The analysis of the mathematical stability of a cholera disease model

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 01 بهمن 1403   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.88685.1464

نویسندگان

I. Sahib^{* 1}؛ M. Baroudi¹؛ H. Gourram¹؛ B. Khajji²؛ A. Labzai²؛ M. Belam¹

¹Laboratory LMACS, Sultan Moulay Slimane University, MATIC Research Team: Applied Mathematics and Information and Communication Technologie, Department of Mathematics and Computer Science, Khouribga Polydisciplinary Faculty, Morocco.

²Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, Morocco.

چکیده

In this study, we develop a deterministic model for cholera transmission dynamics, incorporating vaccination campaigns, treatment of infected individuals, and water sanitation initiatives. A novel feature of our model is the inclusion of healthcare centers, which enhances the simulation of treatment dynamics, offering new insights into cholera  anagement. The model’s central metric is the basic reproduction number $R0$, derived from the disease-free equilibrium $(DF E)$ condition. Stability analysis shows that when $R0 ≤ 1$, the $DFE$ is asymptotically stable, ensuring cholera eradication, while $R0 > 1$ leads to an endemic equilibrium. Sensitivity analysis highlights that vaccination, treatment, sanitation, and public awareness campaigns are critical for reducing $R0$. The inclusion of healthcare centers further improves the model’s effectiveness by ensuring timely treatment. Numerical simulations, validated using $M AT LAB$, confirm that omprehensive public health strategies, including expanded vaccination campaigns and healthcare infrastructure, are essential for combating cholera outbreaks. This model underscores the importance of timely medical intervention in reducing infection rates and fatalities. 

کلیدواژه‌ها

Stability؛ Sensitivity؛ Optimal Control؛ Disease Cholera

مراجع

[1] Agarwal, R.P. Difference equations and inequalities: Theory, methods, and applications, CRC Press, 2000.
[2] Akinsulie, O.C., Adesola, R.O., Aliyu, V.A., Oladapo, I.P. and Hamzat, A. Epidemiology and Transmission Dynamics of Viral Encephali-tides in West Africa. Infect Dis Rep. 2023 Sep 5;15(5):504-517. doi: 10.3390/idr15050050.
[3] Bani-Yaghoub, M., Gautam, R., Shuai, Z., Van Den Driessche, P. and Ivanek, R.Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn. 6 (2) (2012) 923–940.
[4] Bouaine, A., Rachik, M. and Hattaf, K. Optimization strategies are applied to discrete epidemic models with specific nonlinear incidence rates, Int. J. Math. Appl. 4 (2016) 73–80.
[5] Chitnis, N., Hyman, J.M. and Cushing, J.M. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol. 70 (2008) 1272–1296.
[6] Codeço, C.T. Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis. 1 (2001) 1–14.
[7] Falaye, A.J., Awonusi, F., Eseyin, O. and Eboigbe, G. The weak form market efficiency and the Nigerian stock exchange, Afro Asian Journal of Social Sciences, 9 (4) (2018).

[8] Ferguson, N. Capturing human behaviour. Nature 446, (7137) (2007) 733–733.
[9] Gumel, A.B., Shivakumar, P.N. and Sahai, B.M. A mathematical model for the dynamics of HIV-1 during the typical course of infection, Nonlinear Anal. Theory Methods Appl. 47 (3) (2001) 1773–1783.
[10] Hartley, T.W. Public perception and participation in water reuse, De-salination 187 (1-3) (2006): 115–126.
[11] Hu, Z., Teng, Z. and Jiang, H. Stability analysis in a class of discrete SIRS epidemic models,Nonlinear Anal. Real World Appl. 13 (5) (2012) 2017–2033.
[12] Karrakchou, J., Rachik, M. and Gourari, S. Optimal control and infectiology: application to an HIV/AIDS model, Appl.Math. Comput. 177 (2) (2006) 807–818.
[13] Kot, M. Elements of mathematical ecology, Cambridge University Press, 2001.
[14] Kumar, P., Mishra, D.K., Deshmukh, D.G., Jain, M., Zade, A.M., In-gole, K.V., Goel, A.K. and Yadava. P.K. Vibrio cholerae O1 Ogawa El Tor strains with the ctxB7 allele driving cholera outbreaks in south-western India in 2012, Infect. Genet. Evol. 25 (2014) 93–96.
[15] Liu, L., Wong, Y.S. and Lee, B.H.K. Application of the Centre Manifold Theory in Non-Linear Aeroelasticity, Journal of Sound and Vibration. 234 (4) (2000) 641–659.
[16] Mwasa, A. and Tchuenche, J.M. Mathematical Analysis of a Cholera Model with Public Health Interventions. Biosystems 105 (3) (2011) 190–200.
[17] Nelson, E.J., Harris, J.B., Glenn Morris Jr, J., Calderwood, S.B. and Camilli, A. Cholera transmission: The host, pathogen and bacteriophage dynamic, Nat. Rev. Microbiol. 7(10) (2009) 693–702.

[18] Okolo, P.N., Magaji, A.S., Joshua, I. and Useini, P.F. Mathematical modelling and analysis of cholera disease dynamics with control FUDMA J. Sci. 4 (4) (2020) 363–381.
[19] Samanta, S., Rana, S., Sharma, A., Misra, A.K. and Chattopadhyay, J. Effect of awareness programs by media on the epidemic outbreaks: A mathematical model, Appl. Math. Comput. 219 (12) (2013) 6965–6977.
[20] Sebastian, E. and Victor, P. Optimal control strategy of a discrete-time svir epidemic model with immigration of infectives, Int. J. Pure Appl. Math. 113 (8) (2017) 55–63.
[21] Tahir, M., Inayat Ali Shah, S., Zaman, G. and Muhammad, S. Ebola virus epidemic disease its modeling and stability analysis required abstain strategies, Cogent Biol. 4 (1) (2018) 1488511.
[22] Van den Driessche, P. and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (1-2) (2002) 29–48.
[23] Vashist, A., Verma, J., Narendrakumar, L. and Das, B. Molecular In-sights into Genomic Islands and Evolution of Vibrio cholerae, In Micro-bial Genomic Islands in Adaptation and Pathogenicity: Springer, 2023.
[24] Wang, J. and Modnak, C. Modeling cholera dynamics with controls, Can. Appl. Math. Q. 19 (3) (2011) 255–273.
[25] Watson, A.P., Armstrong, A.Q., White, G.H. and Thran, B.H. Health-based ingestion exposure guidelines for Vibrio cholerae: Technical basis for water reuse applications, Sci. Total Environ. 613 (2018) 379–387.