×

A mathematical model to study herbal and modern treatments against COVID-19. (English) Zbl 07877768

Summary: In this paper, we propose a two-group deterministic COVID-19 model which takes into account educational campaigns and the fact that people infected with COVID-19 may choose either modern (allopathic) medicine, traditional medicine or may combine the two modes of treatment. The model is analysed in the case where modern medicine is the only mode of treatment and when traditional medicine is taken as an adjuvant (or another mode of treatment). We prove in the first case that the model has a disease-free equilibrium (DFE), globally asymptotically stable when the control reproduction number is less than one and whenever it is greater than one, we prove the local asymptotic stability of the endemic equilibrium. In the second case, we prove that, misconceptions in the population lead to a backward bifurcation phenomenon, which makes the control of the disease more difficult. We derive using the Lyapunov method that a threshold \(\mathcal{T}\) ensures the global asymptotic stability of DFE in some cases when its value is less than one. Both models are fitted using daily COVID-19 cumulative cases reported from January to February 2022 in South Africa. We found a control reproduction number less than one, meaning that COVID-19 will be eliminated. Comparison of the two models fits highlights that misconceptions should be taken into account to accurately describe the dynamics of COVID-19 in South Africa. Numerically, we prove that educational campaigns should focus on preventive measures and both traditional and allopathic medicine health care systems should complement each other in the fight against COVID-19.

MSC:

92-XX Biology and other natural sciences
91-XX Game theory, economics, finance, and other social and behavioral sciences
Full Text: DOI

References:

