Reproduction numbers of infectious disease models

Abstract

This primer article focuses on the basic reproduction number, ${ℛ}_0$ , for infectious diseases, and other reproduction numbers related to ${ℛ}_0$ that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of ${ℛ}_0$ determining whether or not the disease dies out. The next generation matrix method of calculating ${ℛ}_0$ in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate ${ℛ}_0$ are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.

Keywords: Anthrax; Basic reproduction number; Cholera; Disease control; West Nile virus; Zika virus.

Fig. 1

Flow diagram for the $SIR$ model.

Fig. 2

Flowchart for the $SEIR$ model.

Fig. 3

Flowchart for the West Nile virus model by Wonham and Lewis (2008).

Fig. 4

Flowchart of the cholera model by Tien and Earn (2010).