Discrete epidemic models

Abstract

The mathematical theory of single outbreak epidemic models reallybegan with the work of Kermack and Mackendrick about $8$ decadesago. This gave a simple answer to the long-standing question of whyepidemics woould appear suddenly and then disappear just as suddenlywithout having infected an entire population. Therefore it seemednatural to expect that theoreticians would immediately proceed toexpand this mathematical framework both because the need to handlerecurrent single infectious disease outbreaks has always been apriority for public health officials and because theoreticians oftentry to push the limits of exiting theories. However, the expansionof the theory via the inclusion of refined epidemiologicalclassifications or through the incorporation of categories that areessential for the evaluation of intervention strategies, in thecontext of ongoing epidemic outbreaks, did not materialize. It wasthe global threat posed by SARS in $2003$ that caused theoreticiansto expand the Kermack-McKendrick single-outbreak framework. Mostrecently, efforts to connect theoretical work to data have explodedas attempts to deal with the threat of emergent and re-emergentdiseases including the most recent H1N1 influenza pandemic, havemarched to the forefront of our global priorities. Since data arecollected and/or reported over discrete units of time, developingsingle outbreak models that fit collected data naturally isrelevant. In this note, we introduce a discrete-epidemic frameworkand highlight, through our analyses, the similarities betweensingle-outbreak comparable classical continuous-time epidemic modelsand the discrete-time models introduced in this note. The emphasisis on comparisons driven by expressions for the final epidemic size.

Keywords

• epidemic
• final size
• single outbreak
• continuous time epidemic models.
• discrete epidemic models