Stability of HHV-8 and HIV-1 co-infection model with latent reservoirs and multiple distributed delays

Human immunodeficiency virus type 1 (HIV-1) gradually destroys the CD4 ${}^{+}$ T cells leading to immune system dysfunction. HIV-1 can result in acquired immunodeficiency syndrome (AIDS) if antiretroviral drugs are not used. HIV/AIDS patients are more vulnerable to opportunistic infections or cancers. Human herpesvirus 8 (HHV-8) targets B cells and causes an AIDS-related cancer known as kaposi sarcoma (KS). Numerous investigations have demonstrated co-infection instances between HIV-1 and HHV-8. In this research, we investigated the co-dynamics of HIV-1 and HHV-8 in vivo using a system of delay differential equations (DDEs). The model explained the interactions between uninfected CD4 ${}^{+}$ T cells, latently/actively HIV-1-infected CD4 ${}^{+}$ T cells, free HIV-1 particles, uninfected B cells, latently/actively HHV-8-infected B cells, and free HHV-8 particles. Eight distributed-time delays were incorporated into the model to account for the delays that arose during the generation of both actively and latently infected cells, the activation of latent reservoirs, and the maturation of freshly discharged virions. By examining the nonnegativity and boundedness of the solutions, we demonstrated that the model was both mathematically and biologically well-posed. We calculated the model's equilibria and threshold numbers. We studied the global asymptotic stability of the model's equilibria by building appropriate Lyapunov functionals and applying the Lyapunov-LaSalle asymptotic stability theorem. Numerical simulations were used to display the results. For the basic reproduction numbers of HHV-8 single-infection ( $R_1$) and HIV-1 single-infection ( $R_2$), sensitivity analysis was carried out. Comparing HIV-1 or HHV-8 single infections with co-infections of HHV-8 and HIV-1 was shown. It's interesting to note that we detected larger amounts of HHV-8 and HIV-1 when they co-infect than when they are infected alone. This outcome aligned with several findings seen in the literature. The effect of antiviral drugs and time delays on the co-dynamics of HIV-1 and HHV-8 was investigated. We found that the delay parameter and drug effectiveness both contributed to a decrease in the basic reproduction numbers, $R_1$ and $R_2$. Less treatment efficacies will be needed to keep the system at the infection-free equilibrium and remove HIV-1 and HHV-8 from the body if a model with time delays is employed.