Mathematical modeling and optimal control strategies of Buruli ulcer in possum mammals

Muhammad Altaf Khan; E. Bonyah; Yi-Xia Li; Taseer Muhammad; K. O. Okosun; Muhammad Altaf Khan; E. Bonyah; Yi-Xia Li; Taseer Muhammad; K. O. Okosun

doi:10.3934/math.2021572

AIMS Mathematics

2021, Volume 6, Issue 9: 9859-9881. doi: 10.3934/math.2021572

Previous Article Next Article

Research article

Mathematical modeling and optimal control strategies of Buruli ulcer in possum mammals

1.
Institute for Groundwater studies, Faculty of Natural and Agricultural Sciences, University of the Free State South Africa
2.
Department of Mathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi, Ghana
3.
College of Mathematics and Finance, Xiangnan University, Chenzhou 423000, China
4.
Department of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi Arabia
5.
Department of Mathematics, Vaal University of Technology, Vanderbijlpark, South Africa

Received: 17 February 2021 Accepted: 13 May 2021 Published: 29 June 2021
MSC : 92J15, 34C08

Buruli is a neglected tropical disease that can now be found in many countries including developed countries like Australia. It is a skin disorder that usually occurs in the arms and legs. The disease has been identified in a number of mammals, particularly possum. Adequate eradication and control programs are needed to minimize infection and its spread to less developed countries before it becomes an epidemic. In this work, a SIR type possum epidemic model is proposed. The properties of the model are thoroughly studied and obtained its stability results. We determine the stability of the model at its fixed points and show that the model is locally and globally asymptomatically stable. The stability of the disease-free case is shown for $ \mathcal{R}_0 < 1 $ and the endemic case is examined for $ \mathcal{R}_0 > 1$. We further extend the model using the control variables and obtain an optimal control system and using the optimal control theory to characterize the necessary condition for controlling the spread of Buruli ulcer (BU). The model results are plotted to determine the best strategies for disease elimination. Numerical simulation has shown that the useful strategy consists in implementing all suggested controls.
- possum mammal,
- optimal control,
- Buruli ulcer,
- Mycobacterium ulcerans,
- stability analysis
Citation: Muhammad Altaf Khan, E. Bonyah, Yi-Xia Li, Taseer Muhammad, K. O. Okosun. Mathematical modeling and optimal control strategies of Buruli ulcer in possum mammals[J]. AIMS Mathematics, 2021, 6(9): 9859-9881. doi: 10.3934/math.2021572

Related Papers:

Abstract

Buruli is a neglected tropical disease that can now be found in many countries including developed countries like Australia. It is a skin disorder that usually occurs in the arms and legs. The disease has been identified in a number of mammals, particularly possum. Adequate eradication and control programs are needed to minimize infection and its spread to less developed countries before it becomes an epidemic. In this work, a SIR type possum epidemic model is proposed. The properties of the model are thoroughly studied and obtained its stability results. We determine the stability of the model at its fixed points and show that the model is locally and globally asymptomatically stable. The stability of the disease-free case is shown for $ \mathcal{R}_0 < 1 $ and the endemic case is examined for $ \mathcal{R}_0 > 1$. We further extend the model using the control variables and obtain an optimal control system and using the optimal control theory to characterize the necessary condition for controlling the spread of Buruli ulcer (BU). The model results are plotted to determine the best strategies for disease elimination. Numerical simulation has shown that the useful strategy consists in implementing all suggested controls.

