK[u−q(· | F ′(h))] For the optimal control problem of the treatment frequency of TB, we are going to define the following iteration scheme for n = 0, 1, ...{ (v | un+1 2 ) = (v | un)−qn(v | F ′n(un)), ∀v∈U un+1 = PK(un+ 1 2 ), (26) where F ′n is the functional approached to the n th iteration of F ′n. The convergence of this scheme, and the calcu- lation of F ′n. are given in [17],[13]. For u(·) an optimal control and X(·), the optimal stat corre- sponding to X(·) and for v(·) ∈ U ⊂ L2(0,T) such that vp = u(·) + qv(·), 0 < q < 1, then we have for all v ∈ L2(0,T), F ′n(u)(v) = lim q−→0 Fn(u + qv) −Fn(u) q = E[ ∫ T 0 ϕ′(X)D(X)(v)dt] + ∫ T 0 h′(u)dt, (27) where D(X)(v) = ∫ t 0 [ f ′X(s,X,u)D(X)(v)+f ′ u(s,X,u)v ] ds Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis + ∫ t 0 G′X(s,X)D(X)(v)dWs, d(D(X)(v)) = [f ′X(t,X,u)D(X)(v)+f ′ u(t,X,u)v]dt +G′X(t,X)D(X)(v)dWt. We define a adjoint functional p, Ft-adapted and defined by  −dp = [ϕ′(X) + pf ′X(t,X,u) −pG′X(t,X) .(G′X(t,X)) tr]dt + pG′X(t,X)dWt, p(T ) = 0 (28) such that E[ ∫ T 0 |pt|2dt] < +∞. The right hand side of the equation (27) permits to get finally F ′n(u)(v) from (28), that reads as F ′n(u)(v) = ∫ T 0 E[p(f ′(t,X,u) + h′(u)]vdt, (29) The Projection Gradient Method applied to the stochastic model of TB with control, consist there- fore in considering the system (30) of two equa- tions (22) and (28) in order to solve it numerically,   dXt =f(t,Xt,ut)dt+G(t,Xt,ut)dWt, t∈[0,T ], −dp= [ϕ′(Xt)+pf ′X(t,Xt,ut)−pG ′ X(t,Xt) .(G′X(t,X)) tr]dt + pG′X(t,Xt)dWt, X0 = X(0) ∈ R4 p(T ) = 0. (30) The numerical resolution of (30) uses the iteration scheme (31) below for n = 0, 1... and then the Euler scheme for the two equations of (30) (see [17]) ,  (v|un+ 1 2 ) = (v|un)−qn(v|E[pn(f ′u(t,Xn,un))] + h′(un)), ∀v ∈U un+1 = PK(un+ 1 2 ), (31) where Xn, un and pn are the present steps of the functions constructed. D. Numerical Simulations Algorithm[17]: Stage 1 To choose the arbitrary initial control For n = 0, 1, · · ·, let u = un, to make the buckle iteration of Stage 1 to Stage 5; Stage 2 To use the implicit Euler scheme for the discretization in time of the SDE (22) Stage 3 To use the implicit Euler scheme for the discretization in time of the adjoint equation; (28) Stage 4 To use the iteration scheme (31) of the gradient method to update the controls;  um n+ 1 2 =um−qn(E[pm(f ′u(tm,Xm,um))] + h′(um)), m = 0, 1, · · ·,mmax umn+1 = PK(un+ 1 2 ); Stage 5 Calculate en = ‖un − un+1‖. If en is small enough, then exit. Otherwise; let u = un+1 repeat the buckle iteration from Stage 2 to Stage 5. TABLE II PARAMETER VALUES AND REFERENCES Parameters Values References Λ variable Estimate µ 0.02 [18] σ 1/33 [18] ρ 0.04 [18] δT 0.2 [18] n Variables Estimates ηr, ηT 0.4, 0.06 [18] βT , τ Variables Estimates For the following figures, we take ϕ(x,u) = (x2 + u2)exp( −t x2 + u2 ), h(x) = x2, n = 0.05, en < 10 −7, p0 = 0.01 and the rest Λ, βT , τ, X0 are variable. Fig.2 give a schematic plot of the model (1) not depending of u. The aim is to show, for a initial condition given, the asymptotic behavior of the so- lution around a random endemic equilibrium when the hard epidemic a long time Rτ0 > 1. While, Fig.3(resp. Fig.4) shows a numerical illustration of optimal control u, see (a) and (b) (resp. of cost F(u), see (c) and (d)). The orthogonal projection of the minimum point of F(u) on the closed subset [−1; 1], gives a numerical value of optimal control u∗; e.g. the minimum point • of F(u) represented in (c), is valued as F(u∗) = 2.7066 giving u∗ = 0 if u0 = 1. Thanks to Matlab, we can value the Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis cost F(u∗) and the optimal control u∗ for a control initially chosen u0 as Fig.6. In Fig.5, the trajectory without control, see (t.1), (resp. with control, see (t.2)) of the active infected individuals T is creasing between 0 and 2 years (resp. decreasing between 0 and 1 year and is annulling constantly thereafter). (t.3) and (t.4) show that a control of treatment intervened 0.5 years equal to 6 months after the infection, permits to reduce to nothing numbers of the active infected individuals of TB. 0 5 10 15 20 0 10 20 30 40 50 60 70 80 90 100 Time (years) S , L , T a n d R in d iv id u a ls S without control L without control T without control R without control (i) 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 Time (years) S , L , T a n d R in d iv id u a ls S without control L without control T without control R without control (ii) Fig. 2. Numerical Simulations of the model (1) without control (i.e. not depending of u), showing the asymptotic behavior of the solution when Rτ0 > 1 at different initial con- dition: (i) : X0 = (S0,L0,T0,R0) = (50, 12, 5, 10), Λ = 10, βT = 0.8, τ = 0.08, R τ 0 = 2.3710 > 1 and, (ii) X0 = (S0,L0,T0,R0) = (50, 1, 1, 1), Λ = 8, βT = 0.9, τ = 0.08. Rτ0 = 3.5565 > 1. 0 0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 p tim a l c o n tr o l u Time (years) Optimal control u for u 0 =0 (a) 0 0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (years) 0 p tim a l c o n tr o l u w ith d iff e re n te in iti a l v a lu e Optimal control u for u 0 =0 Optimal control u for u 0 =0.5 Optimal control u for u 0 =0.8 Optimal control u for u 0 =1 (b) Fig. 3. Numerical simulations of a control u: (a) and (b) −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Optimal ontrol u C o st f u n ct io n F (u ) F(u) for u 0 =1 with minimum point • u*=0 (c) −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Optimal ontrol u C o st f u n ct io n F (u ) F(u) for u 0 =0.5 F(u) for u 0 =0.8 F(u) for u 0 =1 (d) Fig. 4. Numerical Simulation of a cost functionF(u) (c) and (d) Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9 10 Time (years) S , L , T a n d R in d iv id u a ls S withoit control L withoit control T withoit control R withoit control (t.1) 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9 10 Time (years) S , L , T ,a n d R S stochastic L stochastic T stochastic R stochastic (t.2) Fig. 5. Trajectories without and with control of the model (1). For Λ = 5, βT = 0.08, τ = 0.08, u0 = 0.08, X0 = (1, 1, 1, 1). Let’s note that initial value of sequence qn is chosen as q0 = 0.1 for Fig.6 and (d); q0 = 0.6 for Fig.4 (c). IV. CONCLUSION The stochastic model (1) of TB without control admits for an initial state X(0), a positive and unique solution Xt ∈ Ω of probability one. It exist for this model an unique disease equilibrium free (DEF) exponentially 2-stable and globally asymp- totically stable (in Lyapunov sense). Under a given condition, the model (1) admits a random endemic equilibrium