LOCAL STABILITY AND SENSITIVITY ANALYSIS OF CHOLERA DISEASE DYNAMICS WITH LOGISTIC GROWTH
Keywords:
Mathematical model, Cholera, Effective reproduction number, Stability, Sensitivity analysis
Abstract
Cholera remains a significant public health threat, particularly in regions with poor sanitation and
limited healthcare access. This paper presents and analyses a deterministic compartmental model
for cholera disease dynamics, incorporating logistic growth of Vibrio cholerae. The model is
formulated as a system of ordinary differential equations stratified into susceptible, exposed,
infectious, vaccinated, and recovered human compartments, with a compartment for the bacteria
concentration. The positivity of solutions is established to confirm the mathematical well-posedness of the model. The effective reproduction number Re is computed using the next
generation matrix operator, and the cholera-free equilibrium is shown to be locally asymptotically
stable whenever Re < 1. Normalised forward sensitivity analysis is performed to quantify the
relative influence of model parameters on Re. Parameters associated with vaccination, recovery
rate, and bacteria death rate yield negative sensitivity indices, indicating that their increase drives
Re below unity. Conversely, parameters associated with transmission probability, human
recruitment, and bacteria shedding rate yield positive sensitivity indices, such that their reduction
is effective in curtailing transmission.
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Communications in Mathematical Biology and Neuroscience, 2023:135.
https://doi.org/10.28919/cmbn/8285
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[19] Jackob, O. K., Akinyi, O., and Tireito, F. (2023). A Mathematical Model on the Dynamics of In-Host Infection Cholera Disease with Vaccination. Discrete Dynamics in Nature and Society, 2023: 11 Article ID 1465228. https://doi.org/10.1155/2023/1465228
[20] Resmawan, R., Yahya, L., Mahmud, S. L., Nuha, A. R., and Laita, N. H. (2023). Dynamic Analysis of the Mathematical Model for the Cholera Disease Spread Involving Medication and Environmental Sanitation. BAREKENG: Journal of Mathematics and Its Applications, 17(1), 341–360. https://doi.org/10.30598/barekengvol17iss1pp0341-0360 [21] Mushanyu, J., Matsebula, L., and Nyabadza, F. (2024). Mathematical Modeling of Cholera Dynamics in the Presence of Antimicrobial Utilization Strategy. Scientific Reports, 14, 30128. https://doi.org/10.1038/s41598-024-77834-4
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16
[24] Olaniyi, S., Falowo, O.D., Oladipo, A.T., Gogovi. K.G., and Sangotola, A.O. (2025).
Stability Analysis of Rift Valley Fever Transmission Model with Efficient and Cost-
Effective Interventions, Scientific Report, 15;14036.
https://doi.org/10.1038/s41598-025-98722-5
[25] Olaniyi, S., Chuma, F.M. (2023). Lyapunov stability and economic analysis of monkeypox
dynamics with vertical transmission and vaccination. International Journal of Applied and
Computational Mathematics 9, 85. https://doi.org/10.1007/s40819-023-01572-w
[26] Van den Driessche, P., and Watmough, J. (2002). Reproduction Numbers and Sub-
Threshold Endemic Equilibria for Compartmental Models of Disease
Transmission. Mathematical Biosciences, 180 (1-2), 29-48.
[27] Obabiyi, O.S., and Olaniyi, S. (2016). Asymptotic stability of malaria dynamics with
vigilant human compartment. International Journal of Applied Mathematics, 29(1), 127-
144. http://dx.doi.org/10.12732/ijam.v29i1.10
[28] Chitnis, N., Hyman, J. M. and Cushing, J. M. (2008). Determining Important Parameters
in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model.
Bulletin of Mathematical Biology. 70: 1272-1296.
https://doi.org/10.1007/s11538008-9299-0
[29] Sahib, I., Barouidi, M., Gourram, H., Khajji, B., Labzai, A., and Belam, M. (2024). The
Mathematical Modeling and Analysis of the Cholera Disease Model.
