Reproduction numbers of infectious disease models

This primer article focuses on the basic reproduction number, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ℛ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> , for infectious diseases, and other reproduction numbers related to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ℛ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ℛ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> determining whether or not the disease dies out. The next generation matrix method of calculating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ℛ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ℛ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.