Dynamics of spatio-temporal HIV–AIDS model with the treatments of HAART and immunotherapy

In this paper, we focus on the study of HIV–AIDS model in space and time that is adapted from the previous study in Ammi et al. (Sci Rep 12:5751, 2022) without the fractional-order derivative. The fixed controls of highly antiretroviral and immunotherapy are considered for the interaction between susceptible and infected CD4+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^+$$\end{document}T cell. The equilibrium points of disease-free and endemic, positivity, boundedness and basic reproduction number of dynamical system are provided in the standard ways. For the local stability, the Fourier series is firstly employed to obtain the Jacobian matrix which is then used for further analysis of stability. Moreover, the classical numerical scheme of standard finite difference is applied to approximate our model. The stability, positivity, and consistency of numerical scheme are very important in the numerical analysis. At the last section of numerical analysis, we provide the experiments of our model numerically by varying values for the parameters of treatment HAART and Immunotherapy. We can conclude that the combinations of HAART and Immunotherapy at once are the most efficient in decreasing the infected CD4+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^+$$\end{document}T cells and the treatment of immunotherapy is more effective than the treatment of HAART. Our dynamical system is eligible to predict the spread of HIV–AIDS based on the validation results with the actual data by using least square technique. Moreover, we apply the ARIMA(1,1,0) model in this paper to predict infected profile and the result has the similar trend (decreasing trend) with the HIV–AIDS model (obtained from the least square technique) and the actual data. Moreover, we employ the neural network for dynamical system, due to the significant results of best validation performance, error histogram, and regression.