“…In order to perform the numerical simulations, the optimization system will be solved numerically using a discretized scheme on the basis of forward and backward finite‐difference approximation method. () Indeed, we will have the following numerical algorithm: The parameters of our numerical simulations are taken from the works of Manna and Meskaf et al, ie, s =2.6×10 7 , k =1.67×10 −12 , μ =0.01, δ =0.053, a =150, β =0.87, u =3.8, τ =5, λ=1.1×10 −2 , p =0.01, b =0.2, c =0.03, A 1 =50 000, and A 2 =5000. The role of these 2 last positive parameters, ie, A 1 and A 2 , is to balance the terms size in the equations.…”
Section: Numerical Simulationsmentioning
confidence: 99%
“…The parameters of our numerical simulations are taken from the works of Manna 25 and Meskaf et al, 26 ie, s = 2.6 × 10 7 , k = 1.67 × 10 −12 , = 0.01, = 0.053, a = 150, = 0.87, u = 3.8, = 5, λ = 1.1 × 10 −2 , p = 0.01, b = 0.2, c = 0.03, A 1 = 50 000, and A 2 = 5000. The role of these 2 last positive parameters, ie, A 1 and A 2 , is to balance the terms size in the equations.…”
Summary
This paper deals with an optimal control problem of a time‐delayed differential equation model that describes the interactions between hepatitis B virus (HBV) with HBV DNA‐containing capsids, liver cells (hepatocytes), and cytotoxic T‐lymphocyte immune response. Both the treatment and the intracellular delay are incorporated into the model. Furthermore, the existence of the optimal control pair is studied, and Pontryagin's minimum principle is used to characterize these 2 optimal controls. The first of them represents the efficiency of drug treatment in preventing new infections, whereas the second stands for the efficiency of drug treatment in inhibiting viral production. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are established to show the role of optimal therapy in controlling viral replication.
“…In order to perform the numerical simulations, the optimization system will be solved numerically using a discretized scheme on the basis of forward and backward finite‐difference approximation method. () Indeed, we will have the following numerical algorithm: The parameters of our numerical simulations are taken from the works of Manna and Meskaf et al, ie, s =2.6×10 7 , k =1.67×10 −12 , μ =0.01, δ =0.053, a =150, β =0.87, u =3.8, τ =5, λ=1.1×10 −2 , p =0.01, b =0.2, c =0.03, A 1 =50 000, and A 2 =5000. The role of these 2 last positive parameters, ie, A 1 and A 2 , is to balance the terms size in the equations.…”
Section: Numerical Simulationsmentioning
confidence: 99%
“…The parameters of our numerical simulations are taken from the works of Manna 25 and Meskaf et al, 26 ie, s = 2.6 × 10 7 , k = 1.67 × 10 −12 , = 0.01, = 0.053, a = 150, = 0.87, u = 3.8, = 5, λ = 1.1 × 10 −2 , p = 0.01, b = 0.2, c = 0.03, A 1 = 50 000, and A 2 = 5000. The role of these 2 last positive parameters, ie, A 1 and A 2 , is to balance the terms size in the equations.…”
Summary
This paper deals with an optimal control problem of a time‐delayed differential equation model that describes the interactions between hepatitis B virus (HBV) with HBV DNA‐containing capsids, liver cells (hepatocytes), and cytotoxic T‐lymphocyte immune response. Both the treatment and the intracellular delay are incorporated into the model. Furthermore, the existence of the optimal control pair is studied, and Pontryagin's minimum principle is used to characterize these 2 optimal controls. The first of them represents the efficiency of drug treatment in preventing new infections, whereas the second stands for the efficiency of drug treatment in inhibiting viral production. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are established to show the role of optimal therapy in controlling viral replication.
“…To our knowledge, there is no study that investigates the adaptive immune response in the early stage of the infection and effect of the early treatment on the progress of the disease. In this work, we are aiming to investigate this issue by considering an augmented model of our recent works [18,19], and we consider the logistic growth only for the healthy hepatocyte cells and the infected hepatocyte cells [11]. This assumption is made to reflect the nature of the growth of these two types of cells in the early stage of the infection.…”
The aim of this paper is to study the early stage of HBV infection and impact delay in the infection process on the adaptive immune response, which includes cytotoxic T-lymphocytes and antibodies. In this stage, the growth of the healthy hepatocyte cells is logistic while the growth of the infected ones is linear. To investigate the role of the treatment at this stage, we also consider two types of treatment: interferon-(IFN) and nucleoside analogues (NAs). To find the best strategy to use this treatment, an optimal control approach is developed to find the possibility of having a functional cure to HBV.
“…In collaboration with this present study, mathematical models formulated to account for the monolytic HIV, HBV infections include [32][33][34][35][36][37][38][39][40]. The inclusion of the vital role of adaptive immune system and delay intracellular as defense mechanism were studied by [1,2,6,15,20,27,[41][42][43][44][45][46][47][48]. Other monolytic models, which had focused on the methodological application of treatment strategies and optimal maximization of healthy population, can be found in [1,2,16,28,29,[49][50][51][52][53][54][55][56][57][58][59][60].…”
It has been of concern for the most appropriate control mechanism associated with the growing complexity of dual HIV-HBV infectivity. Moreso, the scientific ineptitude towards an articulated mathematical model for coinfection dynamics and accompanying methodological application of desired chemotherapies inform this present investigation. Therefore, the uniqueness of this present study is not only ascribed by the quantitative maximization of susceptible state components but opined to an insight into the epidemiological identifiability of dual HIV-HBV infection transmission routes and the methodological application of triple-dual control functions. Using ODEs, the model was formulated as a penultimate 7-Dimensional mathematical dynamic HIV-HBV model, which was then transformed to an optimal control problem, following the introduction of multi-therapies in the presence of dual adaptive immune system and time delay lags. Applying classical Pontryagin's maximum principle, the system was analyzed, leading to the derivation of the model optimality system and uniqueness of the system. Specifically, following the dual role of the adaptive immune system, which culminated into triple-dual application of multi-therapies, the investigation was characterized by dual delayed HIV-HBV virions decays from infected double-lymphocytes in a biphasic manner, accompanied by more complex decay profiles of infectious dual HIV-HBV virions. The result further led to significant triphasic maximization of susceptible double-lymphocytes and dual adaptive immune system (cytotoxic T-lymphocytes and humeral immune response) achieved under minimal systemic cost. Therefore, the model is comparatively a monumental and intellectual accomplishment, worthy of emulation for related and future dual infectivity.
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