Mathematical Model on Optimal Combination of Vaccination and Antiviral Therapy to Curb In uenza in Kenya
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Published
Aug 12, 2020
Page:
134-147
Main Article Content
Derrick M. Nzioki
Department of Mathematics, Kenyatta University, P.O. Box 43844 Nairobi, Kenya.
James K. Gatoto
Department of Mathematics, Kenyatta University, P.O. Box 43844 Nairobi, Kenya.
Abstract
Human influenza is a contagious disease which, if proper precautions are not taken to control the disease, can lead to massive mortality rates and high costs will be incurred to control the disease in case of an outbreak. As a result, we investigate how the cost of implementing both vaccination and antiviral therapy can be minimized and at the same time minimize the number of infected individuals. We have developed a system of ordinary differential equations from our formulated SVIR model and used vaccination and antiviral therapy to study influenza dynamics. We have the basic reproductive number determined using the next generation matrix. The equilibria and stability of the model has also been determined and analyzed. We have used the maximization theory of Pontryagin to define the optimal control rates and then used MATLAB program to do the numerical simulations. The numerical simulations done indicate that an ideal combination of vaccination and antiviral therapy decreases the number of infected individuals which in turn reduces the cost of applying the two control measures.
Keywords:
Influenza, antiviral therapy, vaccination, reproduction number, stability, optimal control, numerical simulation.
Article Details
How to Cite
Nzioki, D. M., & Gatoto, J. K. (2020). Mathematical Model on Optimal Combination of Vaccination and Antiviral Therapy to Curb In uenza in Kenya. Journal of Advances in Mathematics and Computer Science, 35(5), 134-147. https://doi.org/10.9734/jamcs/2020/v35i530287
Section
Original Research Article
References
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Tchuenche JM, Khamis SA, Agusto FB, Mpeshe SC. Optimal control and sensitivity analysis of an influenza model with treatment and vaccination; 2011.
Muhammad AK, Zulfiqar A, Dennis LCC, Ilyas K, Saeed I, Murad U, Taza. Stability analysis of an SVIR Epidemic model with non-linear saturated incidence rate. Applied Mathematical Sciences. 2015;9:23:1145-1158. HIKARI Ltd
Available: www.m-hikari.com
Available: http://dx.doi.org/10.12988/ams.2015.41164
Saiful Islam. Equilibriums and stability of an SVIR Epidemic model. Department of Computer Science and Engineering, Jatiya Kabi Kazi Nazrul Islam University, Trishal, Mymensingh, Bangladesh; 2015.
Joko H, Titik S. Local stability analysis of an SVIR Epidemic model. Mathematics Department, Cenderawasih University; 2017.
Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics.
Proceedings of the Royal Society of London. Series A. 1927;115:700-721.
Van den Driessche, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 2002;180(1-2):29-
DOI:10.1016/s0025-5564(02)00108-6
Lyapunov AM. The general problem of the stability of motion. Taylor and Francis, London; Lasalle J. The stability of dynamical systems. Philadelphia: SIAM; 1976.
Bowman C, Gumel AB. Optimal vaccination strategies for an influenza
Lamb RA. Genes and proteins of the influenza virus . in: Krug RM, Fraenkel-Conrat, H, Wagner RR (Eds.), In the Influenza Viruses, Plenum, New York; 1989.
Oria PA, Matini W, Nelligan I, Emukule G, Scherzer M, Oyier B, et al. Are Kenyan healthcare workers willing to receive the pandemic influenza vaccine? Results from a cross sectional survey of healthcare workers in Kenya about knowledge, attitudes and practices concerning infection with and vaccination against 2009 pandemic influenza A (H1N1), 2010.Vaccine.
;29(19):361722.
Achilla RA, Bulimo WD, Majanja JM, Wadegu MO, et al. Decline of Pandemic (2009) H1N1 Influenza Cases in Sentinel Surveillance Sites in Kenya January 2012 - May 2012. Isirv Conference Abstracts; 2012.
Gasparini R, Amicizia D, Lai PL, Panatto D. Clinical and socioeconomic impact of seasonal and pandemic inuenza in adults and the elderly. Hum. Vaccin. Immunother. 2012;8:2128.
Molinari NA, Ortega-Sanchez IR, Messonnier ML, Thompson WW, Wortley PM, Weintraub E, Bridges CB. The annual impact of seasonal inuenza in the US: Measuring disease burden and costs. Vaccine. 2007;25:50865096.
Lenhart S, Wortman J. Optimal control applied to biological models. Taylor and Francis, Boca Raton; 2007.
Gaff H, Schaefer E. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences and Engineering. 2009;6:469492.
Jan R, Xiao Y. Effect of partial immunity on transmission dynamics of dengue disease with optimal control. Math Meth Appl Sci. 2019;42:19671983
Jan R, Xiao Y. Effect of pulse vaccination on dynamics of dengue with periodic transmission functions. Adv Differ Equ. 2019;(2019):368.
Available:https://doi.org/10.1186/s13662-019-2314-y
Jan R, Khan MA, GmezAguilar JF. Asymptomatic carriers in transmission dynamics of dengue with control interventions. Optimal Control Applications and Methods. 2020;41(2):430-447.
Zaman G, Kang Y, Jung I. Stability analysis and optimal vaccination of a sir epidemic model; Herbert WH. Optimal ages of vaccination for measles. Mathematical Biosciences. 1988;89:2952.
Tchuenche JM, Khamis SA, Agusto FB, Mpeshe SC. Optimal control and sensitivity analysis of an influenza model with treatment and vaccination; 2011.
Muhammad AK, Zulfiqar A, Dennis LCC, Ilyas K, Saeed I, Murad U, Taza. Stability analysis of an SVIR Epidemic model with non-linear saturated incidence rate. Applied Mathematical Sciences. 2015;9:23:1145-1158. HIKARI Ltd
Available: www.m-hikari.com
Available: http://dx.doi.org/10.12988/ams.2015.41164
Saiful Islam. Equilibriums and stability of an SVIR Epidemic model. Department of Computer Science and Engineering, Jatiya Kabi Kazi Nazrul Islam University, Trishal, Mymensingh, Bangladesh; 2015.
Joko H, Titik S. Local stability analysis of an SVIR Epidemic model. Mathematics Department, Cenderawasih University; 2017.
Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics.
Proceedings of the Royal Society of London. Series A. 1927;115:700-721.
Van den Driessche, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 2002;180(1-2):29-
DOI:10.1016/s0025-5564(02)00108-6
Lyapunov AM. The general problem of the stability of motion. Taylor and Francis, London; Lasalle J. The stability of dynamical systems. Philadelphia: SIAM; 1976.
Bowman C, Gumel AB. Optimal vaccination strategies for an influenza