Issue Math. Model. Nat. Phenom. Volume 3, Number 7, 2008 Special issue dedicated to Glenn Webb Page(s) 194 - 228 DOI 10.1051/mmnp:2008050 Published online 23 October 2008

Math. Model. Nat. Phenom. Vol. 3, No. 7, 2008, pp. 194-228
DOI: 10.1051/mmnp:2008050

The Basic Reproduction Number of an Infectious Disease in a Stable Population: The Impact of Population Growth Rate on the Eradication Threshold

H. Inaba¹ and H. Nishiura<sup>2

1</sup>  Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
²  Theoretical Epidemiology, Faculty of Veterinary Medicine, University of Utrecht, Yalelaan 7, 3584 CL, Utrecht, The Netherlands

inaba@ms.u-tokyo.ac.jp

Published online: 23 October 2008

Abstract
Although age-related heterogeneity of infection has been addressed in various epidemic models assuming a demographically stationary population, only a few studies have explicitly dealt with age-specific patterns of transmission in growing or decreasing population. To discuss the threshold principle realistically, the present study investigates an age-duration-structured SIR epidemic model assuming a stable host population, as the first scheme to account for the non-stationality of the host population. The basic reproduction number R₀ is derived using the next generation operator, permitting discussions over the well-known invasion principles. The condition of endemic steady state is also characterized by using the effective next generation operator. Subsequently, estimators of R₀ are offered which can explicitly account for non-zero population growth rate. Critical coverages of vaccination are also shown, highlighting the threshold condition for a population with varying size. When quantifying R₀ using the force of infection estimated from serological data, it should be remembered that the estimate increases as the population growth rate decreases. On the contrary, given the same R₀, critical coverage of vaccination in a growing population would be higher than that of decreasing population. Our exercise implies that high mass vaccination coverage at an early age would be needed to control childhood vaccine-preventable diseases in developing countries.

Mathematics Subject Classification. 92D25, 92D30

Key words: epidemiological model -- stable age distribution -- threshold -- basic reproduction number -- SIR model

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