Analysis of the HIV/AIDS transmission dynamics model using Caputo fractional-order derivative

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Résumé

Although national and international institutions, such as the World Health Organization (WHO) and UNAIDS, are making significant efforts to eradicate HIV by 2030, it remains a major threat to global public health. Despite its low prevalence, HIV continues to claim lives and remains a major public health issue, especially in developing countries. Thanks to the accessibility of antiretroviral drugs, the prevalence of this scourge has been gradually declining worldwide in recent years. Thus, the present article investigates antiretroviral therapy's effectiveness in controlling viral transmission through a fractional-order extension of a deterministic model. We study the boundedness of the model's solution by applying the Laplace transform to solve the fractional Gronwall inequality. To ensure the existence and uniqueness of the model's solution, we rely on the Picard-Lindelöf theorem. We also study the stability of the disease-free equilibrium point to qualitatively analyze the behavior of the model. Next, we perform a sensitivity analysis of the basic reproduction number $\mathcal{R}_0$ to evaluate its robustness concerning the model parameters. Finally, we simulate the approximate solutions of the fractional-order model in MATLAB for different values of the fractional order and present the results of the sensitivity analysis and numerical simulation. Our results demonstrate that the fractional model provides real added value in modeling, thanks to its ability to incorporate memory effects and finely tune transmission dynamics according to the fractional order, thereby allowing for a more realistic representation of epidemiological processes.

Mots-clés

HIV/AIDS, deterministic model, fractional order model, sensitivity analysis, numerical simulation