On a Reaction-Diffusion Model of COVID-19

Rebecca Walo Omana; Issa Ramadhani Issa; Francis-Didier Tshianyi Mwana Kalala

doi:doi:10.11648/j.ijssam.20210601.13

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On a Reaction-Diffusion Model of COVID-19

Rebecca Walo Omana, Issa Ramadhani Issa, Francis-Didier Tshianyi Mwana Kalala

Published in International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 1)

Received: 23 November 2020 Accepted: 11 January 2021 Published: 26 March 2021

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Abstract

Nowadays mathematical models play a major role in epidemiology since they can help in predicting the spreading and the evolution of diseases. Many of them are based on ODEs on the assumption that the populations being studied are homogenous sets of fixed points (individuals) but actually populations are far from being homogenous and people are constantly moving. In fact, thanks to science progresses, distances are no longer what they used to be in the past and a disease can travel and reach out even the most remote places on the globe in a matter of hours. HIV and Covid-19 outbreaks are perfect illustrations of how far and fast a disease can now spread. When it comes to studying the spatio-temporal spreading of a disease, instead of ODEs dynamic models the Reaction-Diffusion ones are best suited. They are inspired by the second Fick’s law in physics and are getting more and more used. In this article we make a study of the spatio-temporal spreading of the COVID-19. We first present our SEIR dynamic model, we find the two equilibrium points and an expression for the basic reproduction number (R0), we use the additive compound matrices and show that only one condition is necessary to show the local stability of the two equilibrium points instead of two like it is traditionally done, and we study the conditions for the DFE (Disease Free Equilibrium point) and the EE (Endemic Equilibrium point) to be globally asymptotically stable. Then we construct a diffusive model from our previous SEIR model, we investigate on the existence of a traveling wave connecting the two equilibrium thanks to the monotone iterative method and we give an expression for the minimal wave speed. Then in the last section we use the additive compound matrices to show that the DFE remains stable when diffusion is added whereas there will be appearance of Turing instability for the EE once diffusion is added. The conclusion of our article emphasizes the importance of barrier gestures and the fact that the more people are getting tested the better governments will be able to handle and tackle the spreading of the disease.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 1)
DOI	10.11648/j.ijssam.20210601.13
Page(s)	22-34
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), 2021. Published by Science Publishing Group

Keywords

Reaction-Diffusion, COVID-19, Traveling Wave, Upper-solution, Lower-solution, Turing Instability

References

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APA Style

Rebecca Walo Omana, Issa Ramadhani Issa, Francis-Didier Tshianyi Mwana Kalala. (2021). On a Reaction-Diffusion Model of COVID-19. International Journal of Systems Science and Applied Mathematics, 6(1), 22-34. https://doi.org/10.11648/j.ijssam.20210601.13

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ACS Style

Rebecca Walo Omana; Issa Ramadhani Issa; Francis-Didier Tshianyi Mwana Kalala. On a Reaction-Diffusion Model of COVID-19. Int. J. Syst. Sci. Appl. Math. 2021, 6(1), 22-34. doi: 10.11648/j.ijssam.20210601.13

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AMA Style

Rebecca Walo Omana, Issa Ramadhani Issa, Francis-Didier Tshianyi Mwana Kalala. On a Reaction-Diffusion Model of COVID-19. Int J Syst Sci Appl Math. 2021;6(1):22-34. doi: 10.11648/j.ijssam.20210601.13

