<p>Parameter nonidentifiability is a critical challenge in infectious disease modeling, where infinitely many parameter values produce equally good fits to observed data but lead to significantly different future predictions. Many methods have been developed to address this issue, including mathematical analysis, computational techniques, and statistical approaches. While each provides valuable insights, the integration of computationally efficient identifiability analysis with Bayesian inference for practical parameter estimation has received relatively less attention. In this paper, we incorporate a sensitivity matrix based identifiability analysis into a Bayesian framework to assess parameter identifiability. By examining identifiability under prior distribution, we design Markov Chain Monte Carlo (MCMC) algorithms that integrate identifiability information to enhance the mixing and efficiency of the sampler. Posterior identifiability analysis can then be performed using the sampling results to assess the practical nonidentifiability of a model. By comparing the posterior nonidentifiability results across different models, our method enables principled model selection strategies that penalize nonidentifiable models within a rigorous Bayesian setting. Numerical studies confirm that widely used epidemic models such as SIR, SEIR, and SEIAR are often practically nonidentifiable when calibrated with limited data, underscoring the importance of model parsimony.</p>