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

•A mathematical model of HIV-1 infection with contact rate represented by Crowley-Martin function response is taken into account. The HIV-1 model of CD4 positive T cells is being studied.•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.•At threshold quantity R0 > 1, the steady state of chronic disease is globally stable.•In to reduce the density of infected cells and viruses as well as maximize the density of healthy cells is determined by the objective functional.