American Institute of Mathematical Sciences

Advanced
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

Islam A. Moneim 1, and  David Greenhalgh 2,

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.
Keywords: disease control, periodic vaccination, childhood diseases, basic reproduction number R0; periodicity, uniform strong repeller, mathematical modelling, uniform persis- tence..
Mathematics Subject Classification: 92D3.
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
[1]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[2]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
[3]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[4]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[5]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239
[6]

Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565
[7]

P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213
[8]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[9]

Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477
[10]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[11]

Liumei Wu, Baojun Song, Wen Du, Jie Lou. Mathematical modelling and control of echinococcus in Qinghai province, China. Mathematical Biosciences & Engineering, 2013, 10 (2) : 425-444. doi: 10.3934/mbe.2013.10.425
[12]

Artur Avila, Thomas Roblin. Uniform exponential growth for some SL(2, R) matrix products. Journal of Modern Dynamics, 2009, 3 (4) : 549-554. doi: 10.3934/jmd.2009.3.549
[13]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909
[14]

Hisashi Inaba. The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments. Mathematical Biosciences & Engineering, 2012, 9 (2) : 313-346. doi: 10.3934/mbe.2012.9.313
[15]

Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789
[16]

Xiaorui Wang, Genqi Xu. Uniform stabilization of a wave equation with partial Dirichlet delayed control. Evolution Equations & Control Theory, 2020, 9 (2) : 509-533. doi: 10.3934/eect.2020022
[17]

Sergey Zelik. Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 781-810. doi: 10.3934/dcdsb.2015.20.781
[18]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic & Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018
[19]

Martin Swaczyna, Petr Volný. Uniform motions in central fields. Journal of Geometric Mechanics, 2017, 9 (1) : 91-130. doi: 10.3934/jgm.2017004
[20]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences