Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth

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April 2018, 15(2): 485-505. doi: 10.3934/mbe.2018022

Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth

Markus Thäter , Kurt Chudej and Hans Josef Pesch ^,

Chair of Mathematics in Engineering Sciences, University of Bayreuth, Bayreuth, D 95440, Germany

* Corresponding author: Hans Josef Pesch

Received July 30, 2016 Accepted May 15, 2017 Published June 2017

In this paper an improved SEIR model for an infectious disease is presented which includes logistic growth for the total population. The aim is to develop optimal vaccination strategies against the spread of a generic disease. These vaccination strategies arise from the study of optimal control problems with various kinds of constraints including mixed control-state and state constraints. After presenting the new model and implementing the optimal control problems by means of a first-discretize-then-optimize method, numerical results for six scenarios are discussed and compared to an analytical optimal control law based on Pontrygin's minimum principle that allows to verify these results as approximations of candidate optimal solutions.

Keywords: SEIR model, logistic growth, infectious diseases, optimal control, vaccination strategy, verification of numerical results.

Mathematics Subject Classification: Primary: 49K15, 49M25, 90C90, 92D30; Secondary: 34A34.

Citation: Markus Thäter, Kurt Chudej, Hans Josef Pesch. Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth. Mathematical Biosciences & Engineering, 2018, 15 (2) : 485-505. doi: 10.3934/mbe.2018022

References:

[1]	F. Bauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Part IV 2^nd edition, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-1686-9. Google Scholar
[2]	M. H. A. Biswas, L. T. Paiva and MdR. de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784. doi: 10.3934/mbe.2014.11.761. Google Scholar
[3]	A. E. Bryson, W. F. Denham and S. E. Dreyfus, Optimal Programming Problems with Inequality Constraints I, AIAA Journal, 1 (1963), 2544-2550. doi: 10.2514/3.2107. Google Scholar
[4]	O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princton University Press, Princton, 2013. Google Scholar
[5]	R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming Duxbury Press, Pacific Grove, 2002.Google Scholar
[6]	W. E. Hamilton, On nonexistence of boundary arcs in control problems with bounded state variables, IEEE Transactions on Automatic Control, AC-17 (1972), 338-343. Google Scholar
[7]	R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Review, 37 (1995), 181-218. doi: 10.1137/1037043. Google Scholar
[8]	K. Ito and K. Kunisch, Asymptotic properties of receding horizont optimal control problems, SIAM J. Control Optim., 40 (2002), 1585-1610. doi: 10.1137/S0363012900369423. Google Scholar
[9]	D. H. Jacobson, M. M. Lele and J. L.. Speyer, New necessary conditions of optimality for control problems with state variable inequality constraints, Journal of Mathematical Analysis and Applications, 35 (1971), 255-284. doi: 10.1016/0022-247X(71)90219-8. Google Scholar