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@article{10.11648/j.ijssam.20210601.13,
  author = {Rebecca Walo Omana and Issa Ramadhani Issa and Francis-Didier Tshianyi Mwana Kalala},
  title = {On a Reaction-Diffusion Model of COVID-19},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {22-34},
  doi = {10.11648/j.ijssam.20210601.13},
  url = {https://doi.org/10.11648/j.ijssam.20210601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210601.13},
  abstract = {Nowadays mathematical models play a major role in epidemiology since they can help in predicting the spreading and the evolution of diseases. Many of them are based on ODEs on the assumption that the populations being studied are homogenous sets of fixed points (individuals) but actually populations are far from being homogenous and people are constantly moving. In fact, thanks to science progresses, distances are no longer what they used to be in the past and a disease can travel and reach out even the most remote places on the globe in a matter of hours. HIV and Covid-19 outbreaks are perfect illustrations of how far and fast a disease can now spread. When it comes to studying the spatio-temporal spreading of a disease, instead of ODEs dynamic models the Reaction-Diffusion ones are best suited. They are inspired by the second Fick’s law in physics and are getting more and more used. In this article we make a study of the spatio-temporal spreading of the COVID-19. We first present our SEIR dynamic model, we find the two equilibrium points and an expression for the basic reproduction number (R0), we use the additive compound matrices and show that only one condition is necessary to show the local stability of the two equilibrium points instead of two like it is traditionally done, and we study the conditions for the DFE (Disease Free Equilibrium point) and the EE (Endemic Equilibrium point) to be globally asymptotically stable. Then we construct a diffusive model from our previous SEIR model, we investigate on the existence of a traveling wave connecting the two equilibrium thanks to the monotone iterative method and we give an expression for the minimal wave speed. Then in the last section we use the additive compound matrices to show that the DFE remains stable when diffusion is added whereas there will be appearance of Turing instability for the EE once diffusion is added. The conclusion of our article emphasizes the importance of barrier gestures and the fact that the more people are getting tested the better governments will be able to handle and tackle the spreading of the disease.},
 year = {2021}
}

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TY  - JOUR
T1  - On a Reaction-Diffusion Model of COVID-19
AU  - Rebecca Walo Omana
AU  - Issa Ramadhani Issa
AU  - Francis-Didier Tshianyi Mwana Kalala
Y1  - 2021/03/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210601.13
DO  - 10.11648/j.ijssam.20210601.13
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 22
EP  - 34
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210601.13
AB  - Nowadays mathematical models play a major role in epidemiology since they can help in predicting the spreading and the evolution of diseases. Many of them are based on ODEs on the assumption that the populations being studied are homogenous sets of fixed points (individuals) but actually populations are far from being homogenous and people are constantly moving. In fact, thanks to science progresses, distances are no longer what they used to be in the past and a disease can travel and reach out even the most remote places on the globe in a matter of hours. HIV and Covid-19 outbreaks are perfect illustrations of how far and fast a disease can now spread. When it comes to studying the spatio-temporal spreading of a disease, instead of ODEs dynamic models the Reaction-Diffusion ones are best suited. They are inspired by the second Fick’s law in physics and are getting more and more used. In this article we make a study of the spatio-temporal spreading of the COVID-19. We first present our SEIR dynamic model, we find the two equilibrium points and an expression for the basic reproduction number (R0), we use the additive compound matrices and show that only one condition is necessary to show the local stability of the two equilibrium points instead of two like it is traditionally done, and we study the conditions for the DFE (Disease Free Equilibrium point) and the EE (Endemic Equilibrium point) to be globally asymptotically stable. Then we construct a diffusive model from our previous SEIR model, we investigate on the existence of a traveling wave connecting the two equilibrium thanks to the monotone iterative method and we give an expression for the minimal wave speed. Then in the last section we use the additive compound matrices to show that the DFE remains stable when diffusion is added whereas there will be appearance of Turing instability for the EE once diffusion is added. The conclusion of our article emphasizes the importance of barrier gestures and the fact that the more people are getting tested the better governments will be able to handle and tackle the spreading of the disease.
VL  - 6
IS  - 1
ER  -

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Author Information

Rebecca Walo Omana

Faculty of Sciences, Regional Center for Doctoral Education in Mathematics and Computer Science, University of Kinshasa, Kinshasa, Democratic Republic of the Congo
Issa Ramadhani Issa

Faculty of Sciences, Regional Center for Doctoral Education in Mathematics and Computer Science, University of Kinshasa, Kinshasa, Democratic Republic of the Congo
Francis-Didier Tshianyi Mwana Kalala

Faculty of Sciences, Regional Center for Doctoral Education in Mathematics and Computer Science, University of Kinshasa, Kinshasa, Democratic Republic of the Congo

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APA Style

Rebecca Walo Omana, Issa Ramadhani Issa, Francis-Didier Tshianyi Mwana Kalala. (2021). On a Reaction-Diffusion Model of COVID-19. International Journal of Systems Science and Applied Mathematics, 6(1), 22-34. https://doi.org/10.11648/j.ijssam.20210601.13

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ACS Style

Rebecca Walo Omana; Issa Ramadhani Issa; Francis-Didier Tshianyi Mwana Kalala. On a Reaction-Diffusion Model of COVID-19. Int. J. Syst. Sci. Appl. Math. 2021, 6(1), 22-34. doi: 10.11648/j.ijssam.20210601.13

