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A delayed model describing the dynamics of HIV (Human Immunodeficiency Virus) with CTL (Cytotoxic T Lymphocytes) immune response is investigated. The model includes four nonlinear differential equations describing the evolution of uninfected, infected, free HIV viruses, and CTL immune response cells. It includes also intracellular delay and two treatments (two controls). While the aim of first treatment consists to block the viral proliferation, the role of the second is to prevent new infections. Firstly, we prove the well-posedness of the problem by establishing some positivity and boundedness results. Next, we give some conditions that insure the local asymptotic stability of the endemic and disease-free equilibria. Finally, an optimal control problem, associated with the intracellular delayed HIV model with CTL immune response, is posed and investigated. The problem is shown to have an unique solution, which is characterized via Pontryagin's minimum principle for problems with delays. Numerical simulations are performed, confirming stability of the disease-free and endemic equilibria and illustrating the effectiveness of the two incorporated treatments via optimal control.

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The mathematical model of the human immunodeficiency virus (HIV) pathogenesis with the adaptive immune response, two saturated rates and therapy is presented and studied in this work. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies; the two saturated rates describe the viral infection and the CTL proliferation. Two kinds of treatments are incorporated into the model; the objective of the first one is to reduce the number of infected cells, while the aim of the second one is to block the free viruses. The positivity and boundedness of solutions are established. The local stability of the disease free steady state and the infection steady states are studied. Numerical simulations are performed to show the behavior of solutions and the effectiveness of the incorporated therapy in controlling the HIV replication which can improves significantly the patient's life quality.

In this paper, we will study mathematically and numerically the dynamics of the hepatitis C virus disease with the consideration of two fundamental modes of transmission of the infection, namely virus-to-cell and cell-to-cell. In our model, we will take into account the role of cure rate of the infected cells and the effect of the adaptive immunity. The model consists of five nonlinear differential equations, describing the interaction between the uninfected cells, the infected cells, the hepatitis C virions and the adaptive immunity. This immunity will be represented by the humoral and cellular immune responses. This work begins with proving the non-negativity and the boundedness of solutions and determining the basic reproduction number. Secondly, five equilibria are established, the local stability analysis for all the equilibria is demonstrated theoretically and numerically. Finally, we have concluded that the numerical results are coherent with our theoretical postulations.