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A Reaction-diffusion System Modeling the Transmission of Typhoid Fever in a Periodic Environment

Received: 13 March 2024     Accepted: 27 March 2024     Published: 21 April 2024
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Abstract

Typhoid fever is a life-threatening infection caused by the bacterium Salmonella Typhi, and it is still an important issue in developing countries. There are two infection routes of Typhoid fever, namely, the human-to-human transmission and the environment-to-human transmission. It is evident that people living near rivers may have a higher rate of typhoid infection, and temperature changes also have significant impacts on Typhoid transmission dynamics. In the model, the population of human will be divided into susceptible individuals, infected individuals, carrier individuals, individuals under treatment, and recovered individuals. Then a periodic dispersion-reaction system is used to describe the transport and the interactions between human and bacteria in the environment. The solution maps of the proposed periodic dispersion-reaction system lack the compactness since the population under treatment has no diffusion term, which makes analysis more difficult. After the feasible domain is chosen carefully, the eventually boundedness of the solutions can be established, and the loss of compactness is overcome if the initial data is chosen from the feasible domain. In order to introduce the reproduction number R0, the linearized system around the disease-free state is constructed, and the basic reproduction number is defined as the spectral radius of the next generation operator. Then the comparison principle and persistence theory can be utilized to establish that the index R0 completely determines the threshold behavior of the typhoid spread. Brief mathematical and biological interpretations are also presented.

Published in Applied and Computational Mathematics (Volume 13, Issue 2)
DOI 10.11648/j.acm.20241302.12
Page(s) 38-52
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

Typhoid Fever, Spatial Variations, Seasonality, Basic Reproduction Number, Global Dynamics, Reaction-diffusion Model, Noncompact Solution Maps

References
[1] N. Bacaër, S. Guernaoui. The epidemic threshold of vector-borne diseases with seasonality, Journal of Mathematical Biology, 53(2006), 421–436.
[2] K. Deimling, Nonlinear Functional Analysis, Springer- Verlag, Berlin, 1988.
[3] A. M. Dewan, R. Corner, M. Hashizume, E. T. Ongee, Typhoid Fever and its association with environmental factors in the Dhaka Metropolitan Area of Bangladesh: a spatial and time-series approach, PLoS Negl. Trop. Dis., 7 (2013), e1998.
[4] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), pp. 365–382.
[5] J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society Providence, RI, 1988.
[6] P. Hess, Periodic-parabolic Boundary Value Problem and Positivity, Pitman Res. Notes Math., 247, Longman Scientific and Technical, 1991.
[7] S. B. Hsu, F. B. Wang, and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, Journal of Dynamics and Differential Equations, 23 (2011), 817-842.
[8] T. W. Hwang and F.-B. Wang, Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation, Discrete and Continuous Dynamical System Series-B, vol. 18 (2013), pp. 147–161.
[9] L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Commun. Partial Diff. Eqns 22 (1997), pp. 413–433.
[10] S. P. Luby, M. S. Islam, R. Johnston, Chlorine spot treatment of flooded tube wells, an efficacy trail, J. Appl. Microbiol. 100 (2006), pp. 1154-1158.
[11] S. P. Luby, S. K. Gupta, M. A. Sheikh, R. B. Johnston, P. K. Ram, et al., Tubewell water quality and predictors of contamination in three flood-prone areas of Bangladesh, J. Appl. Microbiol. 105 (2008), pp. 1002-1008.
[12] H. L. Lin, K.H. Lin, Y.C. Shyu and F.B. Wang, Impacts of Seasonal and Spatial Variations on the Transmission of Typhoid Fever, Applied and Computational Mathematics, 12:2 (2023), pp. 26–41.
[13] H. C. Li, R. Peng and F.-B. Wang, Varying Total Population Enhances Disease Persistence: Qualitative Analysis on a Diffusive SIS Epidemic Model, J. Diff. Eqns. 262 (2017), pp. 885–913.
[14] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. of Math. Biol., 62 (2011), pp. 543-568.
[15] X. Liang, L. Zhang and X.-Q. Zhao, The Principal Eigenvalue for Degenerate Periodic Reaction-Diffusion Systems, SIAM J. Math. Anal., 49 (2017), pp. 3603– 3636.
[16] X. Liang, L. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease), J. Dyn. Differ. Equ., 31 (2019), pp. 1247–1278.
[17] J. Mushanyu, F. Nyabadza, G. Muchatibaya, P. Mafuta, G. Nhawu, Assessing the potential impact of limited public health resources on the spread and control of typhoid, J. Math. Biol., 77 (2018), pp. 647-670.
[18] R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A.M.S., 321 (1990), pp. 1–44.
[19] P. Magal, and X. -Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal 37 (2005), pp. 251-275.
[20] H. L. Smith, Monotone Dynamical Systems:An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.
[21] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics 118, American Mathematical Society, Providence, 2011.
[22] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal 47 (2001), pp. 6169–6179.
[23] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM, J. Appl. Math. 70 (2009), pp. 188– 211.
[24] P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), pp. 29–48.
[25] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984.
[26] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ. 20 (2008), pp. 699– 717.
[27] X.-Q. Zhao, Dynamical Systems in Population Biology, second ed., Springer, New York, 2017.
Cite This Article
  • APA Style

    Lin, H., Shyu, Y., Lin, C., Wang, F. (2024). A Reaction-diffusion System Modeling the Transmission of Typhoid Fever in a Periodic Environment. Applied and Computational Mathematics, 13(2), 38-52. https://doi.org/10.11648/j.acm.20241302.12

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    ACS Style

    Lin, H.; Shyu, Y.; Lin, C.; Wang, F. A Reaction-diffusion System Modeling the Transmission of Typhoid Fever in a Periodic Environment. Appl. Comput. Math. 2024, 13(2), 38-52. doi: 10.11648/j.acm.20241302.12

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    AMA Style

    Lin H, Shyu Y, Lin C, Wang F. A Reaction-diffusion System Modeling the Transmission of Typhoid Fever in a Periodic Environment. Appl Comput Math. 2024;13(2):38-52. doi: 10.11648/j.acm.20241302.12

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  • @article{10.11648/j.acm.20241302.12,
      author = {Huei-Li Lin and Yu-Chiau Shyu and Chih-Lang Lin and Feng-Bin Wang},
      title = {A Reaction-diffusion System Modeling the Transmission of Typhoid Fever in a Periodic Environment},
      journal = {Applied and Computational Mathematics},
      volume = {13},
      number = {2},
      pages = {38-52},
      doi = {10.11648/j.acm.20241302.12},
      url = {https://doi.org/10.11648/j.acm.20241302.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241302.12},
      abstract = {Typhoid fever is a life-threatening infection caused by the bacterium Salmonella Typhi, and it is still an important issue in developing countries. There are two infection routes of Typhoid fever, namely, the human-to-human transmission and the environment-to-human transmission. It is evident that people living near rivers may have a higher rate of typhoid infection, and temperature changes also have significant impacts on Typhoid transmission dynamics. In the model, the population of human will be divided into susceptible individuals, infected individuals, carrier individuals, individuals under treatment, and recovered individuals. Then a periodic dispersion-reaction system is used to describe the transport and the interactions between human and bacteria in the environment. The solution maps of the proposed periodic dispersion-reaction system lack the compactness since the population under treatment has no diffusion term, which makes analysis more difficult. After the feasible domain is chosen carefully, the eventually boundedness of the solutions can be established, and the loss of compactness is overcome if the initial data is chosen from the feasible domain. In order to introduce the reproduction number R0, the linearized system around the disease-free state is constructed, and the basic reproduction number is defined as the spectral radius of the next generation operator. Then the comparison principle and persistence theory can be utilized to establish that the index R0 completely determines the threshold behavior of the typhoid spread. Brief mathematical and biological interpretations are also presented.},
     year = {2024}
    }
    

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  • TY  - JOUR
    T1  - A Reaction-diffusion System Modeling the Transmission of Typhoid Fever in a Periodic Environment
    AU  - Huei-Li Lin
    AU  - Yu-Chiau Shyu
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    DO  - 10.11648/j.acm.20241302.12
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    EP  - 52
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20241302.12
    AB  - Typhoid fever is a life-threatening infection caused by the bacterium Salmonella Typhi, and it is still an important issue in developing countries. There are two infection routes of Typhoid fever, namely, the human-to-human transmission and the environment-to-human transmission. It is evident that people living near rivers may have a higher rate of typhoid infection, and temperature changes also have significant impacts on Typhoid transmission dynamics. In the model, the population of human will be divided into susceptible individuals, infected individuals, carrier individuals, individuals under treatment, and recovered individuals. Then a periodic dispersion-reaction system is used to describe the transport and the interactions between human and bacteria in the environment. The solution maps of the proposed periodic dispersion-reaction system lack the compactness since the population under treatment has no diffusion term, which makes analysis more difficult. After the feasible domain is chosen carefully, the eventually boundedness of the solutions can be established, and the loss of compactness is overcome if the initial data is chosen from the feasible domain. In order to introduce the reproduction number R0, the linearized system around the disease-free state is constructed, and the basic reproduction number is defined as the spectral radius of the next generation operator. Then the comparison principle and persistence theory can be utilized to establish that the index R0 completely determines the threshold behavior of the typhoid spread. Brief mathematical and biological interpretations are also presented.
    VL  - 13
    IS  - 2
    ER  - 

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Author Information
  • Department of Natural Science in the Center for General Education, Chang Gung University, Taoyuan, Taiwan

  • Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung, Taiwan; Department of Nursing, Chang Gung University of Science and Technology, Taoyuan, Taiwan

  • Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung, Taiwan; Department of Gastroenterology and Hepatology, Chang Gung Memorial Hospital, Keelung Branch, Keelung, Taiwan

  • Department of Natural Science in the Center for General Education, Chang Gung University, Taoyuan, Taiwan; Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung, Taiwan; National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan

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