[1] T. Zheng, Y. Luo, X. Zhou, L. Zhang, and Z. Teng, “Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence,” Commun. Pure Appl. Anal., vol. 22, no. 2, pp. 365-396, 2021, doi:10.3934/cpaa.2021154. · Zbl 1515.35289 · doi:10.3934/cpaa.2021154
[2] S. Saha, G. Samanta, and J. J. Nieto, “Epidemic model of COVID-19 outbreak by inducing behavioural response in population,” Nonlinear Dyn., vol. 102, no. 1, pp. 455-487, 2020. doi:10.1007/s11071-020-05896-w. · Zbl 1517.92040 · doi:10.1007/s11071-020-05896-w
[3] WHO Coronavirus (COVID-19) Dashboard, 2023 [Online]. Available at: https://covid19.who.int Accessed: Jul. 8, 2023.
[4] S. Saha and G. Samanta, “Modelling the role of optimal social distancing on disease prevalence of COVID-19 epidemic,” Int. J. Dyn. Control, vol. 9, no. 3, pp. 1053-1077, 2021. doi:10.1007/s40435-020-00721-z. · doi:10.1007/s40435-020-00721-z
[5] P. M. Mphekgwana, M. Makgahlela, and T. M. Mothiba, “Use of traditional medicines to fight COVID-19 during the South African nationwide lockdown: a prevalence study among university students and academic staff,” Open Public Health J., vol. 14, no. 1, pp. 441-445, 2021, doi:10.2174/1874944502114010441. · doi:10.2174/1874944502114010441
[6] A. W. Adunimay and T. A. Ojo, “Western centric medicine for Covid-19 and its contradictions: can African alternate solutions be the cure?” Front. Polit. Sci., vol. 4, p. 835238, 2022, doi:10.3389/fpos.2022.835238. · doi:10.3389/fpos.2022.835238
[7] E. O. J. Ozioma and O. A. N. Chinwe, “Herbal medicines in African traditional medicine,” Herbal Med., vol. 10, pp. 191-214, 2019, doi:10.5772/intechopen.80348. · doi:10.5772/intechopen.80348
[8] S. Mutola, N. V. Pemunta, and N. V. Ngo, “Utilization of traditional medicine and its integration into the healthcare system in Qokolweni, South Africa; prospects for enhanced universal health coverage,” Complement. Therap. Clin. Pract., vol. 43, pp. 1-4, May 2021. doi:10.1016/j.ctcp.2021.101386. · doi:10.1016/j.ctcp.2021.101386
[9] V. P. Titanji, “COVID-19 Response: the case for phytomedicines in Africa with particular focus on Cameroon,” J. Cameroon Acad. Sci., vol. 17, no. 2, pp. 163-175, 2021, doi:10.4314/jcas.v17i2.6. · doi:10.4314/jcas.v17i2.6
[10] WHO Traditional Medicine Strategy: 2014-2023, 2013 [Online]. Available at: https://www.who.int/publications/i/item/9789241506096 Accessed: May 15, 2023.
[11] K. H. Ndukong, COVID-19 Treatment: Endogenous Cures Can Also Do the Trick! 2020 [Online]. Available at: http://en.people.cn/n3/2020/0526/c90000-9694588.html Accessed: Nov. 27, 2022.
[12] S. Saha, P. Dutta, and G. Samanta, “Dynamical behavior of SIRS model incorporating government action and public response in presence of deterministic and fluctuating environments,” Chaos Solit. Fractals, vol. 164, p. 112643, November 2022. doi:10.1016/j.chaos.2022.112643. · doi:10.1016/j.chaos.2022.112643
[13] P. Harjule, V. Tiwari, and A. Kumar, “Mathematical models to predict COVID-19 outbreak: an interim review,” J. Interdis. Math., vol. 24, no. 2, pp. 259-284, 2021, doi:10.1080/09720502.2020.1848316. · doi:10.1080/09720502.2020.1848316
[14] M. Massard, R. Eftimie, A. Perasso, and B. Saussereau, “A multi-strain epidemic model for COVID-19 with infected and asymptomatic cases: application to French data,” J. Theor. Biol., vol. 545, p. 111117, July 2022, doi:10.1016/j.jtbi.2022.111117. · Zbl 1491.92118 · doi:10.1016/j.jtbi.2022.111117
[15] C. H. Nkwayep, S. Bowong, J. Tewa, and J. Kurths, “Short-term forecasts of the COVID-19 pandemic: a study case of Cameroon,” Chaos Solit. Fractals, vol. 140, p. 110106, November 2020, doi:10.1016/j.chaos.2020.110106. · doi:10.1016/j.chaos.2020.110106
[16] T. Piasecki, P. B. Mucha, and M. Rosińska, “On limits of contact tracing in epidemic control,” Plos One, vol. 16, no. 8, pp. 1-21, 2021, doi:10.1371/journal.pone.0256180. · doi:10.1371/journal.pone.0256180
[17] C. Tadmon and S. Foko, “A transmission dynamics model of Covid-19: case of Cameroon,” Infect. Dis. Model., vol. 7, no. 2, pp. 211-249, 2022. doi:10.1016/j.idm.2022.05.002. · doi:10.1016/j.idm.2022.05.002
[18] Coronavirus Disease 2019 (COVID-19) Treatment Guidelines, 2022 [Online]. Available at: https://www.covid19treatmentguidelines.nih.gov/overview/overview-of-covid-19 Accessed: Jul. 16, 2022.
[19] S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, vol. 23, Berlin, Germany, Springer Science & Business Media, 2012.
[20] R. Ouifki and J. Banasiak, “Epidemiological models with quadratic equation for endemic equilibria—a bifurcation atlas,” Math. Methods Appl. Sci., vol. 43, no. 18, pp. 1-17, 2020, doi:10.1002/mma.6389. · Zbl 1471.34097 · doi:10.1002/mma.6389
[21] COVID-19 Statistics in South Africa, 2022 [Online]. Available at: https://sacoronavirus.co.za Accessed: Apr. 4, 2022.
[22] M. Chapwanya, J. Lubuma, Y. Terefe, and B. Tsanou, “Analysis of war and conflict effect on the transmission dynamics of the Tenth Ebola outbreak in the Democratic Republic of Congo,” Bull. Math. Biol., vol. 84, no. 12, p. 136, 2022, doi:10.1007/s11538-022-01094-4. · Zbl 1503.92061 · doi:10.1007/s11538-022-01094-4
[23] South Africa Population, 2022 [Online]. Available at: https://countrymeters.info/en/South_Africa#population_2022 Accessed: Jul. 21, 2022.
[24] M. M. Ojo, T. O. Benson, O. J. Peter, and E. F. D. Goufo, “Nonlinear optimal control strategies for a mathematical model of COVID-19 and influenza co-infection,” Phys. A Stat. Mech. Appl., vol. 607, p. 128173, 1 December 2022, doi:10.1016/j.physa.2022.128173. · Zbl 07614922 · doi:10.1016/j.physa.2022.128173
[25] S. Marino, I. B. Hogue, C. J. Ray, and D. E. Kirschner, “A methodology for performing global uncertainty and sensitivity analysis in systems biology,” J. Theor. Biol., vol. 254, no. 1, pp. 178-196, 2008, doi:10.1016/j.jtbi.2008.04.011. · Zbl 1400.92013 · doi:10.1016/j.jtbi.2008.04.011
[26] R. Taylor, “Interpretation of the correlation coefficient: a basic review,” J. Diagn. Med. Sonogr., vol. 6, no. 1, pp. 35-39, 1990, doi:10.1177/875647939000600106. · doi:10.1177/875647939000600106
[27] J. P. LaSalle, The Stability of Dynamical Systems, Philadelphia, SIAM, 1976. · Zbl 0364.93002
[28] C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Math. Biosci. Eng, vol. 1, no. 2, pp. 361-404, 2004, doi:10.3934/mbe.2004.1.361. · Zbl 1060.92041 · doi:10.3934/mbe.2004.1.361
[29] S. Saha, G. Samanta, and J. J. Nieto, “Impact of optimal vaccination and social distancing on COVID-19 pandemic,” Math. Comput. Simul., vol. 200, pp. 285-314, 2022. doi:10.1016/j.matcom.2022.04.025. · Zbl 1540.92231 · doi:10.1016/j.matcom.2022.04.025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.