Volume 4, Issue 4, August 2015, Page: 264-274
Epidemiological Modeling of Measles Infection with Optimal Control of Vaccination and Supportive Treatment
Okey Oseloka Onyejekwe, Computational Science Program, Addis Ababa University, Arat Kilo Campus, Addis Ababa Ethiopia
Esayas Zewdie Kebede, Computational Science Program, Addis Ababa University, Arat Kilo Campus, Addis Ababa Ethiopia
Received: May 19, 2015;       Accepted: Jun. 9, 2015;       Published: Jul. 1, 2015
DOI: 10.11648/j.acm.20150404.15      View  3690      Downloads  190
Abstract
We consider an SEIR model with constant population size and formulate an optimal control problem subject to vaccination and supportive treatment as controls. Our aim is to find the optimal combination of vaccination and supportive treatment strategies that will minimize the cost of the two control measures as well as the number of infectives while efficiently balancing vaccination and management of measles applied to the models with various cost scenarios. We used Pontryagin’s maximum principle to characterize the optimal levels of the two controls. The resulting optimality system is solved numerically by forward-backward sweep method. The results show that the optimal combination of the strategies required to achieve the set objective will depend on the relative cost of each of the control measures and the resulting optimality system showed that, the use of vaccinating and supportive treating at the same time at the highest possible rate to the population as early as possible is essential for controlling measles epidemic. The results from our simulation are discussed.
Keywords
Measles, Optimal Control, Pontryagin’s Maximum Principle, Adjoint Condition, Transversality Condition, Hamiltonian, Optimality System
To cite this article
Okey Oseloka Onyejekwe, Esayas Zewdie Kebede, Epidemiological Modeling of Measles Infection with Optimal Control of Vaccination and Supportive Treatment, Applied and Computational Mathematics. Vol. 4, No. 4, 2015, pp. 264-274. doi: 10.11648/j.acm.20150404.15
Reference
[1]
CDC, Progress in global measles control, Measles (http://www.cdc.gov/measles/pubs-mmwr.html)
[2]
CDC, Morbidity and Mortality Weekly Report 2008-2015(MMWR) (https://surv.esr.cri.nz/surveillance/WeeklyMeaslesRpt.php)
[3]
CDC, Global Measles Mortality, 2000—2008, MMWR (http://www.cdc.gov/mmwr/preview/mmwrhtml/mm5847a2.htm)
[4]
WHO/UNICEF, Strengthening Immunization Services through Measles Control, Joint Annual Measles Report 2010.
[5]
WHO/UNICEF, Global Measles and Rubella Strategic Plan 2012- 2020 (2012).
[6]
WHO/UNICEF, Measles and Rubella Initiative annual report, 2012.
[7]
R.M. Anderson, R.M. May, Infectious Diseases of Humans Dynamics and Control, Oxford University Press, Oxford, 1991.
[8]
Helena Sofia Ferreira Rodrigues, Optimal Control and Numerical Optimization Applied to Epidemiological Models, 2012.
[9]
Herbert W. Hethcote, “The Mathematics of Infectious Diseases”: SIAM Review, Vol. 42, No. 4. (December 2000).
[10]
M.J. Keeling, B.T. Grenfell, Disease extinction and community size: modeling the persistence of measles, Science 275 (1997)
[11]
M.J. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2007.
[12]
M.J.Keeling and P.Rohani. Modeling infectious diseases in humans and animals. Princeton, NJ: Princeton University Press. xiii, 2008
[13]
Ethiopian health and nutrition research institute federal democratic republic of Ethiopia - Guideline on Measles Surveillance and Outbreak Management, January 2012.
[14]
H. Trottier, P. Philippe. Deterministic Modeling of Infectious Diseases: Applications To Measles And Other Similar Infections. The Internet Journal of Infectious Diseases, Volume 2 Number 1, 2001.
[15]
Suzzan Lenhart and John T.Workman, “Optimal Control Applied to Biological Models”, (2007)
[16]
D.J.D. Earn, P. Rohani, B.M. Bolker, B.T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science 287 (2000)
[17]
M.J. Ferrari, R.F. Grais, N. Bharti, A.J.K. Conlan, O.N. Bjrnstad, L.J. Wolfson, P.J. Guerin, A. Djibo, B.T. Grenfell, The dynamics of measles in sub-SaharanAfrica, Nature 451 (2008) .
[18]
Elsa Hansen and Troy Day, “Optimal control of epidemics with limited resources “, 2000
[19]
Ousmane MOUSSA TESSA, Mathematical model for control of measles by vaccination, 2013.
[20]
Sofia Rodrigues, M. Teresa T. Monteiro, Delm F. M. Torres, Bio-economic Perspectives to an Optimal Control Dengue Model Helena, January 2013.
[21]
Adriana Johnson, Optimal Control of Cell-Cycle-Specific Chemotherapy: An Examination of the Effects of Necrosis, 2010.
[22]
S. Nanda, H. Moore, S. Lenhart. Optimal control of treatment in a mathematical model of chronic myelogenous leukemia. Mathematical Biosciences, may 2012.
[23]
Federal Democratic Republic of Ethiopia, Population Census Commission. Summary statistical report of the 2007 population and housing census, Addis Ababa: Central Statistics Agency, 2008.
[24]
Federal Ministry of Health of Ethiopia, Annual report 2014/2015.
[25]
Abta Abdelhadi and Laarabi Hassan, Optimal Control Strategy for SEIR with Latent Period and a Saturated Incidence, may 2013.
[26]
Abta, Hassan Laarabi, and Hamad Talibi Alaoui, “The Hopf Bifurcation Analysis and Optimal Control of a Delayed SIR Epidemic Model Abdelhadi”, may 2014.
[27]
Cristiana J. Silva and Delfim F. M. Torres, Optimal Control of Tuberculosis, 2010
[28]
O. Diekmann, P. Heesterbeek, and M. Roberts. The construction of next-generation matrices for compartmental epidemic models. Journal of Royal Society Interface, 2010.
[29]
L. Edelstein-Keshet, Mathematical models in biology, 2005
[30]
Dorfman, Robert. “An Economic Intepretation of Optimal Control Theory.” American Economic Review (December 1969).
[31]
Mose Ongau Fred, Johana K. Sigey, Jeconiah A. Okello, James M. Okwoyo & Giterere J.Kang’ethe, “Mathematical Modeling on the Control of Measles by Vaccination”, Vol. 2, No. 3, May 2014
[32]
Federal Democratic Republic of Ethiopia, Population Census Commission. Summary statistical report of the 2007 population and housing census, Addis Ababa: Central Statistics Agency, 2008.
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