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Mathematical Biosciences and Engineering (MBE)
 

Epidemic threshold conditions for seasonally forced SEIR models

Pages: 161 - 172, Volume 3, Issue 1, January 2006      doi:10.3934/mbe.2006.3.161

 
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Junling Ma - Department of Mathematics and Statistics, McMaster University, Hamilton, ON Canada L8S 4K1, Canada (email)
Zhien Ma - Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China (email)

Abstract: In this paper we derive threshold conditions for eradication of diseases that can be described by seasonally forced susceptible-exposed-infectious-recovered (SEIR) models or their variants. For autonomous models, the basic reproduction number $\mathcal{R}_0 < 1$ is usually both necessary and sufficient for the extinction of diseases. For seasonally forced models, $\mathcal{R}_0$ is a function of time $t$. We find that for models without recruitment of susceptible individuals (via births or loss of immunity), max$_t{\mathcal{R}_0(t)} < 1$ is required to prevent outbreaks no matter when and how the disease is introduced. For models with recruitment, if the latent period can be neglected, the disease goes extinct if and only if the basic reproduction number $\bar{\mathcal{R}}$ of the time-average systems (the autonomous systems obtained by replacing the time-varying parameters with their long-term time averages) is less than 1. Otherwise, $\bar{\mathcal{R}} < 1$ is sufficient but not necessary for extinction. Thus, reducing $\bar{\mathcal{R}}$ of the average system to less than 1 is sufficient to prevent or curtail the spread of an endemic disease.

Keywords:  epidemic models, seasonal forcing, basic reproduction number.
Mathematics Subject Classification:  92D30.

Received: January 2005;      Accepted: April 2005;      Available Online: November 2005.