Communications in Mathematical Biology and Neuroscience, 2024: 93.
https://doi.org/10.28919/cmbn/8785
[30] Namawejje H., Obuya E., Luboobi L.S. (2018). Modeling Optimal Control of Cholera
Disease Under the Interventions of Vaccination, Treatment and Education Awareness, J.
Math. Res. 10: 137–152. https://doi.org/10.553 9/jmr.v10n5p137.
[31] Wang, J., and Modnak, C. (2011). Modeling Cholera Dynamics with Controls. Canadian
Applied Mathematics Quarterly, 19(3), 255–273.
[32] Harris, J. B. (2018). Cholera: Immunity and Prospects in Vaccine Development. The
Journal of Infectious Diseases, 218,141–146. https://doi.org/10.1093/infdis/jiy414
[33] Abimbade, S.F., Chuma, F.M., Olaniyi, S., Sangotola, A.O., Gogovi, K.G. (2026).
Lyapunov stability optimization of a fractional-order HIV/AIDS model coupling vertical
transmission with saturated treatment. International Journal of Mathematics and
Mathematical Sciences 2026, 3059683. https://doi.org/10.1155/ijmm/3059683
[34] Olaniyi, S., Chuma, F.M., Abimbade, S.F. (2025). Asymptotic stability analysis of a
fractional epidemic model for Ebola virus disease in Caputo sense. Journal of the Nigerian
Society of Physical Sciences 7, 2304. https://doi.org/10.46481/jnsps.2025.2304
[2] Gbodogbe, S. O. (2025). Harmonizing Epidemic Dynamics: A Fractional Calculus Approach to Optimal Control Strategies for Cholera Transmission. Scientific African,
27(2025): e02545. https://doi.org/10.1016/j.sciaf.2025.e02545
[3] Siddiqua, S., Chaturvedi, A., and Goswami, N. K. (2022). A Simple Mathematical Model of Cholera Dynamics with Sensitivity Analysis. AIP Conference Proceedings, 2516: 130002. https://doi.org/10.1063/5.0108808
[4] World Health Organization (WHO). (2024). Cholera.
https://www.who.int/news-room/factsheets/detail/cholera
[5] Olaniyi, S., Lawal, I.A., Lebelo, R.S., Chuma, F.M., Abimbade, S.F. (2025). Modelling and optimal control of influenza dynamics with structured populations based on education and isolation. International Journal of Analysis and Applications 23, 81.
https://doi.org/10.28924/2291-8639-23-2025-81
[6] Ademosu, J.A., Olaniyi, S., Abimbade, S.F., Chuma, F.M., Ogbonna, R.C., Lebelo, R.S., Okosun, K.O. (2025). Modelling population dynamics of substance abuse in the presence of addicted immigrant with real data of rehabilitation cases. Journal of Applied Mathematics 2025, 2241780. https://doi.org/10.1155/jama/2241780
[7] Ademosu, J.A., Olaniyi, S., Adewale, S.O. (2021). Stability analysis and optimal measure for controlling eco-epidemiological dynamics of prey-predator model. Advances in
Systems Science and Applications
21(2), 83–103.
https://doi.org/10.25728/assa.2021.21.2.1064
[8] Olaniyi, S., Abimbade, S.F., Ademosu, J.A., Ogbonna, R.C., Lebelo, R.S., Okosun, K.O. (2025). Impact of optimal intervention strategies on psychoactive substance abuse dynamics with addicted immigrants: a mathematical study. Complexity 2025, 5581960.
https://doi.org/10.1155/cplx/5581960
[9] Wangari, I.M., Lebelo, R.S., Sangotola, A.O., Chuma, F.M., Olaniyi, S., Simiyu, H.M.
