There are a number of hadith in both Shi'i and Sunni hadith collections which describe types of honorary martyrdom; that is, they describe people who die in certain ways as "martyrs". (This ranges from things as dying from drowning, to dying with love of Ahl al-Bayt (A), to dying while sleeping with wudu.)
From these narrations, it can be gleaned that there is a certain respect or acknowledgment given to people who pass away in these ways, and/or some of the acts (obviously the voluntary ones, not involuntary ones) are respectable.
However, my understanding that the formal or highest notion of martyrdom is reserved for those who give their lives in the way of Allah (and Allah is the One who knows who they are). Perhaps one could consider someone who is intentionally putting themself at risk of infection to serve the people, and then dies, as this sort of martyr.
In the Sunni hadith collection of Sahih Muslim, it is related that the Prophet (S) described five types of people as martyrs; among those are those people who die from the plague. I am not sure offhand if this narration is transmitted via Shi'i chains of narration or in Shi'i books. However, this narration (or others similar to it) is the source of the idea, being circulated today, that people who die from the plague are martyrs.
However, with respect to Shi'i texts, it is related that Imam Reza (A) told someone: "Every beliver whom God afflicts with a misfortune and who exhibits patience over it, shall surely come to possess the rank and recompense of a martyr in the eyes of God.” The person who was told this did not understand why the Imam was telling him this, as he was not going through any major difficulties at the time, but he soon developed a painful and difficult condition that lasted for a number of months, and then he died - the implication here being that his patience over his illness granted him the status of martyrdom.
God knows best.