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AMA Style

Rebecca Walo Omana, Issa Ramadhani Issa, Francis-Didier Tshianyi Mwana Kalala. On a Reaction-Diffusion Model of COVID-19. Int J Syst Sci Appl Math. 2021;6(1):22-34. doi: 10.11648/j.ijssam.20210601.13

Copy | Download

@article{10.11648/j.ijssam.20210601.13,
  author = {Rebecca Walo Omana and Issa Ramadhani Issa and Francis-Didier Tshianyi Mwana Kalala},
  title = {On a Reaction-Diffusion Model of COVID-19},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {22-34},
  doi = {10.11648/j.ijssam.20210601.13},
  url = {https://doi.org/10.11648/j.ijssam.20210601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210601.13},
  abstract = {Nowadays mathematical models play a major role in epidemiology since they can help in predicting the spreading and the evolution of diseases. Many of them are based on ODEs on the assumption that the populations being studied are homogenous sets of fixed points (individuals) but actually populations are far from being homogenous and people are constantly moving. In fact, thanks to science progresses, distances are no longer what they used to be in the past and a disease can travel and reach out even the most remote places on the globe in a matter of hours. HIV and Covid-19 outbreaks are perfect illustrations of how far and fast a disease can now spread. When it comes to studying the spatio-temporal spreading of a disease, instead of ODEs dynamic models the Reaction-Diffusion ones are best suited. They are inspired by the second Fick’s law in physics and are getting more and more used. In this article we make a study of the spatio-temporal spreading of the COVID-19. We first present our SEIR dynamic model, we find the two equilibrium points and an expression for the basic reproduction number (R0), we use the additive compound matrices and show that only one condition is necessary to show the local stability of the two equilibrium points instead of two like it is traditionally done, and we study the conditions for the DFE (Disease Free Equilibrium point) and the EE (Endemic Equilibrium point) to be globally asymptotically stable. Then we construct a diffusive model from our previous SEIR model, we investigate on the existence of a traveling wave connecting the two equilibrium thanks to the monotone iterative method and we give an expression for the minimal wave speed. Then in the last section we use the additive compound matrices to show that the DFE remains stable when diffusion is added whereas there will be appearance of Turing instability for the EE once diffusion is added. The conclusion of our article emphasizes the importance of barrier gestures and the fact that the more people are getting tested the better governments will be able to handle and tackle the spreading of the disease.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - On a Reaction-Diffusion Model of COVID-19
AU  - Rebecca Walo Omana
AU  - Issa Ramadhani Issa
AU  - Francis-Didier Tshianyi Mwana Kalala
Y1  - 2021/03/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210601.13
DO  - 10.11648/j.ijssam.20210601.13
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 22
EP  - 34
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210601.13
AB  - Nowadays mathematical models play a major role in epidemiology since they can help in predicting the spreading and the evolution of diseases. Many of them are based on ODEs on the assumption that the populations being studied are homogenous sets of fixed points (individuals) but actually populations are far from being homogenous and people are constantly moving. In fact, thanks to science progresses, distances are no longer what they used to be in the past and a disease can travel and reach out even the most remote places on the globe in a matter of hours. HIV and Covid-19 outbreaks are perfect illustrations of how far and fast a disease can now spread. When it comes to studying the spatio-temporal spreading of a disease, instead of ODEs dynamic models the Reaction-Diffusion ones are best suited. They are inspired by the second Fick’s law in physics and are getting more and more used. In this article we make a study of the spatio-temporal spreading of the COVID-19. We first present our SEIR dynamic model, we find the two equilibrium points and an expression for the basic reproduction number (R0), we use the additive compound matrices and show that only one condition is necessary to show the local stability of the two equilibrium points instead of two like it is traditionally done, and we study the conditions for the DFE (Disease Free Equilibrium point) and the EE (Endemic Equilibrium point) to be globally asymptotically stable. Then we construct a diffusive model from our previous SEIR model, we investigate on the existence of a traveling wave connecting the two equilibrium thanks to the monotone iterative method and we give an expression for the minimal wave speed. Then in the last section we use the additive compound matrices to show that the DFE remains stable when diffusion is added whereas there will be appearance of Turing instability for the EE once diffusion is added. The conclusion of our article emphasizes the importance of barrier gestures and the fact that the more people are getting tested the better governments will be able to handle and tackle the spreading of the disease.
VL  - 6
IS  - 1
ER  -

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