(2025). Analysis of backward bifurcation induced by nonlinear vertical transmission in COVID-19 dynamics with optimal control. Computational and Mathematical Biophysics
13(1), 20250030. https://doi.org/10.1515/cmb-2025-0030
15
[10] Okolo, P. N., Magaji, A. S., Isaac Joshua and Useini P.F. (2020). Mathematical Modeling and Analysis of Cholera Disease Dynamics with Control. FUDMA Journal of Sciences, 4(4):363–381. https://doi.org/10.33003/fjs-2020-0404-494
[11] Hidayati, N., Sari, E. R., and Waryanto, N. H. (2021). Mathematical Model of Cholera Spread Based on SIR: Optimal Control. Pythagoras: Jurnal Pendidikan Matematika,
16(1): 70–83. https://doi.org/10.21831/pg.v16i1.35729
[12] Maliki, O. S., and Chibuike, O. C. (2021). A Mathematical Model for the Control of Cholera Epidemic Without Natural Recovery. Applied Mathematics, 12: 655–668.
https://doi.org/10.4236/am.2021.128046
[13] Onuorah, M. O., Atiku, F., and Juuko H. (2022). Mathematical Model for Prevention and Control of Cholera Transmission in a Variable Population. Research in Mathematics.
9(1):1-13 2018779. https://doi.org/10.1080/27658449.2021.2018779
[14] Demir, M. (2022). The Effect of Population Migration on the Diffusion of Cholera Outbreaks in Metapopulations. Preprint, bioRxiv.
https://doi.org/10.1101/2022.04.12.22273810
[15] Abdul, N. S., Yahya, L., Resmawan, R., and Nuha, A. R. (2022). Dynamic Analysis of the Mathematical Model of the Spread of Cholera with Vaccination Strategies. BAREKENG: Jurnal Ilmu Matematika dan Terapan, 16(1): 281–292.
https://doi.org/10.30598/barekengvol16iss1pp281-292
[16] Abdelghani, M., Bakheet, M., and Bashier, E. (2023a). AMathematical Model of Cholera Outbreak with Type III Functional Response. European Chemical Bulletin, 12(10): 4599 - 4625.
[17] Issam, S., Bouchaib, K., Labzai, A., Hicham, G., and Mohamed, B. (2023). Mathematical Modeling and Optimal Control Strategy for a Discrete-Time Cholera Model.
Communications in Mathematical Biology and Neuroscience, 2023:135.
https://doi.org/10.28919/cmbn/8285
[18] Hameed, H. H., Al-Saedi, H. M., and Al-Mayyah, M. H. (2023). Mathematical Modelling for Cholera Epidemic in Iraq. 5th International Conference on Problems of Cybernetics and Informatics. https://doi.org/10.1109/PCI60110.2023.10325918
[19] Jackob, O. K., Akinyi, O., and Tireito, F. (2023). A Mathematical Model on the Dynamics of In-Host Infection Cholera Disease with Vaccination. Discrete Dynamics in Nature and Society, 2023: 11 Article ID 1465228. https://doi.org/10.1155/2023/1465228
[20] Resmawan, R., Yahya, L., Mahmud, S. L., Nuha, A. R., and Laita, N. H. (2023). Dynamic Analysis of the Mathematical Model for the Cholera Disease Spread Involving Medication and Environmental Sanitation. BAREKENG: Journal of Mathematics and Its Applications, 17(1), 341–360. https://doi.org/10.30598/barekengvol17iss1pp0341-0360 [21] Mushanyu, J., Matsebula, L., and Nyabadza, F. (2024). Mathematical Modeling of Cholera Dynamics in the Presence of Antimicrobial Utilization Strategy. Scientific Reports, 14, 30128. https://doi.org/10.1038/s41598-024-77834-4
[22] Brhane, K. W., Ahmad, A. G., Hina, H., and Emadifar, H. (2024). Mathematical Modeling of Cholera Dynamics with Intrinsic Growth Considering Constant Interventions. Scientific Reports, (14): 4616. https://doi.org/10.1038/s41598-024-55240-0 [23] Mgumba, E. S., and Opoku, N. K.-D. O. (2025). Modeling the Influence of Rural Community Information Dissemination Centers in the Transmission Dynamics of Cholera. [Preprint]. Research Square. https://doi.org/10.21203/rs.3.rs-7094348/v1
16
[24] Olaniyi, S., Falowo, O.D., Oladipo, A.T., Gogovi. K.G., and Sangotola, A.O. (2025).
