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

Huei-Li Lin; Yu-Chiau Shyu; Chih-Lang Lin; Feng-Bin Wang

doi:doi:10.11648/j.acm.20241302.12

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

Huei-Li Lin, Yu-Chiau Shyu, Chih-Lang Lin, Feng-Bin Wang

Published in Applied and Computational Mathematics (Volume 13, Issue 2)

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

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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
AU  - Chih-Lang Lin
AU  - Feng-Bin Wang
Y1  - 2024/04/21
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241302.12
DO  - 10.11648/j.acm.20241302.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 38
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

Huei-Li Lin

Department of Natural Science in the Center for General Education, Chang Gung University, Taoyuan, Taiwan
Yu-Chiau Shyu

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

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
Feng-Bin Wang

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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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

Copy | Download

@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}
}

Copy | Download

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
AU  - Chih-Lang Lin
AU  - Feng-Bin Wang
Y1  - 2024/04/21
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241302.12
DO  - 10.11648/j.acm.20241302.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 38
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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