Invariant Region, Endemic Equilibria and Stability Analysis

“…This gives N(t) = N(0) as t→0 and N(t)0 as t→. Applying Birkh and Rota's theorem on differential inequality as obtained in (Mafuta et al, 2014), for all t→, we arrive at 0…”

Section: Proofmentioning

confidence: 86%

Optimal Control Techniques for the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics

Bassey¹,

Henry²

2022

Journal of Mathematics and Statistics

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From the studies of HIV/AIDS transmission and treatment dynamics using mathematical modeling, literature reviews have shown that attention had not been given to the behavioral attitude of screen-aware infectives not ready to receive treatment, HIV-aware infectives that initiated treatment but truncated only to resume treatment later (therapy abuse) and those on consistent treatment protocols. Moreso, following the non-outright eradication of the deadly HI-virus, recommendations have been geared towards exploring optimal control theory for the maximization of healthy uninfected CD4 + T-cells. Therefore, this present investigation seeks and formulated an optimal control 6-Dimensional deterministic mathematical dynamic model, which accounted for the Role of Antiretroviral Therapy (ART) abuse in the treatment dynamics of the HIV/AIDS epidemic. The materials and methods for this model are constituted by a set of 6-Dimensional varying subpopulations interacting with concentrated HI-viral load. Interactions are investigated using bilinear control functions (condom use and ART) with empirically generated data. The model assumed a deterministic approach and was formulated using the fundamental theory of differential equations. Theoretical optimal predictions explored classical numerical methods with optimal control techniques (Pontryagin's maximum principle in conjunction with Hessian matrix) as a basis. Numerical simulations were conducted using in-built Runge-Kutta of the order of precision 4 in a Mathcad surface. Following the derived model for both offoptimal control and onset-optimal control functions and model optimal control pair as well as model optimality system, results of simulations indicated that at off-optimal control function, near zero population extinction was observed. From the application of optimal control functions under optimal control techniques, there exists tremendous rejuvenation of susceptible populations vindicated by a reduction in the rate of ART abuse under a minimal proportion of bilinear control functions. The study concluded that adopting optimal control techniques for the investigation of the role of ART abuse in HIV/AIDS treatment yield highly significant recovery of healthy CD4 + T-Cells at minimal systemic cost when compared with off-optimal control outcome. Therefore, the study not only affirmed the vital concept of optimal control strategy but also, instituted the viability of the model. Thus, this model can be extensively used in Bio-system and applied mathematics.

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“…Using differential inequality theorem from [17] and the Birkhoff and Rota version as used in [11,18],…”

mentioning

confidence: 99%

Optimal Control Analysis of Treatment Strategies of the Dynamics of Cholera

Abubakar

Ibrahim

2022

Journal of Optimization

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A nine-compartment deterministic cholera model was formulated, and the model describes interactions between human, Vibrio cholerae bacteria, and the enviroment that warrant the interaction. Realities and socioeconomic burden influence spread and control mechanism of the disease. The model investigated some effective ways of hindering cholera outbreak and spread. The existence and uniqueness of solution of the system of equations that the model comprises were ascertained. The basic reproduction number R 0 of the model was obtained using “next-generation matrix” method, and the most sensitive parameters were identified using “normalised forward sensitivity index” method. Three controls, hygiene consciousness denoted by X1, cholera vaccine X2, and cholera awareness programme X3, were chosen. Optimal control theory is applied to ascertain the level of effect of the controls in reducing susceptible, exposed, infected individuals and causative pathogen population. Pontryagin’s maximum principle is used to prove the optimal solution of the model, and the optimal system was derived and numerically solved. Simulations were made with graphs that show the effects of the controls on susceptible, exposed, infected, and Vibrio cholerae population. The findings are that simultaneous application of the three controls can be one of the fast and effective ways of controlling cholera. If two controls are to be selected, hygiene consciousness and vaccine are the best combination.

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Invariant Region, Endemic Equilibria and Stability Analysis

Cited by 4 publications

References 6 publications

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

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

Optimal Control Techniques for the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics

Optimal Control Analysis of Treatment Strategies of the Dynamics of Cholera

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