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Mathematical Modeling and Treatment Impacts of Water Borne Disease Spread

Received: 19 August 2022     Accepted: 14 September 2022     Published: 11 November 2022
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Abstract

In this work, the treatment impacts of water borne disease is modeled and analyzed from a mathematical perspective via a deterministic SEIR model. The total human population is partitioned into four sub-classes namely susceptible individuals, exposed individuals, infected individuals and recovered individuals. The stability theory of non-linear differential equations and the basic reproductive number represents the epidemic indicator which is obtained from the largest eigen value of the next-generation matrix. The model explored invariant region, equilibrium condition, basic reproduction number, and stability analysis. The invariant region was proved to be positive and bounded that confirm the feasible model solution. It is also observed that the water borne disease is free equilibrium is locally asymptotically stable if the basic reproduction number is less than one. In this situation it is found that the disease is controlled whenever the treatment is allowable in the community. The disease is endemic equilibrium and globally asymptotically stable in the invariant region if the basic reproduction number is greater than one. The sensitivity analysis revealed that the rate of transmission and the rate at which exposed individuals become infectious are the most sensitive parameters. The numeric results have been illustrated through figures for different values of sensitive parameters by use of MATLAB simulation method. The findings indicate that effective treatment is adequate in eradicating and controlling water borne disease.

Published in Science Journal of Applied Mathematics and Statistics (Volume 10, Issue 5)
DOI 10.11648/j.sjams.20221005.12
Page(s) 90-97
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), 2022. Published by Science Publishing Group

Keywords

Water Borne, Basic Reproduction Number, Mathematical Model, Numerical Simulation

References
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[2] John Wiley and Sons, Chichester, 2000, Model building, Analysis and Interpretation.
[3] WHO/UNICEF: World Diarrhea report 2008, Report series.
[4] Hoogendoorn, S. State of the art of Vehicular Traffic Flow Modeling. Special Issue on Road Traffic Modelling and Control of the Journal of Systems and Control Engineering.
[5] BLACK AND M. SCHOLES, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973).
[6] Black-Sholes option valuation for scientific computing students (January, 2004).
[7] Dr. A. Chernov; Numerical and Analytic Methods in option pricing, Journal, (20015).
[8] E. Shim. A note on epidemic models with infective immigrants and vaccination. Math. Biosci. Engg, 3 (2006): 557566.
[9] Federal Democratic Republic of Ethiopia Ministry Of Health Ethiopia National Diarrhea Indicator Survey, Addis Ababa, 2008.
[10] O. Diekmann, J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases.
[11] LaSalle, J. P.: The stability of dynamical systems. In: Regional Conference Series in Applied Mathematics, SIAM, Philadelphia (1976).
[12] Bayor College of Medicine, Department of Molecular Virology and Micro biology, Research âA¸ S Emerging Infectious Diseases. (Accessed: 19 May ˘2014) Available: https://www.bcm.edu/departments/molecular-virology-and microbiology/research
[13] Chaturvedi O, Jeffrey M, Lungu E, Masupe S. Epidemic model formulation and analysis for diarrheal infections caused by salmonella. Simulation Journal. 2017; 93: 543-552.
[14] Gerald T. Keusch, Olivier Fontaine, AlokBhargava, Cynthia BoschiPinto, Zulfiqar A. Bhutta, Eduardo Gotuzzo, Juan Rivera, Jeffrey Chow, Sonbol A. Shahid-Salles, and Ramanan Laxminarayan; on Diarrheal Diseases.
[15] Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 2948 (2002).
[16] Jian-quan Li, Jie L., Mei-zhi L., Some discrete SI and SIS models, Applied Mathematics and Mechanics, 2012.
[17] S. O. Adewale, I. A. Olopade, S. O. Ajao and G. A. Adeniran;mathematical analysis of diarrhea in the presence of vaccine December-2015.
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  • APA Style