Stability Analysis of Rift Valley Fever Transmission Model with Efficient and Cost-
Effective Interventions, Scientific Report, 15;14036.
https://doi.org/10.1038/s41598-025-98722-5
[25] Olaniyi, S., Chuma, F.M. (2023). Lyapunov stability and economic analysis of monkeypox
dynamics with vertical transmission and vaccination. International Journal of Applied and
Computational Mathematics 9, 85. https://doi.org/10.1007/s40819-023-01572-w
[26] Van den Driessche, P., and Watmough, J. (2002). Reproduction Numbers and Sub-
Threshold Endemic Equilibria for Compartmental Models of Disease
Transmission. Mathematical Biosciences, 180 (1-2), 29-48.
[27] Obabiyi, O.S., and Olaniyi, S. (2016). Asymptotic stability of malaria dynamics with
vigilant human compartment. International Journal of Applied Mathematics, 29(1), 127-
144. http://dx.doi.org/10.12732/ijam.v29i1.10
[28] Chitnis, N., Hyman, J. M. and Cushing, J. M. (2008). Determining Important Parameters
in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model.
Bulletin of Mathematical Biology. 70: 1272-1296.
https://doi.org/10.1007/s11538008-9299-0
[29] Sahib, I., Barouidi, M., Gourram, H., Khajji, B., Labzai, A., and Belam, M. (2024). The
Mathematical Modeling and Analysis of the Cholera Disease Model.
Communications in Mathematical Biology and Neuroscience, 2024: 93.
https://doi.org/10.28919/cmbn/8785
[30] Namawejje H., Obuya E., Luboobi L.S. (2018). Modeling Optimal Control of Cholera
Disease Under the Interventions of Vaccination, Treatment and Education Awareness, J.
Math. Res. 10: 137–152. https://doi.org/10.553 9/jmr.v10n5p137.
[31] Wang, J., and Modnak, C. (2011). Modeling Cholera Dynamics with Controls. Canadian
Applied Mathematics Quarterly, 19(3), 255–273.
[32] Harris, J. B. (2018). Cholera: Immunity and Prospects in Vaccine Development. The
Journal of Infectious Diseases, 218,141–146. https://doi.org/10.1093/infdis/jiy414
[33] Abimbade, S.F., Chuma, F.M., Olaniyi, S., Sangotola, A.O., Gogovi, K.G. (2026).
Lyapunov stability optimization of a fractional-order HIV/AIDS model coupling vertical
transmission with saturated treatment. International Journal of Mathematics and
Mathematical Sciences 2026, 3059683. https://doi.org/10.1155/ijmm/3059683
[34] Olaniyi, S., Chuma, F.M., Abimbade, S.F. (2025). Asymptotic stability analysis of a
fractional epidemic model for Ebola virus disease in Caputo sense. Journal of the Nigerian
Society of Physical Sciences 7, 2304. https://doi.org/10.46481/jnsps.2025.2304
Published
2026-06-30
How to Cite
Ogunniyi, T. D., Ademosu, J. A., & Lawal, I. A. (2026). LOCAL STABILITY AND SENSITIVITY ANALYSIS OF CHOLERA DISEASE DYNAMICS WITH LOGISTIC GROWTH. IJO - International Journal of Mathematics (ISSN: 2992-4421 ), 9(06), 01-16. Retrieved from https://www.ijojournals.com/index.php/m/article/view/1323
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