Ultrasound elasticity images, which enable the visualization of quantitative maps of tissue stiffness, can be reconstructed by solving an inverse problem. Classical model-based approaches for ultrasound elastography use deterministic finite element methods (FEMs) to incorporate the governing physical laws leading to poor performance in low SNR conditions. Moreover, these approaches utilize approximate linear forward models discretized by FEMs to describe the underlying physics governed by partial differential equations (PDEs). To achieve highly accurate stiffness images, it is essential to compensate the error induced by noisy measurements and inaccurate forward models. In this regard, we propose a joint model-based and learning-based framework for estimating the elasticity distribution by solving a regularized optimization problem. To address noise, we introduce a statistical representation of the imaging system, which incorporates the noise statistics as a signal-dependent correlated noise model. Moreover, in order to compensate for the model errors, we introduce an explicit data-driven correction model, which can be integrated with any regularization term. This constrained optimization problem is solved using fixed-point gradient descent where the analytical gradient of the inaccurate data-fidelity term is corrected using a neural network, while regularization is achieved by data-driven unrolled regularization by denoising (RED). Both networks are jointly trained in an end-to-end manner.