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This dissertation uses statistical physics based mathematical models to investigate and predict the emergent properties of biological systems in both healthy and diseased states. Three projects are presented, each exploring a different aspect of biological systems. The first project focuses on modeling cartilage-like tissue systems as a double network consisting of two interconnected networks. By using rigidity percolation theory, we study the tunable mechanics and fracture resistance of such biological tissues in healthy and degraded states. Our findings show that the secondary network density can be tuned to facilitate stress relaxation, leading to robust tissue properties when the primary network density is just above its rigidity percolation threshold. However, when the primary network is very dense, the double network becomes stiff and brittle. In the second project, we develop active double network models to investigate the interplay of actin-microtubule interactions and actomyosin dynamics in the cytoskeleton, an active network of proteins in cells. Our study reveals that the rigidity of the composite depends on the interplay between myosin-dependent crosslinking and contraction and the proximity of the actin or microtubule networks to the rigidity percolation threshold. The third project employs a viral quasispecies dynamics model to investigate the effectiveness of antiviral strategies that target different aspects of the lifecycle of cold or flu-like viruses, including COVID-19. Our results demonstrate that antivirals targeting fecundity and reproduction rates decrease the viral load linearly and via a power law, respectively. However, antivirals targeting the infection rate cause a non-monotonic change in the viral load, initially increasing and then decreasing as the infection rate is decreased, particularly for individuals with low immunity. In summary, the findings of these projects underscore the importance of understanding the underlying mechanisms behind the properties of biological cells and tissues. They also provide insights into the development of tunable, resilient, and adaptive biomaterials and the effectiveness of different antiviral strategies for COVID-19 and similar viral diseases.