<p>In this paper, a stochastic HIV/AIDS model with a bilinear incidence rate is established by perturbing the incidence coefficient. Solving this model is crucial for quantitatively studying HIV/AIDS transmission. However, owing to the complexity and nonlinearity of the random terms, an analytical solution is unattainable, making it necessary to find a suitable numerical method. In practical applications, ensuring the positivity of the numerical solution is key. To achieve this, we perform a logarithmic transformation on the model to obtain a new system. It is note that the drift and diffusion coefficients of the new system grow exponentially, failing to meet monotonicity conditions. We then use the truncated Euler–Maruyama (TEM) method to obtain the numerical solution of the transformed system and derive the positivity-preserving log-truncated EM (PPLTEM) scheme for the original system through an inverse transformation. Importantly, the logarithmic transformation alters the coupling intensity among the variables, which has a nontrivial influence on the convergence rate and stability of the numerical solution. Therefore, we begin by showing the existence and uniqueness of the analytical solution, and then carry out a deep and comprehensive analysis of the boundedness and exponential integrability of both the numerical and analytical solutions with the Burkholder–Davis–Gundy inequality, Bihari’s inequality, etc. Through this process, we derive detailed and comprehensive analytical conclusions pertaining to the convergence, convergence rate, and stability of the numerical solution. Finally, the effectiveness of the theoretical results of the numerical method is illustrated via numerical experiments.</p>