Abstract
A transmission and control model for measles infection is presented. The model incorporates vaccinated individuals and the role of treatment for both exposed and infected individuals. We present two main equilibrium points (disease-free and endemic) with the analysis of their stability. The basic reproduction number is calculated and we find that when it is less than unity, the disease-free equilibrium point is both locally and globally stable which means the disease can be eradicated under such condition. When it is greater than one, the infection is uniformly persistent and the endemic equilibrium is globally stable. The sensitivity index of basic reproduction number to the parameters within the model is also determined. Further, by using Pontryagin’s minimum principle, the optimal control problem is constructed with three controls i.e. vaccination, treatment of exposed individuals and treatment of infected individuals. Finally, the numerical simulations are established and our results show that a combination of all three controls gives the best result in reducing the number of measles infected individuals. These results indicate that being vaccinated followed by some treatments for both exposed and infected individuals would make measles eradication more efficient.
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References
Mazer, A., Sankalé, M.: Guide de médecine en Afrique et Océan Indien. EDICEF, Paris (1988)
Tessa, O.M.: Mathematical Model for Control of Measles by Vaccination, pp. 31–36. Department of Mathematics and Computer Science, Abdou Moumouni University, Niamey (2006)
Norrby, E., Oxman, M.N.: Measles virus. In: Fields, B.N., Knipe, D.M. (eds.) Virology, 2nd edn, pp. 1013–44. Raven Press Ltd, New York (1990)
Perry, R.T., Halsey, N.A.: The clinical significance of measles: a review. J. Infect. Dis. 189, S4–16 (2004)
Panum, P.L.: Observations Made During the Epidemic of Measles on the Faroe Islands in the Year 1846. Delta Omega Society, Cleveland (1940)
Mossong, J., Muller, C.P.: Modelling measles re-emergence as a result of waning of immunity in vaccinated populations. Vaccine 21, 4597–4603 (2003)
Ejima, K., Omori, R., Aihara, K., Nishiura, H.: Real-time investigation of measles epidemics with estimate of vaccine efficacy. Int. J. Biol. Sci. 8(5), 620–629 (2012)
WHO, World Health Organization.: Measles. https://www.who.int/news-room/fact-sheets/detail/measles (2018). Accessed 12 March 2019
Ciupe, S.M.: Modeling the dynamics of hepatitis B infection, immunity, and drug therapy. Immunol. Rev. 285, 38–54 (2018). https://doi.org/10.1111/imr.12686
Viriyapong, R., Koompawan, G.: The impact of hygiene care and maternal immunity on stability behaviour of rotavirus infection model for children under the age of five in Thailand. Int. J. Math. Model. Numer. Optim. 8(4), 378–392 (2018). https://doi.org/10.1504/IJMMNO.2018.10015804
Yosyingyong, P., Viriyapong, R.: Global stability and optimal control for a hepatitis B virus infection model with immune response and drug therapy. J. Appl. Math. Comput. 60(1–2), 537–565 (2019). https://doi.org/10.1007/s12190-018-01226-x
Elaiw, A.M., Almuallem, N.A.: Global properties of delayed-HIV dynamics models with differential drug efficacy in co-circulating target cells. Appl. Math. Comput. 265, 1067–1089 (2015). https://doi.org/10.1016/j.amc.2015.06.011
Jia, J., Xiao, J.: Stability analysis of a disease resistance SEIRS model with nonlinear incidence rate. Adv. Differ. Equ. 75, 13 (2018). https://doi.org/10.1186/s13662-018-1494-1
Rahman, G.U., Shah, K., Haq, F., Ahmad, N.: Host vector dynamics of pine wilt disease model with convex incidence rate. Chaos Solitons Fract. 113, 31–39 (2018). https://doi.org/10.1016/j.chaos.2018.05.010