[10]	I. Kornienko, L. T. Paiva and MdR. de Pinho, Introducing state constraints in optimal control for health problems, Procedia Technology, 17 (2014), 415-422. doi: 10.1016/j.protcy.2014.10.249. Google Scholar
[11]	V. Lykina, Beiträge zur Theorie der Optimalsteuerungsprobleme mit unendlichem Zeithorizont, Dissertation, Brandenburgische Technische Universität Cottbus, Germany, 2010, http://opus.kobv.de/btu/volltexte/2010/1861/pdf/dissertationLykina.pdf.Google Scholar
[12]	V. Lykina, S. Pickenhain and M. Wagner, On a resource allocation model with infinite horizon, Applied Mathematics and Computation, 204 (2008), 595-601. doi: 10.1016/j.amc.2008.05.041. Google Scholar
[13]	H. Maurer and H. J Pesch, Direct optimization methods for solving a complex state-constrained optimal control problem in microeconomics, Applied Mathematics and Computation, 204 (2008), 568-579. doi: 10.1016/j.amc.2008.05.035. Google Scholar
[14]	H. Maurer and MdR. de Pinho, Optimal control of epidemiological SEIR models with L¹-objectives and control-state constraints, Pac. J. Optim., 12 (2016), 415-436. Google Scholar
[15]	J. D. Murray, Mathematical Biology: Ⅰ An Introduction 3^rd edition, Springer-Verlag, New York, 2002. Google Scholar
[16]	J. D. Murray, Mathematical Biology: Ⅱ Spatial Models and Biomedical Applications 3^rd edition, Springer-Verlag, New York, 2003. Google Scholar
[17]	R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics, 75 (2010), 67-81. Google Scholar
[18]	H. J. Pesch, A practical guide to the solution of real-life optimal control problems, Control and Cybernetics, 23 (1994), 7-60. Google Scholar
[19]	S. Pickenhain, Infinite horizon optimal control problems in the light of convex analysis in Hilbert Spaces, Set-Valued and Variational Analysis, 23 (2015), 169-189. doi: 10.1007/s11228-014-0304-5. Google Scholar
[20]	M. Plail and H. J. Pesch, The Cold War and the maximum principle of optimal control, Doc. Math. , 2012, Extra vol. : Optimization stories, 331–343. Google Scholar
[21]	H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems of L:2-type objectives and control constraints, Dicrete and Continuous Dynamical Systems Series B, 19 (2014), 2657-2679. doi: 10.3934/dcdsb.2014.19.2657. Google Scholar
[22]	M. Thäter, Restringierte Optimalsteuerungsprobleme bei Epidemiemodellen Master Thesis, Department of Mathematics, University of Bayreuth in Bayreuth, 2014.Google Scholar
[23]	P.-F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance Mathématique et Physique, 10 (1838), 113-121. Google Scholar
[24]	A. Wächter, An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering PhD Thesis, Carnegie Mellon University in Pittsburgh, 2002.Google Scholar
[25]	A. Wächter and L.T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar
[26]	D. Wenzke, V. Lykina and S. Pickenhain, State and time transformations of infinite horizon optimal control problems, Contemporary Mathematics Series of The AMS, 619 (2014), 189-208. doi: 10.1090/conm/619/12391. Google Scholar

