Global sensitivity analysis and optimal control of Typhoid fever transmission dynamics
Abstract
This paper presents a mathematical model aimed at studying the global behaviour and optimal control strategies for Typhoid fever. The primary objective of this study is to identify the most effective control strategy that minimizes the spread of the disease. To achieve this, we calculate the effective and basic reproduction numbers and utilize them to investigate the existence and stability of the equilibria. Furthermore, we investigate the global impact of each model parameter on the variables using Latin Hypercube Sampling and Partial Rank Correlation Coefficient. The necessary conditions of the optimal control problem are analyzed using Pontryagin’s maximum principle, and the numerical values of the model parameters are estimated using the maximum likelihood estimator. The results indicate that the optimal use of vaccination for susceptible individuals, as well as the screening and treatment of asymptomatic infected individuals, have a significant impact on reducing the spread of the disease in endemic regions.
Keyword : global sensitivity analysis, optimal control, screening and treatment, typhoid fever
How to Cite
Nyerere, N., Mpeshe, S. C., Ainea, N., Ayoade, A. A., & Mgandu, F. A. (2024). Global sensitivity analysis and optimal control of Typhoid fever transmission dynamics. Mathematical Modelling and Analysis, 29(1), 141–160. https://doi.org/10.3846/mma.2024.17859
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S.P. Sethi. Optimal quarantine programmes for controlling an epidemic spread. Journal of the Operational Research Society, pp. 265–268, 1978. https://doi.org/10.1057/jors.1978.55
J.J. Tewa, J.L. Dimi and S. Bowong. Lyapunov functions for a dengue disease transmission model. Chaos, Solitons & Fractals, 39(2):936–941, 2009. https://doi.org/10.1016/j.chaos.2007.01.069
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C.J. Uneke. Concurrent malaria and typhoid fever in the tropics: the diagnostic challenges and public health implications. J Vector Borne Dis, 452133:133–142, 2008.
G. Zaman, Y.H. Kang, G. Cho and I.H. Jung. Optimal strategy of vaccination & treatment in an SIR epidemic model. Mathematics and Computers in Simulation, 136:63–77, 2017. https://doi.org/10.1016/j.matcom.2016.11.010
O. Diekmann, J.A.P. Heesterbeek and J.A.J. Metz. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28(4):365–382, 1990. https://doi.org/10.1007/BF00178324
O. Diekmann, J.A.P. Heesterbeek and M.G. Roberts. The construction of nextgeneration matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47):873–885, 2010. https://doi.org/10.1098/rsif.2009.0386
S. Edward and N. Nyerere. Modelling typhoid fever with education, vaccination and treatment. Eng. Math, 1(1):44–52, 2016.
W.H. Fleming and R.W. Rishel. Deterministic and stochastic optimal control, volume 1. Springer Science & Business Media, 2012.
N.K. Gupta and R.E. Rink. Optimum control of epidemics. Mathematical Biosciences,18(3-4):383–396,1973. https://doi.org/10.1016/0025-5564(73)90012-6
T.K. Irena and S. Gakkhar. A dynamical model for HIV-typhoid co-infection with typhoid vaccine. Journal of Applied Mathematics and Computing, pp. 1–30, 2021. https://doi.org/10.1007/s12190-020-01485-7
T.K. Irena and S. Gakkhar. Modelling the dynamics of antimicrobial-resistant typhoid infection with environmental transmission. Applied Mathematics and Computation, 401:126081, 2021. https://doi.org/10.1016/j.amc.2021.126081
T.K. Irena and S. Gakkhar. Optimal control of two-strain typhoid transmission using treatment and proper hygiene/sanitation practices. Journal of Computational Analysis & Applications, 30(2):355–369, 2022. https://doi.org/10.1142/S0218339022500164
D. Kalajdzievska and M.Y. Li. Modeling the effects of carriers on transmission dynamics of infectious diseases. Mathematical Biosciences & Engineering, 8(3):711, 2011. https://doi.org/10.3934/mbe.2011.8.711
J.P. La Salle. The stability of dynamical systems. SIAM, 1976. https://doi.org/10.1137/1.9781611970432
S. Lenhart and J.T. Workman. Optimal control applied to biological models. Chapman and Hall/CRC, 2007. https://doi.org/10.1201/9781420011418
D.L. Lukes. Differential equations: classical to controlled. 1982.
