Bifurcation analysis of macrophages infection model with delayed immune response

• Macrophages infection model is considered with two different types of delays.
• The importance of antiretroviral treatment in the macrophage infection by HIV-1 is proved.
• The existence of Hopf bifurcation with multiple parameters is discussed.
• Role of discrete and distributed delay into the model is explored.
• Numerical results are provided to prove the derived analytical results.

Macrophages are capable of producing large amounts of both intracellular and extracellular infected cells without necessarily succumbing to the lethal effects of productive viral infection. In this manuscript, authors studied the macrophages infection by human immunodeficiency virus type 1 in the form of mathematical models along with two different types of delays. The asymptotic stability of the constructed model without time delay is proved by utilizing the roots of the characteristic quasi-polynomial which is obtained by applying Jacobian matrix method. Based on Routh–Hurwitz criterion, the dynamical properties of model with a delay is investigated. Instead of discrete time delay, the effect of distribution of delays in the immune activation is also analyzed by using linear chain trick technique. In particular, the model undergoes Hopf bifurcation depending on the critical value of single delay in the activation of immune response. Anti-retroviral treatments have been used to control the replication of HIV-1 virus in infected patients is investigated along with time delay. Moreover, it is proved that the existence of two different types of delays may cause the solutions of the model to become stable or unstable depending on the conditions of chosen bifurcation parameter. Numerical simulations are performed to validate the derived theoretical results.