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References:

[1]	F. Bauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Part IV 2^nd edition, Springer-Verlag, New York, 2012. doi: 10.1007/978-1-4614-1686-9. Google Scholar
[2]	M. H. A. Biswas, L. T. Paiva and MdR. de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784. doi: 10.3934/mbe.2014.11.761. Google Scholar
[3]	A. E. Bryson, W. F. Denham and S. E. Dreyfus, Optimal Programming Problems with Inequality Constraints I, AIAA Journal, 1 (1963), 2544-2550. doi: 10.2514/3.2107. Google Scholar
[4]	O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princton University Press, Princton, 2013. Google Scholar
[5]	R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming Duxbury Press, Pacific Grove, 2002.Google Scholar
[6]	W. E. Hamilton, On nonexistence of boundary arcs in control problems with bounded state variables, IEEE Transactions on Automatic Control, AC-17 (1972), 338-343. Google Scholar
[7]	R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Review, 37 (1995), 181-218. doi: 10.1137/1037043. Google Scholar
[8]	K. Ito and K. Kunisch, Asymptotic properties of receding horizont optimal control problems, SIAM J. Control Optim., 40 (2002), 1585-1610. doi: 10.1137/S0363012900369423. Google Scholar
[9]	D. H. Jacobson, M. M. Lele and J. L.. Speyer, New necessary conditions of optimality for control problems with state variable inequality constraints, Journal of Mathematical Analysis and Applications, 35 (1971), 255-284. doi: 10.1016/0022-247X(71)90219-8. Google Scholar
[10]	I. Kornienko, L. T. Paiva and MdR. de Pinho, Introducing state constraints in optimal control for health problems, Procedia Technology, 17 (2014), 415-422. doi: 10.1016/j.protcy.2014.10.249. Google Scholar
[11]	V. Lykina, Beiträge zur Theorie der Optimalsteuerungsprobleme mit unendlichem Zeithorizont, Dissertation, Brandenburgische Technische Universität Cottbus, Germany, 2010, http://opus.kobv.de/btu/volltexte/2010/1861/pdf/dissertationLykina.pdf.Google Scholar
[12]	V. Lykina, S. Pickenhain and M. Wagner, On a resource allocation model with infinite horizon, Applied Mathematics and Computation, 204 (2008), 595-601. doi: 10.1016/j.amc.2008.05.041. Google Scholar
[13]	H. Maurer and H. J Pesch, Direct optimization methods for solving a complex state-constrained optimal control problem in microeconomics, Applied Mathematics and Computation, 204 (2008), 568-579. doi: 10.1016/j.amc.2008.05.035. Google Scholar
[14]	H. Maurer and MdR. de Pinho, Optimal control of epidemiological SEIR models with L¹-objectives and control-state constraints, Pac. J. Optim., 12 (2016), 415-436. Google Scholar
[15]	J. D. Murray, Mathematical Biology: Ⅰ An Introduction 3^rd edition, Springer-Verlag, New York, 2002. Google Scholar
[16]	J. D. Murray, Mathematical Biology: Ⅱ Spatial Models and Biomedical Applications 3^rd edition, Springer-Verlag, New York, 2003. Google Scholar
[17]	R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics, 75 (2010), 67-81. Google Scholar
[18]	H. J. Pesch, A practical guide to the solution of real-life optimal control problems, Control and Cybernetics, 23 (1994), 7-60. Google Scholar
[19]	S. Pickenhain, Infinite horizon optimal control problems in the light of convex analysis in Hilbert Spaces, Set-Valued and Variational Analysis, 23 (2015), 169-189. doi: 10.1007/s11228-014-0304-5. Google Scholar
[20]	M. Plail and H. J. Pesch, The Cold War and the maximum principle of optimal control, Doc. Math. , 2012, Extra vol. : Optimization stories, 331–343. Google Scholar
[21]	H. Schättler, U. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems of L:2-type objectives and control constraints, Dicrete and Continuous Dynamical Systems Series B, 19 (2014), 2657-2679. doi: 10.3934/dcdsb.2014.19.2657. Google Scholar
[22]	M. Thäter, Restringierte Optimalsteuerungsprobleme bei Epidemiemodellen Master Thesis, Department of Mathematics, University of Bayreuth in Bayreuth, 2014.Google Scholar
[23]	P.-F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance Mathématique et Physique, 10 (1838), 113-121. Google Scholar
[24]	A. Wächter, An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering PhD Thesis, Carnegie Mellon University in Pittsburgh, 2002.Google Scholar
[25]	A. Wächter and L.T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y. Google Scholar
[26]	D. Wenzke, V. Lykina and S. Pickenhain, State and time transformations of infinite horizon optimal control problems, Contemporary Mathematics Series of The AMS, 619 (2014), 189-208. doi: 10.1090/conm/619/12391. Google Scholar

Figure 1. The SEIR model with vaccination; cp. [2]

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Figure 2. Scenario 0: The progress of the population without control

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Figure 3. cenario 1: The progress of the population for W_M = ∞

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Figure 4. Scenario 1: Discrete and verified optimal control for W_M = ∞

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Figure 5. Scenario 2: "Almost bang-bang" control for W_M = 2 500 when vaccination is stopped

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Figure 6. Scenario 2: The infected and the recovered population. The latter exhibits an "almost kink" when vaccination is stopped

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Figure 7. Scenario 3: The upper picture shows the two competing control candidates according to (33) and (36), the latter for V₀ = 125. The optimal control is the boundary control (36) of the mixed control-state constraint (34) almost over the entire time interval. Only at the end, the control (32) in the interior of the admissble set becomes active. In the lower picture the discrete and the verified control values are compared showing again a perfect coincidence. The upright bar marks the switching time.

