The Volterra–Lyapunov matrix theory for global stability analysis of a model of the HIV/AIDS

Abstract

In this paper, we analyze a nonlinear mathematical model of the HIV/AIDS and screening of unaware infectives on the transmission dynamics of the disease in a homogeneous population with constant immigration of susceptibles incorporating use of condom, screening of unaware infectives and treatment of the infected. We consider constant controls and thereafter by incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis of HIV/AIDS. The analysis and results presented in this paper make building blocks toward a comprehensive study and deeper understanding of the fundamental mechanism in HIV/AIDS. A numerical study of the model is also carried out to investigate the analytical results.

Global stability analysis of the role of multi-therapies and non-pharmaceutical treatment protocols for COVID-19 pandemic

“…If define the endemic equilibrium point, then with Eq. (8) , the differential sum of the system (1) at C-19EE, is derived as: [24] or which corresponds to the fact that at equilibrium, . If , then there exists disease endemicity, which in terms of , we have or provided with as the disease constant derived from Eqs (9) to (13) .…”

“…Here, with the incorporation of the LaSalle's invariant principle, we shall adopt the Lyapunov function for the investigation of the global stability of system endemic equilibrium (C-19EE). This approach, which had been explored by [ 24 , 31 ] is found useful in compartmental epidemic models. Then, the following theorem establishes the system global stability of C-19EE.…”

Global Stability Analysis of the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics

“…For instance, using mathematical model, the effect of screening of unaware HIV/AIDS infectives on the spread of HIV infection without any control function (treatments) have been investigated, [21]. Improving on the above finding while sustaining the screening method, similar model with the incorporation of condom use by the susceptible and application of control function (ART) by aware infective class was proposed and studied, [22]. Notably, a review of these models shows that the behavioral attitude to treatment consistency by those who were screened to become aware and receiving treatment was not considered.…”

“…Presumably, taking into account of the fact that the above model was developed in the absence of treatment and postexposure intervention but rather with the effect of screening of unaware infectives, similar model with the introduction of treatment functions both at the susceptible stage (condom use) and those who were screen and placed under ART treatment was proposed and studied, [22]. For clarity of presentation, while urging readers to access the model details and assumptions, the differential equations for the model was derived as:…”

“…Here, no direct incorporation of treatment function, noting that upon been screened and becoming aware, sexual interaction were bound to change resulting to reduction of the endemicity of the infection. On the other hand, improving on the above findings led to a study of similar model with the incorporation of condom use by the susceptible and application of treatment control function (ART) by the aware infectives class, while sustaining the screening method, [22]. The model was analyzed using the theory of Volterra-Lyapunov stable matrices on classical method of Lyapunov functions.…”