Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon
The goal of this work is to derive and study a new model which traduces the transmission dynamics of the Buruli ulcer (BU), in which we replace mass action incidences with standard incidences, consider latent period, and by using both integer and fractional derivatives. We first present reported dat...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Elsevier
2023-12-01
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| Series: | Partial Differential Equations in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S266681812300102X |
| _version_ | 1797390354940428288 |
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| author | Rubin Fandio Hamadjam Abboubakar Henri Paul Ekobena Fouda Anoop Kumar Kottakkaran Sooppy Nisar |
| author_facet | Rubin Fandio Hamadjam Abboubakar Henri Paul Ekobena Fouda Anoop Kumar Kottakkaran Sooppy Nisar |
| author_sort | Rubin Fandio |
| collection | DOAJ |
| description | The goal of this work is to derive and study a new model which traduces the transmission dynamics of the Buruli ulcer (BU), in which we replace mass action incidences with standard incidences, consider latent period, and by using both integer and fractional derivatives. We first present reported data of Buruli in Cameroon between 01/01/2001 and 31/12/2014, and the statistical model that approximates real data as closely as possible, and thus make a forecasting in the next sixteen years. After that, we formulate a Susceptible humans-Exposed humans-Infectious humans-Recovered humans-Susceptible humans; Susceptible vectors-Infectious vectors (SEIRS-SI) type compartmental model of Buruli Ulcer using integer derivative. We compute the basic reproduction number denoted by R0 and prove the asymptotic stability of the Buruli-free equilibrium whenever R0<1. Then, for R0>1, we prove the existence of an unique endemic equilibrium point and its global stability using the general theory of Lyapunov. We perform parameter estimation to calibrate our model using real data from Cameroon, while sensitivity analysis is conducted to determine important parameters in the BU dynamics. From this model calibration, we obtain R0=2.0843 which is greater than one and implies that BU ulcer is endemic in the country. To determine whether if or not the BU admits one or multiple waves, we compute the strength number denoted by A0. We find that A0≈17>0 which means that Bu admits multiple epidemic waves. We then replace integer derivative with the Caputo fractional derivative. Asymptotic stability of equilibrium points are also performed for the fractional model. This follows by the proof of existence and the uniqueness of solutions of the fractional model. We then construct a numerical scheme based on the Adams–Bashforth–Moulton (ABM) Method. Theoretical results are validated from numerical simulations. These last also permits to evaluate the impact of varying fractional order α in the BU dynamics. This permits to conclude that for α<1, reported cases are closer to model prediction. |
| first_indexed | 2024-03-08T23:10:30Z |
| format | Article |
| id | doaj.art-86535940b06848fabcfc9933be280f3f |
| institution | Directory Open Access Journal |
| issn | 2666-8181 |
| language | English |
| last_indexed | 2024-03-08T23:10:30Z |
| publishDate | 2023-12-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj.art-86535940b06848fabcfc9933be280f3f2023-12-15T07:26:53ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812023-12-018100589Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of CameroonRubin Fandio0Hamadjam Abboubakar1Henri Paul Ekobena Fouda2Anoop Kumar3Kottakkaran Sooppy Nisar4The University of Yaoundé I, Laboratory of Biophysics, P.O. Box 812, CameroonThe University of Ngaoundéré, University Institute of Technology, Department of Computer Engineering, P.O. Box 455, Ngaoundéré, Cameroon; Corresponding author.The University of Yaoundé I, Laboratory of Biophysics, P.O. Box 812, Cameroon; The University of Yaoundé I, Faculty of Science, Department of Physics, P.O. Box 812, CameroonCentral University of Punjab, School of Basic and Applied Sciences, Department of Mathematics and Statistics, Bathinda, Punjab 151001, IndiaDepartment of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi ArabiaThe goal of this work is to derive and study a new model which traduces the transmission dynamics of the Buruli ulcer (BU), in which we replace mass action incidences with standard incidences, consider latent period, and by using both integer and fractional derivatives. We first present reported data of Buruli in Cameroon between 01/01/2001 and 31/12/2014, and the statistical model that approximates real data as closely as possible, and thus make a forecasting in the next sixteen years. After that, we formulate a Susceptible humans-Exposed humans-Infectious humans-Recovered humans-Susceptible humans; Susceptible vectors-Infectious vectors (SEIRS-SI) type compartmental model of Buruli Ulcer using integer derivative. We compute the basic reproduction number denoted by R0 and prove the asymptotic stability of the Buruli-free equilibrium whenever R0<1. Then, for R0>1, we prove the existence of an unique endemic equilibrium point and its global stability using the general theory of Lyapunov. We perform parameter estimation to calibrate our model using real data from Cameroon, while sensitivity analysis is conducted to determine important parameters in the BU dynamics. From this model calibration, we obtain R0=2.0843 which is greater than one and implies that BU ulcer is endemic in the country. To determine whether if or not the BU admits one or multiple waves, we compute the strength number denoted by A0. We find that A0≈17>0 which means that Bu admits multiple epidemic waves. We then replace integer derivative with the Caputo fractional derivative. Asymptotic stability of equilibrium points are also performed for the fractional model. This follows by the proof of existence and the uniqueness of solutions of the fractional model. We then construct a numerical scheme based on the Adams–Bashforth–Moulton (ABM) Method. Theoretical results are validated from numerical simulations. These last also permits to evaluate the impact of varying fractional order α in the BU dynamics. This permits to conclude that for α<1, reported cases are closer to model prediction.http://www.sciencedirect.com/science/article/pii/S266681812300102XBuruli ulcerCaputo derivativeAsymptotic stabilityParameter estimationStrength number |
| spellingShingle | Rubin Fandio Hamadjam Abboubakar Henri Paul Ekobena Fouda Anoop Kumar Kottakkaran Sooppy Nisar Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon Partial Differential Equations in Applied Mathematics Buruli ulcer Caputo derivative Asymptotic stability Parameter estimation Strength number |
| title | Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon |
| title_full | Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon |
| title_fullStr | Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon |
| title_full_unstemmed | Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon |
| title_short | Mathematical modelling and projection of Buruli ulcer transmission dynamics using classical and fractional derivatives: A case study of Cameroon |
| title_sort | mathematical modelling and projection of buruli ulcer transmission dynamics using classical and fractional derivatives a case study of cameroon |
| topic | Buruli ulcer Caputo derivative Asymptotic stability Parameter estimation Strength number |
| url | http://www.sciencedirect.com/science/article/pii/S266681812300102X |
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