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Figure 8. Scenario 3: Time behaviour of the infected and recovered population. The terminal value of the recovered population is a bit higher than in the model with a finite amount of vaccines

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Figure 9. Scenario 4: Susceptible population and optimal control for $S_{\max} = 1\,200$. The upright lines mark the switching points from unconstrained boundary arcs on state-constrained ones or vice versa. Three boundary arcs occur here; the last one is extremely short; see the zoom. Note that a touch point cannot exist here [6]. The optimal control is discontinuous at entry and/or exit points due to the jump discontinuities of $\lambda_S$ at these junction points; see [7]

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Figure 11. Scenario 5: Susceptible and infected population: entry and exit point to the state constrained arc are marked in green resp.red. The maximal allowed values $ S_{\max} $ and $ V_0/u(t) $ are marked in blue and purple respectively

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Figure 10. Scenario 5: The approximate candidate optimal control: the verification test yields $ 8.5 \cdot 10^{-8} $, hence indicating again a perfect coincidence

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Figure 12. Scenario 6: Discounted functional. The progress of the population for $ W_M = \infty $

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Figure 13. Scenario 6: Discounted functional. Discrete and verified optimal control for $ W_M = \infty $

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Table 1. Values from [17], also chosen in [2]¹

Parameters	Definitions	Units	Values
$ b $	natural birth rate	unit of time$^{-1}$	0.525
$ d $	natural death rate	unit of time$^{-1}$	0.5
$ c $	incidence coefficient	$\frac{1}{\mbox{ unit of capita}\,\cdot\,\mbox{unit of time}}$	0.001
$ e $	exposed to infectious rate	unit of time$^{-1}$	0.5
$ g $	natural recovery rate	unit of time$^{-1}$	0.1
$ a $	disease induced death rate	unit of time$^{-1}$	0.1
$ u_{\max} $	maximum vaccination rate	unit of time$^{-1}$	1
$ W_M $	maximum available vaccines	unit of capita	various
$ V_0 $	upper bound in Eq. 34	$\frac{\mbox{ unit of capita}}{\mbox{ unit of time}}$	various
$ S_{\max} $	upper bound in Eq. 38	unit of capita	various
$ A_1 $	weight parameter	$\frac{\mbox{ unit of money}}{\mbox{ unit of capita}\, \cdot\,\mbox{ unit of time}}$	0.1
$ A_2 $	weight parameter	unit of money$\,\cdot\,$unit of time	1
$ t_0 $	initial time	unit of time (years)	0
$ T $	final time	unit of time (years)	20
$ S_0 $	initial susceptible population	unit of capita	1000
$ E_0 $	initial exposed population	unit of capita	100
$ I_0 $	initial infected population	unit of capita	50
$ R_0 $	initial recovered population	unit of capita	15
$ N_0 $	initial total population	unit of capita	1165
$ W_0 $	initial vaccinated population	unit of capita	0

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Parameters	Definitions	Units	Values
$ b $	natural birth rate	unit of time$^{-1}$	0.525
$ d $	natural death rate	unit of time$^{-1}$	0.5
$ c $	incidence coefficient	$\frac{1}{\mbox{ unit of capita}\,\cdot\,\mbox{unit of time}}$	0.001
$ e $	exposed to infectious rate	unit of time$^{-1}$	0.5
$ g $	natural recovery rate	unit of time$^{-1}$	0.1
$ a $	disease induced death rate	unit of time$^{-1}$	0.1
$ u_{\max} $	maximum vaccination rate	unit of time$^{-1}$	1
$ W_M $	maximum available vaccines	unit of capita	various
$ V_0 $	upper bound in Eq. 34	$\frac{\mbox{ unit of capita}}{\mbox{ unit of time}}$	various
$ S_{\max} $	upper bound in Eq. 38	unit of capita	various
$ A_1 $	weight parameter	$\frac{\mbox{ unit of money}}{\mbox{ unit of capita}\, \cdot\,\mbox{ unit of time}}$	0.1
$ A_2 $	weight parameter	unit of money$\,\cdot\,$unit of time	1
$ t_0 $	initial time	unit of time (years)	0
$ T $	final time	unit of time (years)	20
$ S_0 $	initial susceptible population	unit of capita	1000
$ E_0 $	initial exposed population	unit of capita	100
$ I_0 $	initial infected population	unit of capita	50
$ R_0 $	initial recovered population	unit of capita	15
$ N_0 $	initial total population	unit of capita	1165
$ W_0 $	initial vaccinated population	unit of capita	0

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