Khan, S.A., Shah, K., Zaman, G., Jarad, F.: Existence theory and numerical solutions to smoking model under Caputo–Fabrizio fractional derivative. Chaos Interdiscip. J. Nonlinear Sci. 29(1), 013128 (2019). https://doi.org/10.1063/1.5079644
Hag, F., Shah, K., Rahman, G.U., Li, Y., Shazad, M.: Computational analysis of complex population dynamical model with arbitrary order. Complexity. 2018, Article ID 8918541, p. 8. https://doi.org/10.1155/2018/8918541 (2018)
Okyere - Siabouh, S., Adetunde, I.A.: Mathematical model for the study of measles in cape coast metropolis. Int. J. Mod. Biol. Med. 4(2), 110–133 (2013)
Momoh, A.A., Ibrahim, M.O., Uwanta, J.I., Manga, S.B.: Mathematical model for control of measles epidemiology. Int. J. Pure Appl. Math. 87(5), 707–718 (2013)
Garba, S.M., Safi, M.A., Usaini, S.: Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics. Math. Methods Appl. Sci. 40, 6371–6388 (2016)
Bolarin, G.: On the dynamical analysis of a new model for measles infection. Int. J. Math. Trends Technol. 7(2), 2231–5373 (2014)
Edward, S., Raymond, K., Gabriel, K., Nestory, F., Godfrey, M., Arbogast, M.: A mathematical model for control and elimination of the transmission dynamics of measles. Appl. Comput. Math. 4(6), 396–408 (2015)
Obumneke, C., Adamu, I.I., Ado, S.T.: Mathematical model for the dynamics of measles under the combined effect of vaccination and measles therapy. International Journal of Science and Technology 6(6), 862–874 (2017)
Beay, L.K.: Modelling the effects of treatment and quarantine on measles. In: AIP Conference Proceedings (2018)
Ochoche, J.M., Gweryina, R.I.: A mathematical model of measles with vaccination and two phases of infectiousness. IOSR J. Math. (IOSR-JM) 10(1), 95–105 (2014)
van den Driessche, P., Watmough, J.: Reproductive numbers and sub-threshold endemic equilibria for compartment models of disease transmission. Math. Biosci. 180, 29–48 (2002). https://doi.org/10.1016/S0025-5564(02)00108-6
LaSalle, J.P.: The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1976)
Li, M.Y., Muldowney, J.S.: A geometric approach to global-stability problems. SIAM J. Math. Anal. 27(4), 1070–1083 (1996). https://doi.org/10.1137/S0036141094266449
Li, M.Y., Muldowney, J.S.: On Bendixson’s criterion. J. Differ. Equ. 106(1), 27–39 (1993). https://doi.org/10.1006/jdeq.1993.1097
Freedman, H.I., Ruan, S., Tang, M.: Uniform persistence and flows near a closed positively invariant set. J. Dyn. Differ. Equ. 6(4), 583–600 (1994)
Butler, G., Freedman, H.I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96(3), 425–30 (1986)
Samsuzzoha, M.D., Singh, M., Lucy, D.: Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza. Appl. Math. Model. 37, 903–915 (2013). https://doi.org/10.1016/j.apm.2012.03.029
Ngoteya, F.N., Gyekye, Y.N.: Sensitivity analysis of parameters in a competition model. Appl. Comput. Math. 4(5), 363–368 (2015). https://doi.org/10.11648/j.acm.20150405.15
Pontryagin, L.S.V., Boltyanskii, G.R., Gamkrelidze, V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Gordon and Breach Science Publishers, London (1986)
Peter, O., Afolabi, O., Victor, A., Akpan, C., Oguntolu, F.: Mathematical model for the control of measles. J. Appl. Sci. Environ. Manag. 22(4), 571–576 (2018)
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This work has been supported by Department of Mathematics, Faculty of Science, Naresuan University, Thailand.
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Viriyapong, R., Ridbamroong, W. Global stability analysis and optimal control of measles model with vaccination and treatment. J. Appl. Math. Comput. 62, 207–237 (2020). https://doi.org/10.1007/s12190-019-01282-x
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DOI: https://doi.org/10.1007/s12190-019-01282-x