References

[1]	E. Bonyah, I. Dontwi, F. Nyabadza, A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment, Comput. Math. Meth. Med., 2014 (2014), 1–14.
[2]	E. Bonyah, K. Badu, S. Kwes, Optimal control application to an Ebola model, Asian Pac. J. Trop. Biom., 6 (2016), 283–289. doi: 10.1016/j.apjtb.2016.01.012
[3]	F. Nyabadza, E. Bonyah, On the transmission dynamics of Buruli ulcer in Ghana: Insights through a mathematical model, BMC Res. Notes, 8 (2015), 1–15. doi: 10.1186/1756-0500-8-1
[4]	E. Bonyah, I. Dontwi, F. Nyabadza, Optimal Control Applied to the Spread of Buruli Uclcer Disease, Am. J. Comput. Appl. Math., 4 (2014), 61–76.
[5]	K. Blayneh, Y. Cao, H. Kwon, Optimal control of vector-borne diseases: Treatment and Prevention, Disc. Cont. Dyn. Sys. Ser. B, 11 (2009), 587–611.
[6]	O. D. Makinde, K. O. Okosun, Impact of chemo-therapy on optimal control of malaria disease with infected immigrants, BioSystems, 104 (2011), 32–41. doi: 10.1016/j.biosystems.2010.12.010
[7]	E. Bonyah, M. A. Khan, K. O. Okosun, S. Islam, A theoretical model for Zika virus transmission, Plos one, 12 (2017), e0185540. doi: 10.1371/journal.pone.0172713
[8]	K.O. Ouifki, N. Marcus, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, BioSystems, 106 (2011), 136–145. doi: 10.1016/j.biosystems.2011.07.006
[9]	S. Ullah, M. A. Khan, Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study, Chaos Soliton. Fract., 139 (2020), 110075. doi: 10.1016/j.chaos.2020.110075
[10]	E. O. Alzahrani, W. Ahmad, M. A. Khan, M. J. Malebary, Optimal Control Strategies of Zika Virus Model with Mutant, Commun. Nonl. Sci. Num. Simul., 93 (2021), 105532. doi: 10.1016/j.cnsns.2020.105532
[11]	M. A. Khan, L. Ahmed, P. K. Mandal, R. Smith, M. Haque, Modelling the dynamics of Pine Wilt Disease with asymptomatic carriers and optimal control, Sci. Rep., 10 (2020), 1–15. doi: 10.1038/s41598-019-56847-4
[12]	S. Ullah, O. Ullah, M. A. Khan, T. Gul, Optimal control analysis of tuberculosis (TB) with vaccination and treatment, Eur. Phy. Jour. Plus, 135 (2020), 1–27. doi: 10.1140/epjp/s13360-019-00059-2
[13]	M. A. Khan, S. A. A. Shah, S. Ullah, K. O. Okosun, Optimal control analysis of the effect of treatment, isolation and vaccination on hepatitis B virus, J. Biol. Syst., 28 (2020), 351–376. doi: 10.1142/S0218339020400057
[14]	J. P. Lasalle, Stability theroy for difference equations, J. K. Hale (Ed.) Studies in Ordinary Differential Equations, Washington DC: Math. Assoc. of America, 1977.
[15]	E. Bonyah, K. O Okosun, Mathematical modeling of Zika virus, Asi. Paci. Jour. Trop. Dis., 6 (2016), 673–679. doi: 10.1016/S2222-1808(16)61108-8
[16]	W. H. Fleming, R. W. Rishel, Deterministic and stochastic optimal control, Springer Verlag, New York (1975).
[17]	L. S. Pontryagin, The mathematical theory of optimal processes, Wiley, New York, 1962.
[18]	K. Asiedu, F. Portaels, Mycobacterium ulcerans infection, World Health Organisation, Global Buruli Ulcer Initiative, 2000.
[19]	A. Bolliger, B. R. V. Forbes, W. B. Kirkland, Transmission of a recently isolated mycobacterium to phalangers (Trichosurusvulpecula), Science, 12 (1950), 146–147.
[20]	J. Hayman, Postulated epidemiology of Mycobacterium ulcerans infection, Inter. J. Epi., 20 (1991), 1093–1098. doi: 10.1093/ije/20.4.1093
[21]	G. Birkhoff, G. C. Rota, Ordinary differential equations, Ginn, Boston, 1982.
[22]	P. V. D. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. doi: 10.1016/S0025-5564(02)00108-6
[23]	U. D. Purwati, F. Riyudha, H. Tasman, Optimal control of a discrete age-structured model for tuberculosis transmission, Heliyon, 6 (2020), e03030. doi: 10.1016/j.heliyon.2019.e03030
[24]	H. Tasman, Optimal control of HIV resistance and tuberculosis co-infection using treatment intervention, Asi. Pac. Jour. Trop. Dis., 7 (2017), 366–373. doi: 10.12980/apjtd.7.2017D6-400
[25]	Fatmawati, D. Utami, M. U. I. Purwati, C. Alfiniyah, Y. Prihartini, The dynamics of tuberculosis transmission with optimal control analysis in Indonesia, Commun. Math. Biol. Neur., 2020 (2020), 1–17.
[26]	L. Marsollier, R. Robert, J. Aubry, J. P. Saint Andre, H. Kouakou, P. Legras, et al., Aquatic insects as a vector for Mycobacterium ulcerans, Appl. Environ. Microbiol., 68 (2002), 4623–4628. doi: 10.1128/AEM.68.9.4623-4628.2002
[27]	J. A. Hayman, J. Hibble, Mycobacterium ulcerans in wild animals, Rev. sci. tech. Off. Int. Epiz., 20 (2001), 252–264.
[28]	D. L. Lukes, Differential equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, New York, 1982.
[29]	J. A. M. Fyfe, C. J. Lavender, K. A. Handasyde, A. R. Legione, C. R. O'Brien, T. P. Stinear, et al., A Major Role for Mammals in the Ecology of Mycobacterium ulcerans, PLoS. Negl. Trop. Dis., 4 (2010), e791. doi: 10.1371/journal.pntd.0000791
[30]	L. Marsollier, R. Robert, J. Aubry, J. Saint Andre, H. Kouakou, et al., Aquatic insects as a vector for Mycobacterium ulcerans, Appl. Environ. Microbiol., 68 (2002), 4623–4628. doi: 10.1128/AEM.68.9.4623-4628.2002
[31]	H. R. Williamson, M. E. Benbow, K. D. Nguyen, D. C. Beachboard, R. K. Kimbirauskas, Distribution of Mycobacterium ulcerans in Buruli Ulcer Endemic and NonEndemic Aquatic Sites in Ghana, PLoS. Negl. Trop. Dis., 2 (2008), 1–15.
[32]	F. Portaels, W. M. Meyers, A. Ablordey, A. G. Castro, K. Chemlal, et al., First Cultivation and Characterization of Mycobacterium ulcerans from the Environment, PLoS. Negl. Trop. Dis., 2 (2008), 40–55.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)