exponentially p-stable (p ≥ 2) and globally stable. The introduction of a treatment control function in model (1) gives an optimal control problem governed by model (22). The Projection Gradient method permits to determine numerically the optimal control as well as the cost function corresponding to this problem. −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Optimal ontrol u C o st f u n ct io n F (u ) F(u) for u 0 =0.2 F(u) for u 0 =0.5 F(u) for u 0 =0.8 F(u) for u 0 =1 Fig. 6. Numerical Simulations of the control u and the function cost F(u) for the different initials values of u0. We obtain: If u0 = 0.2 then F(u∗) = 2.7073 and u∗ = uop = 0.03606; if u0 = 0.5 then F(u∗) = 2.9166 and u∗ = uop = 0.01607; if u0 = 0.8 then F(u∗) = 3.3071 and u∗ = uop = 0.0000; and if u0 = 1 then F(u∗) = 3.6673 and u∗ = uop = 0; For example, with a treatment rate equal to τ = 8% and with an initial value equal to u0 = 0.2 of the function control, we obtain u∗ = 0.3606, the admissible optimal control and F(u∗) = 2.7073, the cost. Also with τ = 8% and u0 = 1, we obtain u∗ = 0. We therefore deduce that the optimal control is without effect when u0, the initial dose of the medicines taken by a patient ranges from 80% to 100 %. On the other hand the optimal control is efficient admissible when this initial dose is lower to 50 %. Thanks to the presence of the optimal control in the stochastic model (1) of TB, we can reduce considerably and quickly (less than one year) the number of the active infected individuals. As in Fig.5(t.2) and Fig.7 (t.3)-(t.4), the trajectory with control of the active infected individuals T is decreasing between 0 and 1 year and becomes null constantly thereafter. This work is therefore a contribution that enters well in the same line of struggle against mortality due to the infections that several governments as well as Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (years) T T B in fe ct e d a ct iv e T without control T with control (t.3) 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (years) T T B in fe ct e d a ct iv e T without control T with control (t.4) Fig. 7. Trajectory without and with control of TB active infected individuals T. With βT = 0.08 for t.3 and βT = 0.8 for t.4. humanitarian associations advocated so much. APPENDIX A: PROOF OF THEOREM 1 Let Nt = St +Lt +Tt +Rt, the random variable giving the total number of the population at the time t. We have dNt = (Λ −µNt −δTTt)dt− ξ. where ξ = ( √ µSt + √ µLt + √ (µ + δT )Tt+√ µRt − √ Λ)d$(t), with $ = Wi i = 1, ..., 10. because Wi follow the same law of probability, namely the normal law. We need to show that if Xt = (St,Lt,Tt,Rt)T ∈ R4+ for all t ∈ [0; t�[ where t� is the explosion time, then we have for P -almost surely (P−as) Nt < Λ µ . In fact, if Xt ∈ R4+ for all t ∈ [0; t�[, then Nt is given such that for P−as.: dNt = (Λ −µNt −δTTt − ξ)dt ≤ (Λ −µNt)dt According to the lemma of Gronwall, we obtain: Nt ≤ Λ µ + (N0 − Λ µ )e−µt P −as. And as by hypothesis (S0,L0,T0,R0) ∈ Ω i.e. N0 − Λ µ ≤ 0, we have then Nt < Λ µ P −as. The terms f(t,Xt) and G(t,Xt) of the stochastic model (1) being locally Lipschitz, there is an unique local solution Xt = (St,Lt,Tt,Rt)T for all t ∈ [0; t�[ fixed. Therefore, the unique local solution Xt = (St,Lt,Tt,Rt)T ∈ R4+. In the sequel