<p>In this paper, we create a mathematical model for herpes transmission dynamics that includes the impact of vaccination using a compartmental modeling approach. The model is expressed as a system of differential equations, and we prove its key theoretical aspects, such as the existence and uniqueness of solutions. To solve the problem efficiently, we offer a new deep learning-based method that takes advantage of a dual-layer neural network architecture. This method combines two different activation functions radial basis and hyperbolic tangent to improve accuracy and prediction performance. By combining two complementing functions, our method achieves excellent precision when approximating solutions for the herpes disease model, proving its promise as a dependable tool for epidemiological analysis and decision making.</p>