In addition to starting broadly and narrowing down to your thesis, another common way to begin an essay of the sort you describe is to start with something opposite of what you're writing about, or, to rephrase and explain, to start with something similar yet different.

In your case, you could start with the traditional flu.  If you know the history, symptoms, and treatment of traditional flue, you can start with them, then lead into the swine flu.

Another option is to start with the common cold.  This would be particularly relevant, since people seem to struggle with the ability, or inability, to tell the difference.  This is vital because a cold doesn't usually require a doctor's visit or hospitalization, while the swine flu might. 

Starting with something similar yet different, can be a way to naturally introduce your topic and thesis.

Generally speaking, most introductions start with broad, general information about the topic itself and then as the paragraph progresses the scope of the ideas narrow until the thesis statement is reached. Think of a funnel for a moment. At the top of the funnel the opening is wide, but at the bottom of the funnel the opening is far narrower. Many introductions follow this same patter as well. To catch your reader's attention at the very beginning you might try utilizing a startling statistic, or a thought-provoking question, or profound quotation from an expert. Comb through your research one last time and look for anything that you believe would grab the attention of a potential reader. Try starting with that idea.

Avoiding "I," "me," and "myself," etc. is often just a matter of focusing on the topic and not on one's self. So instead of writing "I believe the Swine Flu is the most dangerous epidemic of the 21st century," you could simply write, "The Swine Flue is the most dangerous epidemic of the 21st century." Because you are writing an essay, we know this is, in part, the opinion of the author.