    Mideksa Tola Jiru. (2022). Mathematical Modeling and Treatment Impacts of Water Borne Disease Spread. Science Journal of Applied Mathematics and Statistics, 10(5), 90-97. https://doi.org/10.11648/j.sjams.20221005.12

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    ACS Style

    Mideksa Tola Jiru. Mathematical Modeling and Treatment Impacts of Water Borne Disease Spread. Sci. J. Appl. Math. Stat. 2022, 10(5), 90-97. doi: 10.11648/j.sjams.20221005.12

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    AMA Style

    Mideksa Tola Jiru. Mathematical Modeling and Treatment Impacts of Water Borne Disease Spread. Sci J Appl Math Stat. 2022;10(5):90-97. doi: 10.11648/j.sjams.20221005.12

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  • @article{10.11648/j.sjams.20221005.12,
      author = {Mideksa Tola Jiru},
      title = {Mathematical Modeling and Treatment Impacts of Water Borne Disease Spread},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {10},
      number = {5},
      pages = {90-97},
      doi = {10.11648/j.sjams.20221005.12},
      url = {https://doi.org/10.11648/j.sjams.20221005.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20221005.12},
      abstract = {In this work, the treatment impacts of water borne disease is modeled and analyzed from a mathematical perspective via a deterministic SEIR model. The total human population is partitioned into four sub-classes namely susceptible individuals, exposed individuals, infected individuals and recovered individuals. The stability theory of non-linear differential equations and the basic reproductive number represents the epidemic indicator which is obtained from the largest eigen value of the next-generation matrix. The model explored invariant region, equilibrium condition, basic reproduction number, and stability analysis. The invariant region was proved to be positive and bounded that confirm the feasible model solution. It is also observed that the water borne disease is free equilibrium is locally asymptotically stable if the basic reproduction number is less than one. In this situation it is found that the disease is controlled whenever the treatment is allowable in the community. The disease is endemic equilibrium and globally asymptotically stable in the invariant region if the basic reproduction number is greater than one. The sensitivity analysis revealed that the rate of transmission and the rate at which exposed individuals become infectious are the most sensitive parameters. The numeric results have been illustrated through figures for different values of sensitive parameters by use of MATLAB simulation method. The findings indicate that effective treatment is adequate in eradicating and controlling water borne disease.},
     year = {2022}
    }
    

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  • TY  - JOUR
    T1  - Mathematical Modeling and Treatment Impacts of Water Borne Disease Spread
    AU  - Mideksa Tola Jiru
    Y1  - 2022/11/11
    PY  - 2022
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    DO  - 10.11648/j.sjams.20221005.12
    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
    SP  - 90
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    PB  - Science Publishing Group
    SN  - 2376-9513
    UR  - https://doi.org/10.11648/j.sjams.20221005.12
    AB  - In this work, the treatment impacts of water borne disease is modeled and analyzed from a mathematical perspective via a deterministic SEIR model. The total human population is partitioned into four sub-classes namely susceptible individuals, exposed individuals, infected individuals and recovered individuals. The stability theory of non-linear differential equations and the basic reproductive number represents the epidemic indicator which is obtained from the largest eigen value of the next-generation matrix. The model explored invariant region, equilibrium condition, basic reproduction number, and stability analysis. The invariant region was proved to be positive and bounded that confirm the feasible model solution. It is also observed that the water borne disease is free equilibrium is locally asymptotically stable if the basic reproduction number is less than one. In this situation it is found that the disease is controlled whenever the treatment is allowable in the community. The disease is endemic equilibrium and globally asymptotically stable in the invariant region if the basic reproduction number is greater than one. The sensitivity analysis revealed that the rate of transmission and the rate at which exposed individuals become infectious are the most sensitive parameters. The numeric results have been illustrated through figures for different values of sensitive parameters by use of MATLAB simulation method. The findings indicate that effective treatment is adequate in eradicating and controlling water borne disease.
    VL  - 10
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Author Information
  • Department of Mathematics, Hawassa College of Teacher Education, Hawassa, Ethiopia

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