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By: Betty J. Dong PharmD, FASHP, FCCP

• Professor of Clinical Pharmacy and Clinical Professor of Family and Community Medicine
• Department of Clinical Pharmacy and Department of Family and Community Medicine, Schools of Pharmacy and Medicine
• University of California, San Francisco https://pharmacy.ucsf.edu/betty-dong

Let t ∈ [0 discount benicar 20mg on-line arteria gastroepiploica sinistra, τ] purchase benicar 40 mg on line blood pressure medication starting with n, from the ﬁrst equation of system (6) order 40 mg benicar mastercard heart attack 3 stents, we obtain dX ≤ s − µX(t), dt so, s −µt s X(t) ≤ X(0) − e + , µ µ this means that X is bounded. From the second equation of (6), we obtain dY −λτ ≤ e k(1 − η1(t))V(t − τ)X(t − τ) − δI(t), dt since (1 − η (t)) ≤ 1 and e−λτ ≤ 1, it follows 1 dY ≤ kV(t − τ)X(t − τ) − δY(t), dt therefore, Z t −δt δ(ξ−t) Y(t) ≤ Y(0)e + kV(ξ − τ)X(ξ − τ)e dξ, 0 since (t − τ) ∈ [−τ, 0] and from (2) and (3), we have the fact that V(t − τ)X(t − τ) is bounded, then Y is also bounded. From the third equation of (6), we obtain dD = (1 − η2(t))aY(t) − βD(t) − δD(t), dt since (1 − η2(t)) ≤ 1, it follows dD ≤ aY(t) − βD(t) − δD(t), dt this inequality implies that Z t −(δ+β)t (δ+β)(ξ−t) D(t) ≤ D(0)e + aI(ξ)e dξ, 0 from the boundedness result of I, one can conclude that D is bounded. From the fourth equation of (6), we obtain dV ≤ βD(t) − uV(t), dt then, Z t −ut u(ξ−t) V(t) ≤ e V(0) + βD(ξ)e dξ, 0 from the boundedness result of D, we conclude that V is also bounded. From both the fourth and the ﬁfth equations of (6), we obtain dW g + hW(t) = gV(t)W(t) = βD(t) − uV(t) − V˙ , dt q High-Throughput 2018, 7, 35 7 of 16 then Z g t −ht h(ξ−t) W(t) ≤ W(0)e + (βD(ξ) + (h − u)V(ξ))e dξ q 0 −ht − V(t) + V(0)e , from the boundedness results of D and V, we deduce that W is bounded. From the second and the last equation of system (6), we obtain dZ + bZ(t) = cI(t)Z(t) dt c −λτ = ke (1 − η (t))V(t − τ)H(t − τ) − δI(t) − I˙ , 1 p then Z c t −bt b(ξ−t) Z(t) ≤ Z(0)e + (kV(ξ − τ)X(ξ − τ) + (b − δ)Y(ξ))e dξ p 0 −bt − Y(t) + Y(0)e , from the boundedness results of X, Y and V, it follows the result that Z is bounded. By following the same analysis as before, for each single interval [nτ, (n + 1)τ] with n ≥ 1, one can conclude that all the solutions are bounded for all t ≥ 0. Therefore, every local solution can be prolonged up to any time tm > 0, which means that the solution exists globally. Let us consider the following objective functional: Z t n h io f A1 2 A2 2 J (η1, η2) = X(t) + W(t) + Z(t) − η1(t) + η2(t) dt, (7) 0 2 2 where tf is the time period of therapy and the two positive constants A1 and A2 are based on the beneﬁt-cost of the therapy η1 and η2, respectively. The two control functions, η1(t) and η2(t) are supposed to be bounded and also Lebesgue integrable. That means, we are seeking an optimal control pair (η∗, η∗) such that 1 2 ∗ ∗ (8) J (η1, η2) = max{J (η1, η2) : (η1, η2) ∈ U}, where U is the control set given by U = {(η1(t), η2(t)) : ηi(t) measurable, 0 ≤ ηi(t) ≤ 1, t ∈ [0, tf ], i = 1, 2}. An Optimal Control Existence Result the two optimal control pair existence result can be obtained via the results [24,25]. There exists an optimal control (η∗, η∗) ∈ U such that 1 2 ∗ ∗ J (η1, η2) = max J (η1, η2). To use the existence result , we should ﬁrst check the following properties High-Throughput 2018, 7, 35 8 of 16 (C1) the set of the corresponding state variables and controls is nonempty. We can therefore deduce that the set of controls and the corresponding state variables are non-empty, this gives us the condition (C1). The