# American Institute of Mathematical Sciences

2006, 3(1): 161-172. doi: 10.3934/mbe.2006.3.161

## Epidemic threshold conditions for seasonally forced SEIR models

 1 Department of Mathematics and Statistics, McMaster University, Hamilton, ON Canada L8S 4K1, Canada 2 Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China

Received  January 2005 Revised  April 2005 Published  November 2005

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.
Citation: Junling Ma, Zhien Ma. Epidemic threshold conditions for seasonally forced SEIR models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 161-172. doi: 10.3934/mbe.2006.3.161
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