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In this paper, we rigorously analyze an HIV-1 infection model with CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model with proving the positivity and bound-edness of the solution. For this model, the basic reproduction number R0 and the immune reproduction number R1 are identified. Moreover, we have shown that the model has three e-quilibria, namely the infection-free equilibrium E0, the infectious equilibrium without immune response E1 and the infectious equilibrium with immune response E2. By applying uctuation lemma and Lyapunov functionals, we have demonstrated that the global stability of E0 and E1 are only related to R0 and R1. The local stability of the third equilibrium is obtained under four situations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, some numerical simulations are carried out for illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

In the present paper, we propose and analyze a nonlinear fractional-order SEIR (susceptible-exposed-infected-recovered) epidemic model to transmit HIV. The fixed points of the model and their stability results are obtained. Using the fractional derivatives, we relied on the Caputo fractional derivative. Further, we employed the homotopy analysis method (HAM) to get an approximate solution of the dynamic fractional derivatives of the model. The purpose of using HAM as a solution technique is its reliability, easy to handle, that utilizes a simple process to adjust and control the convergence region of the obtained infinite series solution. It uses an auxiliary parameter and allows to obtain a one-parametric family of explicit series solutions. Firstly, several h-curves are plotted to demonstrate the regions of convergence, then the residual and square residual errors are obtained for different values of these regions. In the end, numerical solutions are presented for various iterations to show the accuracy of the HAM. Besides, the convergence theorem of HAM is also proved. The obtained results show the effectiveness and strength of the applied HAM on the proposed fractional-order SEIR model. Also, from the sensitivity analysis results, it is seen that the parameters µ and σ are more sensitive than ϵ and ρ in disease transmission. [ABSTRACT FROM AUTHOR]