American Journal of Applied Mathematics

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The Impact of Infective Immigrants on the Spread and Dynamics of Zika Viruss

Received: 09 May 2017    Accepted: 27 May 2017    Published: 05 November 2017
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Abstract

In this paper, a compartment model has been built, presented and investigated the dynamics and spread of zika virus in both human and mosquito populations. It is focused to study the impact of symptomatic and asymptomatic infective immigrants on the spread of zika virus. A new mathematical model SI1I2R for human and SI model for vector population has been designed and presented. Here I1 is symptomatic infective and I2 is asymptomatic infective human populations. The present model is developed making some reasonable modifications in the corresponding epidemic SIR model by considering symptomatic and asymptomatic infective immigrants. Susceptible vectors get infection either from symptomatic or asymptomatic infected human populations. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. It is shown that the disease free equilibrium point is locally and globally asymptotically stable if the reproduction number takes a value less than one unit and unstable if it is more than one unit. Simulation study is conducted using MATLAB ode45.

DOI 10.11648/j.ajam.20170506.11
Published in American Journal of Applied Mathematics (Volume 5, Issue 6, December 2017)
Page(s) 145-153
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

SI1I2R Model, Symptomatic Infected, Asymptomatic Infected, Zika Virus, Microcephally

References
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[3] Victor M., Baltazar E., Derdei B., Susan A. Holechek, and Carlos C. Simon. Role of short-term dispersal on the dynamics of Zika virus. Center for Infectious Diseases and Vaccinology, the Biodesign Institute, Arizona State University 16 Mar 2016.
[4] Nidhi Nirwani, V. H. Badshah, R. Khandelwal. A Mathematical Model of Malaria Disease with Vertical Transmission. Published by Canadian Center of Science and Education. 7 August 2015.
[5] Mary K. K., Tomas Allen, Veronika Frank, Ravi S. Santhana, Christopher D. The origin and spread of a mosquito-borne virus. Bulletin of the World Health Organization. 9 February 2016.
[6] T. Alex Perkins, Amir S S., Corrine W. R., Moritz U. G. Kraemer, Andrew J. Tatem. Model based projections of Zika virus infections in childbearing women in the America. 12 Feb.2016 http://dx.doi.org/10.1101/039610.
[7] Maimuna S M., Emily C., Durland F., John S. Estimating a feasible serial interval range for Zika fever. Bulletin of the World Health Organization. 9 February 2016.
[8] Abadi Abay Gebremeskel, Harald Elias Krogstad. Mathematical Modeling of Endemic Malaria Transmission. American Journal of Applied Mathematics. 12 February 2015.
[9] Mikayla C. Chubb Kathryn H. J. Mathematical modeling and the epidemiological research. European Journal of Epidemiology. 27 October 2009.
[10] S. Olaniyi, O. S. Obabiyi. Mathematical model for malaria transmission dynamics in human and Mosquito populations with non-linear force of infection. International Journal of Pure and Applied Mathematics. 15 August 2013.
[11] Syafruddin Side, Salmi Md Noorani. A SIR Model for Spread of Dengue Fever Disease. World Journal of Modeling and Simulation. 14 April 2013.
[12] Zika virus fact sheet. Ethiopian midwives association.
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[14] Katherine F. D., Alexa O., Emily P. Interim Zika Virus Clinical Guidance and Recommendations. Centers for Disease Control and Prevention. 26 January, 2016.
[15] Rapid Risk Assesment, Zika virus disease epidemic: potential association with microcephaly and Guillain-Barré syndrome. 20 January 2016.
[16] Thais D. S, Wanderson K. D. Zika Virus and the Guillain-Barre Syndrome: Case Series from Seven Countries. The new England journal of medicine. 20 October 2016.
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[18] Kelly A. T. Benign bacteria block mosquitoes from transmitting Zika, chikungunya viruses. University of Wisconsin U Madison. 1 Jun 2016.
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Author Information
  • School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

  • School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

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  • APA Style

    Molalegn Ayana, Purnachandra Rao Koya. (2017). The Impact of Infective Immigrants on the Spread and Dynamics of Zika Viruss. American Journal of Applied Mathematics, 5(6), 145-153. https://doi.org/10.11648/j.ajam.20170506.11

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    ACS Style

    Molalegn Ayana; Purnachandra Rao Koya. The Impact of Infective Immigrants on the Spread and Dynamics of Zika Viruss. Am. J. Appl. Math. 2017, 5(6), 145-153. doi: 10.11648/j.ajam.20170506.11

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    AMA Style

    Molalegn Ayana, Purnachandra Rao Koya. The Impact of Infective Immigrants on the Spread and Dynamics of Zika Viruss. Am J Appl Math. 2017;5(6):145-153. doi: 10.11648/j.ajam.20170506.11

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  • @article{10.11648/j.ajam.20170506.11,
      author = {Molalegn Ayana and Purnachandra Rao Koya},
      title = {The Impact of Infective Immigrants on the Spread and Dynamics of Zika Viruss},
      journal = {American Journal of Applied Mathematics},
      volume = {5},
      number = {6},
      pages = {145-153},
      doi = {10.11648/j.ajam.20170506.11},
      url = {https://doi.org/10.11648/j.ajam.20170506.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajam.20170506.11},
      abstract = {In this paper, a compartment model has been built, presented and investigated the dynamics and spread of zika virus in both human and mosquito populations. It is focused to study the impact of symptomatic and asymptomatic infective immigrants on the spread of zika virus. A new mathematical model SI1I2R for human and SI model for vector population has been designed and presented. Here I1 is symptomatic infective and I2 is asymptomatic infective human populations. The present model is developed making some reasonable modifications in the corresponding epidemic SIR model by considering symptomatic and asymptomatic infective immigrants. Susceptible vectors get infection either from symptomatic or asymptomatic infected human populations. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. It is shown that the disease free equilibrium point is locally and globally asymptotically stable if the reproduction number takes a value less than one unit and unstable if it is more than one unit. Simulation study is conducted using MATLAB ode45.},
     year = {2017}
    }
    

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    AU  - Molalegn Ayana
    AU  - Purnachandra Rao Koya
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    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    PB  - Science Publishing Group
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    AB  - In this paper, a compartment model has been built, presented and investigated the dynamics and spread of zika virus in both human and mosquito populations. It is focused to study the impact of symptomatic and asymptomatic infective immigrants on the spread of zika virus. A new mathematical model SI1I2R for human and SI model for vector population has been designed and presented. Here I1 is symptomatic infective and I2 is asymptomatic infective human populations. The present model is developed making some reasonable modifications in the corresponding epidemic SIR model by considering symptomatic and asymptomatic infective immigrants. Susceptible vectors get infection either from symptomatic or asymptomatic infected human populations. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. It is shown that the disease free equilibrium point is locally and globally asymptotically stable if the reproduction number takes a value less than one unit and unstable if it is more than one unit. Simulation study is conducted using MATLAB ode45.
    VL  - 5
    IS  - 6
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