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 R_{0}, 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 R_{0} 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 |
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Typhoid Fever, Spatial Variations, Seasonality, Basic Reproduction Number, Global Dynamics, Reaction-diffusion Model, Noncompact Solution Maps
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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
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
@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} }
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 -