Modeling Epidemics: A Primer and Numerus Software Implementation

Wayne M. Getz; Richard Salter; Oliver Muellerklein; Hyun S. Yoon; Krti Tallam

The properties of Equations 2 have been extensively studied over the past three to four decades [10] [11] [12] 22] , with the most important results pertaining to the pathogen-invasion or disease-outbreak condition. This is informally derived here for the case τ = βI/N , λ(t) = µN (i.e., individual birth and death rates are the same) and ν = 0 from the following considerations. Each infectious individual infects susceptible individuals at a rate βS/N (≈ β when S ≈ N ) over an infectious period that lasts on average for a time 1/(γ + α + µ). However, only a proportion σ/(σ + µ) of infected individuals become infectious, due to natural and diseaseinduced mortality rates while in state E. Thus the number R 0 of susceptible individuals that each infectious individual is expected to infect at the onset of an epidemic is given by