HIV-1 infection dynamics and optimal control with Crowley-Martin function response

Abstract

Background and objective: As we all know, mathematical models provide very important information for the study of the human immunodeficiency virus type. Mathematical model of human immunodeficiency virus type-1 (HIV-1) infection with contact rate represented by Crowley-Martin function response is taken into account. The aims of this novel study is to checkthe local and global stability of the disease and also prevent the outbreak from the community.

Methods: The mathematical model as well as optimal system of nonlinear differential equations are tackled numerically by Runge-Kutta fourth-order method. For global stability we use Lyapunov-LaSalle invariance principle and for the description of optimal control, Pontryagin's maximum principle is used.

Results: Graphical results are depicted and examined with different parameters values versus the basic reproductive number R₀ and also the plots with and without control. The density of infected cells continued to increase without treatment, but the concentration of these cells decreased after treatment. The intensity of the pathogenic virus before and after the optimal treatment. This indicates a sharp drop in the rate of pathogenic viruses after treatment. It prevents the production of viruses by preventing cell infection and minimizing side effects.

Conclusions: We analysed the model by defining the basic reproductive number, showing the boundedness, positivity and permanence of the solution, and proving the local and global stability of the infection-free state. We show that the threshold quantity R₀ < 1, the elimination of HIV-1 infection from the T cell population, is eradicated; while for the threshold quantity R₀ > 1, HIV-1 infection remains in the host. When the threshold quantity R₀ > 1, then it shows that the steady-state of chronic disease is globally stable. Optimal control strategies are developed with the optimal control pair for the description of optimal control. To reduce the density of infected cells and viruses as well as maximize the density of healthy cells is determined by the objective functional.

Keywords: Basic reproductive number; Boundedness; HIV-1 infection; Lyapunov-LaSalle invariance principle; Numerical results; Optimal control pair; Permanence; Positive invariance; Stability.