R₀ and Deterministic Models

Abstract

The basic reproductive number, R₀, is defined as the expected number of new infectious hosts that one typical infectious host will produce during his or her infectious period in a large, completely susceptible population. For example, if R₀ = 5 for mumps in a human population, then one infectious person in that population would be expected to produce five new secondary infectious cases if the population were completely susceptible. If he produced three additional cases who were not infectious, R₀ would still be 5.

For microparasitic infections such as viruses and bacteria, R₀ can be thought of as the product of the contact rate c, the duration of infectiousness d, and the transmission probability per contact with the infectious person, p:

$$R_0 = \begin{array}{*{20}c}{{\rm number \, of}} \\ {{\rm contacts \, per}} \\ {{\rm unit \, time}} \\ \end{array}\,\,\,\begin{array}{*{20}c}\\ \times \\ \end{array}\,\,\,\,\begin{array}{*{20}c} {{\rm transmission}} \\ {{\rm probability}} \\ {{\rm per \, contact}} \\ \end{array}\,\,\,\begin{array}{*{20}c}\\ \times \\ \end{array}\,\,\,\begin{array}{*{20}c} {{\rm duration}} \\ {{\rm of}} \\ {{\rm infectiousness}} \\ \end{array} = cpd.$$

R₀ summarizes many important aspects of an infectious agent in a host population in one parameter. It allows comparison of seemingly disparate diseases from the viewpoint of population biology. A value of R₀ is not specific to an infectious agent, but to an infectious agent population within a particular host population at a particular time. Contact rates relevant for respiratory transmission will be lower in rural areas than in urban areas. So, for example, we expect the R₀ of mumps to be lower in rural than in urban areas. The R₀ of malaria may be low during the season of low mosquito density, but high during the season in which mosquitoes are plentiful. The R₀ of HIV in a sexually active population of single people might be much higher than it is in a population of fairly monogamous married couples.