When a population is first exposed to a disease, the density of infected individuals (denoted I) increases from zero to a low value greater than zero. For the disease to establish and spread among members of the population, the density of infected individuals must continue to rise—otherwise, the disease would die out (not establish) or the density of infected individuals would remain very low and hence the disease would establish but not spread.
We can think of the group of individuals that are infected with a disease as a population that is part of a larger population that also includes individuals that are not infected. When the infected individuals are viewed as a population, we see that for the density of infected individuals to increase over time, the growth rate of this population must be greater than zero. If that were not the case, over time the density of infected individuals would either remain constant (if the growth rate equaled zero) or decrease (if the growth rate was less than zero).
Since the density of infected individuals is represented by I, the growth rate of the population of infected individuals is dI/dt. Thus, expressed mathematically, the growth rate of our population of infected individuals will be greater than zero when
Based on textbook Equation 13.1, we see that
when
This occurs when βSI > dI, or S > d/β.
Hence, the density of infected individuals will increase over time when S > d/β, in which case the disease will establish and spread.