we show that Xt is global solution P−almost surely i.e. t� = ∞. Let n0 > 0, an integer sufficiently large such that (S0,L0,T0,R0) ∈ [ 1 n0 ; n0 ]4 . Set Et = {St,Lt,Tt,Rt} and for all integer n ≥ n0, we define the stop-times tn = inf {Hn} with Hn ={ t∈[0,t�] : min Et∈ [ 0; 1 n ] or max Et∈[n; +∞[ } . (tn)n>0 is an increasing sequence and convergent; denote by t∞ = lim n−→∞ tn then t∞ ≤ t�. Let us show that t∞ = ∞ so that we has t� = ∞. For it, let us suppose by absurd that t∞ < ∞, there is θ > 0 such that for all p ∈]0; 1[ we have P{t∞ ≤ θ} > p. Consequently, there is an integer n1 ≥ n0 such that for all set An = {tn ≤ θ}, we have P{An} > p n ≥ n1. (32) Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis Let us consider the function V defined on R4+ and to values in R+ such that V =−ln ( µSt Λ ) −ln ( µLt Λ ) −ln ( µTt Λ ) −ln ( µRt Λ ) . Using the multidimensional Itô formula on the interval [0; min(τn; θ)], we have for all t ≥ 0 dV = dV (Xt) = [ ∂V (Xt) ∂t + 4∑ i=1 fi(t,Xt) ∂V (Xt) ∂Xit + 1 2 4∑ i,j=1 (GGT )ij ∂2V (Xt) ∂Xt∂X j t ]dt + 4∑ i=1 10∑ j=1 GijdW j t ∂V (Xt) ∂Xit , where for i = 1, 2, ..., 4; j = 1, 2, ..., 10, G = (Gij); and (GG T )ij = 10∑ k=1 Gik.Gkj. Therefore dV = 2[4µ + σ + δT + τ]dt + 5 4 (λI + λr)dt + 1 2 [µ 1 St + (µ + σ) 1 Lt + (µ + δT + τ) 1 Tt + (µ + ρ) 1 Rt ]dt− 1 2 {Λ (4St − 1) S2t + (nλISt+ρRt) (4Lt−1) L2t +[(1−n)λISt + (σ+λr)Lt] (4Tt−1) L2t +τTt (4Rt−1) L2t }dt − 1 St (G11dW 1 t + G12dW 2 t + G13dW 3 t + G14dW 4 t ) − 1 Lt (G23dW 3 t + G25dW 5 t + G26dW 6 t + G27dW 7 t ) − 1 Tt (G34dW 4 t + G36dW 6 t + G38dW 8 t + G39dW 9 t ) − 1 Rt (G47dW 7 t + G49dW 9 t + G410dW 10 t ). We further obtain the following inequations: dV (Xt) ≤ Mdt− 1 St (G11dW 1 t + G12dW 2 t +G13dW 3 t + G14dW 4 t ) − 1 Lt (G23dW 3 t +G25dW 5 t + G26dW 6 t + G27dW 7 t ) − 1 Tt (G34dW 4 t + G36dW 6 t + G38dW 8 t +G39dW 9 t ) − 1 Rt (G47dW 7 t + G49dW 9 t +G410dW 10 t ) P −as. with M = 5 2 [4µ+σ +δT +τ + 1 2 βT (1 +ηT +ηr)] > 0. Which implies by integration that∫ ∧tnθ 0 dV ≤ M ∫ ∧tnθ 0 dt−[ 4∑ k=1 (∫ ∧tnθ 0 G1k St dWkt ) + 7∑ k=3,6=4 (∫ ∧tnθ 0 G2k Lt dWkt ) ] −[ 9∑ k=4,6=7 (∫ ∧tnθ 0 G3k Tt dWkt ) + 10∑ k=7,k 6=8 (∫ ∧tnθ 0 G4k Rt dWkt ) ], (33) where ∧tnθ = tn ∧ θ = min(tn; θ). Taking the mathematical expectations for all terms of inequations (33), we obtain E[V (Xtn∧ θ)] ≤ E[V (X0)] + Mθ (34) Let a set An = {tn ≤ θ}. Denote by IAn (resp. I{An) the indicator function of An (resp. of the complementary {An). Thus E[V (Xtn∧θ)] = E[V (Xtn∧θ)IAn]+E[V (Xtn∧θ)I{An] According to the definition of function V , we have V (Xtn∧ θ) ≥ 0. Hence E[V (X0)]+Mθ≥E[V (X∧tnθ )IAn]+E[V (X∧tnθ )I{An] E[V (X0)] + Mθ ≥ E[V (Xtn)IAn] Biomath 6 (2017), 1705077, http://dx.doi.org/10.11145/j.biomath.2017.05.077 Page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2017.05.077 Bongor Danhree et al., Optimal control of the treatment frequency in a stochastic model of Tuberculosis Thanks to the continuity, it exists at least one of the components Xtn equals to n or to 1 n . So V (Xtn) ≥ min { −ln (µn Λ ) ;−ln ( µ Λn )} V (Xtn) ≥ min { ln ( Λ µn ) ; ln ( Λn µ )} , and consequently E[V (X0)] + Mθ ≥ E[V (Xtn∧ θ)IAn] ≥ P{An}× min { ln ( Λ µn ) ; ln ( Λn µ )} . Hence P{An}=P{tn≤θ}≤ E[V (X0)]+Mθ min { ln ( Λ µn ) ; ln ( Λn µ )}. (35) Taking the limit when n −→ +∞ in (35), we found that 0