Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T05:04:31.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Vaccination, Treatment, and Preventive Campaigns inRegard to the SIR Epidemic Model

Published online by Cambridge University Press:  20 June 2014

E.V. Grigorieva  and
E.N. Khailov
E.V. Grigorieva*
Affiliation:
Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, USA
E.N. Khailov
Affiliation:
Department of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia
*
⋆ Corresponding author. E-mail: egrigorieva@mail.twu.edu
Get access

Abstract

The Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease ina population of constant size is considered. In order to control the spread of infection,we propose the model with four bounded controls which describe vaccination of newborns,vaccination of the susceptible, treatment of infected, and indirect strategies aimed at areduction of the incidence rate (e. g. quarantine). The optimal control problem ofminimizing the total number of the infected individuals on a given time interval is statedand solved. The optimal solutions are obtained with the use of the Pontryagin MaximumPrinciple and investigated analytically. Numerical results are presented to illustrate theoptimal solutions.

Keywords

SIR modelcontrol the spread of infectionnonlinear control systemPontryagin maximum principleRiccati equationgeneralized Rolle’s theorem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 4: Optimal control , 2014 , pp. 105 - 121
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.M. Anderson, R.M. May. Infections Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford, UK, 1992.
S. Anita, V. Arnaǔtu, V. Capasso. An Introduction to Optimal Control Problems in Life Sciences and Economics. Birkhäuser, USA, 2011.
N.T.J. Bailey. The Mathematical Theory of Epidemics. Griffin, London, 1957.
Behncke, H.. Optimal control of deterministic epidemics. Optim. Contr. Appl. Met., 21 (2000), No. 6, 269–285. CrossRefGoogle Scholar
A. Bressan, B. Piccoli. Introduction to the Mathematical Theory of Control. AIMS Series on Applied Mathematics, Vol. 2, AIMS, USA, 2007.
V. Capasso. Mathematical Structures of Epidemical Systems. Lecture Notes in Biomathematics, Vol. 97, Springer, Heidelberg, 2008.
Castilho, C.. Optimal control of an epidemic through educational campaigns. Electron. J. Differential Equations, 2006 (2006), No. 125, 1–11. Google Scholar
D.J. Daley, J. Gani. Epidemic Modelling: An Introduction. Cambridge University Press, Cambridge, 1999.
B.P. Demidovich. Lectures on Stability Theory. Nauka, Moscow, 1967. (Russian)
Dmitruk, A.V.. A generalized estimate of the number of zeros for solutions of a class of linear differential equations. SIAM J. Control Optim., 30 (1992), No. 5, 1087–1091. CrossRefGoogle Scholar
Fister, K.R., Lenhart, S., McNally, J.S.. Optimizing chemotherapy in an HIV model. Electron. J. Differentail Equations, 1998 (1998), No. 32, 1–12. Google Scholar
Grigorieva, E.V., Khailov, E.N.. Attainable set of a nonlinear controlled microeconomic model. J. Dyn. Control Syst., 11 (2005), No. 2, 157–176. CrossRefGoogle Scholar
E. Grigorieva, N. Bondarenko, E. Khailov, A. Korobeinikov. Finite-dimensional methods for optimal control of autothermal thermophilic aerobic digestion. In Industrial Waste, eds. K.Y. Show and X. Guo, Intech, Croatia, 2012, 91–120.
Grigorieva, E., Bondarenko, N., Khailov, E., Korobeinikov, A.. Analysis of optimal control problems for the process of wastewater biological treatment. Revista de Matemática: Teoría y Aplicaciones, 20 (2013), No. 2, 103–118. Google Scholar
Grigorieva, E.V., Khailov, E.N., Korobeinikov, A.. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Math. Biosci. Eng., 10 (2013), No. 4, 1067–1094. Google Scholar
Gupta, N.K., Rink, R.E.. Optimal control of epidemics. Math. Biosci., 18 (1973), 383–396. CrossRefGoogle Scholar
P. Hartman. Ordinary Differential Equations. John Wiley & Sons, New York-London-Sydney, 1964.
Hethcote, H.W., Waltman, P.. Optimal vaccination schedules in a deterministic epidemic model. Math. Biosci., 18 (1973), 365–381. CrossRefGoogle Scholar
Jaquette, D.L.. Mathematical models for controlling growing biological populations: a survey. Oper. Res., 20 (1972), 1142–1151. CrossRefGoogle Scholar
Joshi, H.R.. Optimal control of an HIV immunology model. Optim. Contr. Appl. Met., 23 (2002), No. 4, 199–213. CrossRefGoogle Scholar
Joshi, H.R., Lenhart, S., Li, M.Y., Wang, L.. Optimal control methods applied to disease models. In AMS Volume on Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges, AMS Contemporary Mathematics Series, Vol. 410, 2006, 187–207. CrossRefGoogle Scholar
M.J. Keeling, P. Rohani. Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton, 2008.
U. Ledzewicz, H. Schättler. On optimal sigular controls for a general SIR-model with vaccination and treatment. Discret. Contin. Dyn. S., supplement volume (2011), 981–990.
E.B. Lee, L. Marcus. Foundations of Optimal Control Theory. John Wiley & Sons, New York, 1967.
S. Lenhart, J.T. Workman. Optimal Control Applied to Biological Models. CRC Press, Taylor & Francis Group, London, 2007.
Morton, R., Wickwire, K.H.. On the optimal control of a deterministic epidemic. Adv. Appl. Probab., 6 (1974), No. 4, 622–635. CrossRefGoogle Scholar
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. Mathematical Theory of Optimal Processes. John Wiley & Sons, New York, 1962.
R. Smith?. Modelling Disease Ecology with Mathematics. AIMS Series on Differential Equations & Dynamical Systems, Vol. 2, AIMS, USA, 2008.
A.N. Tikhonov, A.B. Vasil’eva, A.G. Sveshnikov. Differential Equations. Springer-Verlag, Berlin-Heidelberg-New York, 1985.
F.P. Vasil’ev. Optimization Methods. Factorial Press, Moscow, 2002. (Russian)
Wickwire, K.H.. Mathematical models for the control of pests and infectious diseases: a survey. Theor. Popul. Biol., 11 (1977), No. 2, 182–238. CrossRefGoogle ScholarPubMed
Zaric, G.S., Brandeau, M.L.. Resource allocation for epidemic control over short time horizons. Math. Biosci., 171 (2001), 33–58. CrossRefGoogle ScholarPubMed