92D30.

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

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
2. Department of Statistics and Modelling Science, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH

Abstract    Related pages

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.
Figure/Table
Supplementary
Article Metrics

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

• 1. Zhenguo Bai, Yicang Zhou, Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Analysis: Real World Applications, 2012, 13, 3, 1060, 10.1016/j.nonrwa.2011.02.008
• 2. Drew Posny, Jin Wang, Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments, Applied Mathematics and Computation, 2014, 242, 473, 10.1016/j.amc.2014.05.079
• 3. Weiming Wang, Yongli Cai, Jingli Li, Zhanji Gui, Periodic behavior in a FIV model with seasonality as well as environment fluctuations, Journal of the Franklin Institute, 2017, 354, 16, 7410, 10.1016/j.jfranklin.2017.08.034
• 4. Eric Ávila-Vales, Erika Rivero-Esquivel, Gerardo Emilio García-Almeida, Global Dynamics of a Periodic SEIRS Model with General Incidence Rate, International Journal of Differential Equations, 2017, 2017, 1, 10.1155/2017/5796958
• 5. Nicolas Bacaër, Rachid Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Mathematical Biosciences, 2007, 210, 2, 647, 10.1016/j.mbs.2007.07.005
• 6. Nicolas Bacaër, Xamxinur Abdurahman, Resonance of the epidemic threshold in a periodic environment, Journal of Mathematical Biology, 2008, 57, 5, 649, 10.1007/s00285-008-0183-1
• 7. Yangjun Ma, Maoxing Liu, Qiang Hou, Jinqing Zhao, Modelling seasonal HFMD with the recessive infection in Shandong, China, Mathematical Biosciences and Engineering, 2013, 10, 4, 1159, 10.3934/mbe.2013.10.1159
• 8. L. Jódar, R.J. Villanueva, A. Arenas, Modeling the spread of seasonal epidemiological diseases: Theory and applications, Mathematical and Computer Modelling, 2008, 48, 3-4, 548, 10.1016/j.mcm.2007.08.017
• 9. Nicolas Bacaër, Approximation of the Basic Reproduction Number R 0 for Vector-Borne Diseases with a Periodic Vector Population, Bulletin of Mathematical Biology, 2007, 69, 3, 1067, 10.1007/s11538-006-9166-9
• 10. J.V. Greenman, R.A. Norman, Environmental forcing, invasion and control of ecological and epidemiological systems, Journal of Theoretical Biology, 2007, 247, 3, 492, 10.1016/j.jtbi.2007.03.031
• 11. Abraham J. Arenas, Gilberto González-Parra, Benito M. Chen-Charpentier, Dynamical analysis of the transmission of seasonal diseases using the differential transformation method, Mathematical and Computer Modelling, 2009, 50, 5-6, 765, 10.1016/j.mcm.2009.05.005
• 12. I. A. Moneim, Efficiency of Different Vaccination Strategies for Childhood Diseases: A Simulation Study, Advances in Bioscience and Biotechnology, 2013, 04, 02, 193, 10.4236/abb.2013.42028
• 13. Yong Li, Xianning Liu, Lianwen Wang, Xingan Zhang, Hopf bifurcation of a delay SIRS epidemic model with novel nonlinear incidence: Application to scarlet fever, International Journal of Biomathematics, 2018, 1850091, 10.1142/S1793524518500912
• 14. Maia Martcheva, Benjamin M Bolker, Robert D Holt, Vaccine-induced pathogen strain replacement: what are the mechanisms?, Journal of The Royal Society Interface, 2008, 5, 18, 3, 10.1098/rsif.2007.0236