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Our study investigates the dynamics of disease interaction and persistence within populations, exploring various epidemic scenarios, including backward bifurcation and cross-immunity effects. We establish conditions under which the disease-free equilibrium of the model demonstrates local or global asymptotic stability, contingent on the efficacy of quarantine measures. Notably, we find that a strain with a quarantine reproduction number greater than 1 will out-compete a strain with a quarantine reproduction number less than 1, leading to its extinction under complete immunity conditions. Additionally, we identify scenarios where diseases persist in a sub-critical coexistence endemic equilibrium, despite one control reproduction number being below one. Our exploration of backward bifurcation reveals the model's capacity to accommodate the coexistence of the disease-free equilibrium with up to four endemic equilibria. Moreover, we demonstrate that the existence of cross-immunity enhances the coexistence of two strains. However, co-infections and imperfect quarantine measures pose significant challenges in containing outbreaks, sustaining the outbreak potential even with successful control of individual virus strains. Conversely, controlling outbreaks becomes more manageable in the absence of co-infections, especially with perfect quarantine measures. We conclude by advocating for public health strategies that address the complexities posed by co-infections, emphasizing the importance of simultaneously tackling multiple pathogens.