A German version of this content written in a different style can be found here.
The "corona crisis" is mostly a crisis of the current economic system: profit-oriented health systems lead to high death counts and state interventions are needed to prevent an economic breakdown. So let's see how anarchist societies would handle a pandemic. Disclaimer: there are diverse versions of anarchist theories. Their common ground is the love for freedom and solidarity. The following is not a blueprint for an anarchist society but some inspirations for possibilities.
First, there is a lower risk for a pandemic in anarchist societies because the need for business-related intercontinental travels will be reduced, intensive livestock farming will be frowned upon both for ecological and animal rights reasons. Nature gets more into balance again.
However, should there still be a pandemic, people would be prepared: The health system will be freely available for everyone. The capacity of the health system will be optimized for the needs of the people instead of profit. Also, as knowledge and scientific research will be open, everyone who is interested can participate in figuring out action plans for various possible disasters even before they arise.
In general, it will become easier to trust experts, scientists, and media because they will all act in the interest of the people and not in the economic interest of cooperations. Thus, in the event of a pandemic, well-researched information and suggested measures can be provided by domain experts to everyone. Based on that information and their local needs, neighborhood communities and regions can decide what to do. The decision will involve everyone interested and impacted. As these decisions are routine for members of anarchist societies, they can be done quite fast e.g. via the internet. Each region can have its own way of handling the pandemic as long as this does not sabotage the way neighboring regions handle it.
E.g. one community might decide to have quarantine houses for those that fear harsh consequences when getting the virus and let others proceed as usual. Another community might implement an extensive virus detection system to be able to isolate infected people as fast as possible. Again other communities might ask for healthy volunteers to get infected on purpose and reach herd immunity after some time. As solidarity is one of the core principles of anarchy, all of the communities or regions are open to visitors from other places as long as they stick to the agreements of the region they visit. That way, individuals who do not like the decisions and agreements in their region are free to move to another region. E.g. communities isolated in nature might host some elderly people from other communities and sick people of those nature-based communities could be moved to hospitals in the cities.
There will be a different mental model: People would not have to risk their health for economic reasons. Also, there would be no discrimination against people with health issues. On the contrary, people will be used to listening to the needs and concerns of everyone involved, and finding solutions that respect various needs would be a social technique practiced daily.
Thus, each community can decide about their balance between freedom and security and define for themselves the ways of implementing solidarity in times of pandemic.
However, if it turns out that the pandemic can't be stopped with decentral diversity, there might be a need for a global decision. This will not be out of the ordinary. There are topics like climate change, scarce resources, and pandemics which affect the whole planet. Thus, people will have figured out how to make those decisions (e.g. online discussions and decision-making using tools that capture concerns as well as the amount of support for various suggestions). In the case of the pandemic, the global decision might be a planet-wide lockdown for two weeks.
Production of goods in anarchist societies has a focus on local production and decentral planning with cross-regional cooperations and support. Such a system is resilient to crises and can adapt quickly to modified needs in times of a pandemic.
Stay safe and question the system.