control set is convex and closed by deﬁnition, which ensures the condition (C2). Moreover, since the system of state is bi-linear in η1, η2, the right hand-side of (6) veriﬁes condition (C3), using the fact that the solutions are bounded. For the condition (C5), we have 2 2 (14) I(X, W, Z, η1, η2) ≤ c2 − c1(| η1 | + | η2 | ), A A with c depends on the upper bound on X, W, Z, and c = min 1 , 2 > 0. We deduce that there 2 1 2 2 exists an optimal control pair (η∗, η∗) ∈ U such that 1 2 ∗ ∗ J (η1, η2) = max J (η1, η2). The Optimality System To prove the necessary conditions for the optimal control problem, we will use the Pontryagin’s minimum principle . This principle changes (6), (7) and (9) into a problem of maximizing of an Hamiltonian, T, pointwise with respect to η1 and η2: 6 A1 2 A2 2 T(t, X, Y, D, V, W, Z, Xτ, Vτ, η1, η2, λ) = η1 + η2 − X − W − Z + ∑ λi fi, 2 2 i=1 High-Throughput 2018, 7, 35 9 of 16 where the λi for i = 1,.. For any optimal control pair η∗, η∗, and any solutions (X∗, Y∗, D∗, V∗, W∗, Z∗) (6), there exists 1 2 an adjoint variables, λ1, λ2, λ3, λ4,λ5 and λ6 satisfying  λ0 (t) = 1 + λ (t) µ + k 1 − η∗(t) V∗(t)  1 1 1  ∗ −λτ ∗  +χ[0,t −τ](t)λ2 t + τ η1 t + τ − 1 ke V (t),  f  0 ∗ ∗ ∗  λ2(t) = λ2(t)δ − λ3(t)a 1 − η2(t) − cZ (t)λ6(t) + pZ (t)λ2(t),  λ0 (t) = λ (t) δ + β − βλ (t) 3 3 4 0 ∗ ∗ ∗ (16)  λ4(t) = λ1(t) k(1 − η1(t))X (t) + λ4(t)(u + qW (t))   +χ (t)λ (t + τ) ke−λτ(η∗(t + τ) − 1)X∗(t) ,  [0,tf −τ] 2 1   λ0 (t) = 1 + λ (t)qV∗(t) + λ (t) h − cV∗(t)  5 4 5  λ0 (t) = 1 + λ (t)pY∗(t) + λ (t) b − cY∗(t) 6 2 6 where the transversality conditions λi(tf ) = 0, i = 1,.. The transversality conditions and adjoint equations as follows,   λ0 (t) = − ∂T (t) − χ (t) ∂T (t + τ), λ (t ) = 0,  1 ∂X [0,tf −τ] ∂Xτ 1 f   λ0 (t) = − ∂T (t), λ (t ) = 0,  2 ∂Y 2 f   λ0 (t) = − ∂T (t), λ (t ) = 0, 3 ∂D 3 f 0 ∂T ∂T (19)  λ4(t) = − (t) − χ[0,t −τ](t) (t + τ), λ4(tf ) = 0,  ∂V f ∂Vτ   λ0 (t) = − ∂T (t), λ (t ) = 0. A2 If we replace η∗ and η∗ in the systems (6), we have the following optimality system: 1 2 dX∗ ∗ ∗ ∗ ∗ = s − µX (t) − k(1 − η1(t))V (t)X (t), dt dY∗ −λτ ∗ ∗ ∗ ∗ ∗ ∗ = e k(1 − η1(t))V (t − τ)X (t − τ) − δY (t) − pY (t)Z (t), dt dD∗ ∗ ∗ ∗ ∗ = (1 − η2(t))aY (t) − δD (t) − βD (t) dt dV∗ ∗ ∗ ∗ ∗ = βD (t) − uV (t) − qV (t)W (t), dt dW∗ ∗ ∗ ∗ = gV (t)W (t) − hW (t), dt dZ∗ ∗ ∗ ∗ = cY (t)Z (t) − bZ (t), dt then,  λ0 (t) = 1 + λ (t) µ + k 1 − η∗(t) V∗(t)  1 1 1  ∗ −λτ ∗  +χ[0,t −τ](t)λ2 t + τ η1 t + τ − 1 ke V (t),  f  0 ∗ ∗ ∗  λ2(t) = λ2(t)δ − λ3(t)a 1 − η2(t) − cZ (t)λ6(t) + pZ (t)λ2(t),  λ0 (t) = λ (t) δ + β − βλ (t) 3 3 4 0 ∗ ∗ ∗ (20)  λ4(t) = λ1(t) k(1 − η1(t))X (t) + λ4(t)(u + qW (t))   +χ (t)λ (t + τ) ke−λτ(η∗(t + τ) − 1)X∗(t) ,  [0,tf −τ] 2 1   λ0 (t) = 1 + λ (t)qV∗(t) + λ (t) h − cV∗(t)  5 4 5  λ0 (t) = 1 + λ (t)pY∗(t) + λ (t) b − cY∗(t) 6 2 6 λi(tf ) = 0, i = 1,.. Numerical Results To illustrate the numerical simulations, we implement and solved numerically our optimality system by the ﬁnite difference approximation method [27–29]. We obtain the following algorithm (Algorithm 1): High-Throughput 2018, 7, 35 11 of 16 Algorithm 1: the forward-backward ﬁnite difference numerical scheme. The role of the two parameters A1 and A2 is to calibrate the terms size in the system equations. Figure 1 depicts the evolution of the uninfected cells as function of time for both cases with and without control therapy. It is shown that with control the number of the uninfected cells is higher than those observed for the case without control. This result support the fact that the control strategy is to maximize the number of the healthy cells.

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