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Cholera Models with Hyperinfectivity and Temporary Immunity

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Abstract

A mathematical model for cholera is formulated that incorporates hyperinfectivity and temporary immunity using distributed delays. The basic reproduction number \(\mathcal{R}_{0}\) is defined and proved to give a sharp threshold that determines whether or not the disease dies out. The case of constant temporary immunity is further considered with two different infectivity kernels. Numerical simulations are carried out to show that when \(\mathcal{R}_{0}>1\), the unique endemic equilibrium can lose its stability and oscillations occur. Using cholera data from the literature, the quantitative effects of hyperinfectivity and temporary immunity on oscillations are investigated numerically.

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Acknowledgements

This work originated through a workshop at the Banff International Research Station for Mathematical Innovation and Discovery on “Modelling and Analysis of Options for Controlling Persistent Infectious Diseases,” organized by J. Dushoff, D. Earn, and D. Fisman. The research of ZS and PvdD was supported in part by a Natural Science and Engineering Research Council of Canada (NSERC) Postdoctoral Fellowship, an NSERC Discovery Grant, and the Mprime project “Transmission Dynamics and Spatial Spread of Infectious Diseases: Modelling, Prediction and Control.” JHT acknowledges support from the National Science Foundation (OCE-115881), and from the Mathematical Biosciences Institute (NSF DMS 0931642). We thank two anonymous reviewers for helpful comments and some references.

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  1. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    Zhisheng Shuai & P. van den Driessche

  2. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Joseph H. Tien

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  1. Zhisheng Shuai
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Shuai, Z., Tien, J.H. & van den Driessche, P. Cholera Models with Hyperinfectivity and Temporary Immunity. Bull Math Biol 74, 2423–2445 (2012). https://doi.org/10.1007/s11538-012-9759-4

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