# American Institute of Mathematical Sciences

2005, 2(3): 591-611. doi: 10.3934/mbe.2005.2.591

## Use Of A Periodic Vaccination Strategy To Control The Spread Of Epidemics With Seasonally Varying Contact Rate

 1 Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt 2 Department of Statistics and Modelling Science, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

Received  January 2005 Revised  July 2005 Published  August 2005

In this paper, a general periodic vaccination has been applied to control the spread and transmission of an infectious disease with latency. A $SEIRS^1$ epidemic model with general periodic vaccination strategy is analyzed. We suppose that the contact rate has period $T$, and the vaccination function has period $LT$, where $L$ is an integer. Also we apply this strategy in a model with seasonal variation in the contact rate. Both the vaccination strategy and the contact rate are general time-dependent periodic functions. The same SEIRS models have been examined for a mixed vaccination strategy composed of both the time-dependent periodic vaccination strategy and the conventional one. A key parameter of the paper is a conjectured value $R^c_0$ for the basic reproduction number. We prove that the disease-free solution (DFS) is globally asymptotically stable (GAS) when $R^{"sup"}_0 < 1$. If $R^{"inf"}_0 > 1$, then the DFS is unstable, and we prove that there exists a nontrivial periodic solution whose period is the same as that of the vaccination strategy. Some persistence results are also discussed. Necessary and sufficient conditions for the eradication or control of the disease are derived. Threshold conditions for these vaccination strategies to ensure that $R^{"sup"}_0 < 1$ and $R^{"inf"}_0 > 1$ are also investigated.
Citation: Islam A. Moneim, David Greenhalgh. Use Of A Periodic Vaccination Strategy To Control The Spread Of Epidemics With Seasonally Varying Contact Rate. Mathematical Biosciences & Engineering, 2005, 2 (3) : 591-611. doi: 10.3934/mbe.2005.2.591
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