Fractional Complex-order Hopfield Neural Networks to Analyze the Effect of Drug-resistance in the HIV Infection

The present paper uses a complex and fractional-order model for Hopfield neural networks to set a nonlinear model that represents the quantity of infected/uninfected CD4+T cells into the HIV dynamic when an antiviral therapy based on protease inhibitors is applied. By using a mathematical model of the environment associated with CD4+T cells that are progressively infected, it is proposed a closed-loop scheme associated with the HIV dynamic and its antiviral therapy, both acting in the same environment. To this end, the work reported here will use Caputo-type derivatives in order to represent such closed-loop dynamic by assuming that there is a nonlinear model based on Hopfield neural networks (HNN). In this way, the mutual interference between the additive activation dynamics of HNN and the complex-valued fractional-order analysis will be used to study the local and global asymptotic stability of HIV. The effect of drug-resistance will be the main starting point to understand how the resistant CD4+T cells can be reduced. The equilibrium point of HNN model will be studied by using quadratic-type Lyapunov functions and compared with a model based on Grünwald-Letnikov formulation in order to validate the approach proposed. The results show that HNN-based model converges toward a small neighborhood of the origin with better performance than Grünwald-Letnikov model.