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Modelling and analysis of an HIV$/$AIDS model with different window period and treatment
Hai-Feng Huo,Li-Na Gu,Hong Xiang
Keywords:Treatment; Window period; Forward bifurcation; Optimal control
Abstract:
      A novel HIV$/$AIDS model with different window period and treatment is proposed. Window period individuals and latent individuals are divided into two types: treated and untreated in our model. The basic reproduction number $R_{0}$ is obtained by using the next generation matrix. Stability of the disease-free equilibrium and existence of the endemic equilibrium is derived. Using the theory of central manifold, the generation of forward bifurcation is established. Existence of the optimal control pair is analyzed and the mathematical expression of the optimal control is also given by the Pontryagin maximum principle. The best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the AIDS data in Gansu province of China from 2004 to 2019. We also estimate that the basic reproduction number $R_{0}$ is $2.1985$ (95\%CI: (1.3535-3.0435)). Numerical simulation and sensitivity analysis of several parameters are also presented. Our results suggest that individuals who are in the stage of window period for AIDS should receive treatment, and this plays a key role in the prevention and control of HIV$/$AIDS.