A.L. Malisa and H. Nyaki. Prevalence and constraints of typhoid fever and its control in an endemic area of Singida region in Tanzania: Lessons for effective control of the disease. Journal of Public Health and Epidemiology, 2(5):93–99, 2010.
S. Marino, I.B. Hogue, C.J. Ray and D.E. Kirschner. A methodology for performing global uncertainty and sensitivity analysis in systems biology. Journal of Theoretical Biology, 254(1):178–196, 2008. https://doi.org/10.1016/j.jtbi.2008.04.011
S.C. Mpeshe, L.S. Luboobi and Y. Nkansah-Gyekye. Optimal control strategies for the dynamics of rift valley fever, 2014(5):1–18, 2014.
S. Mushayabasa. A simple epidemiological model for typhoid with saturated incidence rate and treatment effect. International Journal of Mathematical and Computational Sciences, 6(6):688–695, 2013.
S. Mushayabasa. Modeling the impact of optimal screening on typhoid dynamics. International Journal of Dynamics and Control, 4(3):330–338, 2016. https://doi.org/10.1007/s40435-014-0123-4
S. Mushayabasa, C. P. Bhunu and E. T. Ngarakana-Gwasira. Mathematical analysis of a typhoid model with carriers, direct and indirect disease transmission. International Journal of Mathematical Sciences and Engineering Applications, 7(1):79–90, 2013. https://doi.org/10.1155/2013/303645
J.P. Ndenda, J.B.H. Njagarah and S. Shaw. Role of immunotherapy in tumorimmune interaction: Perspectives from fractional-order modelling and sensitivity analysis. Chaos, Solitons & Fractals, 148:111036, 2021. https://doi.org/10.1016/j.chaos.2021.111036
N. Nyerere, L.S. Luboobi, S.C. Mpeshe and G.M. Shirima. Optimal control strategies for the infectiology of brucellosis. International Journal of Mathematics and Mathematical Sciences, 2020, 2020. https://doi.org/10.1155/2020/1214391
N. Nyerere, S.C. Mpeshe and S. Edward. Modeling the impact of screening and treatment on the dynamics of typhoid fever. World Journal of Modelling and Simulation, 14(4):298–306, 2018.
D. Okuonghae and A. Korobeinikov. Dynamics of tuberculosis: the effect of direct observation therapy strategy (DOTS) in Nigeria. Mathematical modelling of natural phenomena, 2(1):113–128, 2007. https://doi.org/10.1051/mmnp:2008013
L.S. Pontryagin, V.G. Poltyanskii, R.V. Gramkelidze and E.F. Mishchenko. The mathematical theory of optimal processes, 1962.
S.P. Sethi. Optimal quarantine programmes for controlling an epidemic spread. Journal of the Operational Research Society, pp. 265–268, 1978. https://doi.org/10.1057/jors.1978.55
J.J. Tewa, J.L. Dimi and S. Bowong. Lyapunov functions for a dengue disease transmission model. Chaos, Solitons & Fractals, 39(2):936–941, 2009. https://doi.org/10.1016/j.chaos.2007.01.069
G.T. Tilahun, O.D. Makinde and D. Malonza. Modelling and optimal control of typhoid fever disease with cost-effective strategies. Computational and mathematical methods in medicine, 2017, 2017. https://doi.org/10.1155/2017/2324518
C.J. Uneke. Concurrent malaria and typhoid fever in the tropics: the diagnostic challenges and public health implications. J Vector Borne Dis, 452133:133–142, 2008.
G. Zaman, Y.H. Kang, G. Cho and I.H. Jung. Optimal strategy of vaccination & treatment in an SIR epidemic model. Mathematics and Computers in Simulation, 136:63–77, 2017. https://doi.org/10.1016/j.matcom.2016.11.010