arXiv:2610.36924v1 Announce Type: new Abstract: This paper establishes a fully implicit model to investigate the spread of epidemics and studies its dynamic properties, extending the semi-implicit model in Yang et al.(Commun. Nonlinear Sci. Numer. Simul., 152 (2026) 109140 ). We first employ the split-step backward Euler method to address the non-adaptability arising from backward Euler discretization and ensure the global convergence of the numerical solution. By linearizing the model, the computational cost is reduced at the expense of a slight increase in error, effectively avoiding complex iterative solution processes. Then, we derive the existence, uniqueness, positivity, and boundedness of the numerical solution, and obtain its global convergence order. On this basis, the extinction and persistence of the disease are investigated under three different scenarios. The results indicate that environmental noise is a key factor leading to disease extinction. Finally, numerical simulations visually verify the obtained theoretical conclusions.