Bifurcation and Stability Analysis of HIV-1 Coronavirus Co-infection Model

The co-infection of HIV-1 viruses has emerged as a significant threat to global public health as a result of shared mode of transmission. This article presents a novel mathematical model that addresses the dynamics of this co-infection by extending the SVEIR (Susceptible – Vaccinated – Exposed – Infectious - Recovered) framework to incorporate time-delay, chemotherapy and quarantine compartments. The population is divided into twelve compartments, with infections individuals further subdivided into symptomatic and asymptomatic individuals. The mathematical model developed is constrained to adhere to fundamental epidemiology properties such as non-negativity and boundedness within a feasible. We investigate the fundamental reproduction number that guarantees stability of equilibrium points are disease free and endemic qualitative behavior of models are examined. Stability threshold explicitly state that when reproduction number is less than one the disease free equilibrium is globally asymptotically stable, meaning the infection can be eliminated. Using Lyapunov functions, local and global stability of these states are explored and findings presented graphically. They were used to account for the history dependent nature of time delay. Our research assessed control policies and proposed alternatives, performing bifurcation analysis so as to establish prevention strategies. We investigated Hopf bifurcation analytically and numerically to demonstrate disease dynamics, which is novel to our study.. Numerical simulations, performed using the MATLAB dde23 solver, demonstrate that the introduction of chemotherapy and quarantine significantly reduces the peak of symptomatic infections. Crucially, our Hopf bifurcation analysis identifies a critical delay threshold beyond which stable equilibrium is lost to sustained periodic oscillations, representing recurrent waves of infection or rather viral blips. This offered new insights into the long-term management of HIV-1 co-infection cycles.