# American Institute of Mathematical Sciences

2008, 5(3): 585-599. doi: 10.3934/mbe.2008.5.585

## Calculation of $R_0$ for age-of-infection models

 1 Harvard Graduate School of Education, Harvard University, Cambridge, MA 02138, United States 2 Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2

Received  January 2007 Revised  February 2008 Published  June 2008

We consider age-of-infection epidemic models to describe multiple- stage epidemic models, including treatment. We derive an expression for the basic reproduction number $R_0$ in terms of the distributions of periods of stay in the various compartments. We find that, in the model without treatment, $R_0$ depends only on the mean periods in compartments, and not on the form of the distributions. In treatment models, $R_0$ depends on the form of the dis- tributions of stay in infective compartments from which members are removed for treatment, but the dependence for treatment compartments is only on the mean stay in the compartments. The results give a considerable simplification in the calculation of the basic reproduction number.
Citation: Christine K. Yang, Fred Brauer. Calculation of $R_0$ for age-of-infection models. Mathematical Biosciences & Engineering, 2008, 5 (3) : 585-599. doi: 10.3934/mbe.2008.5.585
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