When the zombiepocalypse comes, most people will likely hole up in their homes in favor of self-preservation. In some areas it is likely that people will be able to survive on their own without assistance from the government for weeks, possibly years. But while these people remain locked behind doors and barricades, who will be handling recyclables and waste management? While eradication of the zombie scourge is a top priority, civil undertakings should not be neglected. Most assuredly people will be making do and reusing what they can under the circumstances, but there will be refuse and other wasted materials no doubt filling the streets. Without regular trash collections and ongoing recycling, garbage will pile up higher than a zombie staircase trying to reach the last fat cat stuck on the second balcony at the Met. By continuing waste disposal service in the midst of adversity, humankind supports sustainability on a number of fronts and proves itself as a species worth saving.
Broadly speaking, sustainability is the ability to endure, so it is doubly appropriate with respect to zombies. As zombies are for the most part unique to humans (depending on the source material), they pose a singular threat to peoplekind. Zombies threaten all aspects of human life, from politics to society, economics, etc. Hence it is appropriate that people would take a stand against zombies to preserve the status quo.
But sustainability also relates to ecology and the environment. Creating a sustainable environment also promotes the status quo insofar as slowing down the ravaging effects of industrialization and civilization in general. Sustainable environments can also ensure a healthier future with reduced greenhouse gases and increased flora and fauna populations over time which adds much to diversity in nature. Should the environment be tainted by an overzealous human race ignorant of what is happening, zombies won't be the only thing threatening their existence.
By continuing to be the custodians of the Earth, humans will prove to be worthy caretakers who deserved to have inherited it from the dinosaurs. Furthermore keeping the streets clean would likely reduce the transmission of fatal infections and diseases that would only serve to increase the zombie state. People really have to do their part in maintaining a sustainable environment if they hope to survive. By focusing solely on the zombie dilemma while neglecting other areas of life, humans will paint themselves into a corner where it will be all to easy for the zombies to trap them and consume their delectable brains.