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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
Chair of Mathematics in Engineering Sciences, University of Bayreuth, Bayreuth, D 95440, Germany |
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
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References:
| [1] |
F. Bauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Part IV 2nd edition, Springer-Verlag, New York, 2012.
doi: 10.1007/978-1-4614-1686-9. |
| [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. |
| [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. |
| [4] |
O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princton University Press, Princton, 2013.
|
| [5] |
R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming Duxbury Press, Pacific Grove, 2002. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [15] |
J. D. Murray, Mathematical Biology: Ⅰ An Introduction 3rd edition, Springer-Verlag, New York, 2002. |
| [16] |
J. D. Murray, Mathematical Biology: Ⅱ Spatial Models and Biomedical Applications 3rd edition, Springer-Verlag, New York, 2003. |
| [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.
|
| [18] |
H. J. Pesch,
A practical guide to the solution of real-life optimal control problems, Control and Cybernetics, 23 (1994), 7-60.
|
| [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. |
| [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. |
| [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. |
| [22] |
M. Thäter, Restringierte Optimalsteuerungsprobleme bei Epidemiemodellen Master Thesis, Department of Mathematics, University of Bayreuth in Bayreuth, 2014. |
| [23] |
P.-F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance Mathématique et Physique, 10 (1838), 113-121. |
| [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. |
| [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. |
| [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. |
show all references
References:
| [1] |
F. Bauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Part IV 2nd edition, Springer-Verlag, New York, 2012.
doi: 10.1007/978-1-4614-1686-9. |
| [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. |
| [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. |
| [4] |
O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princton University Press, Princton, 2013.
|
| [5] |
R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming Duxbury Press, Pacific Grove, 2002. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [15] |
J. D. Murray, Mathematical Biology: Ⅰ An Introduction 3rd edition, Springer-Verlag, New York, 2002. |
| [16] |
J. D. Murray, Mathematical Biology: Ⅱ Spatial Models and Biomedical Applications 3rd edition, Springer-Verlag, New York, 2003. |
| [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.
|
| [18] |
H. J. Pesch,
A practical guide to the solution of real-life optimal control problems, Control and Cybernetics, 23 (1994), 7-60.
|
| [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. |
| [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. |
| [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. |
| [22] |
M. Thäter, Restringierte Optimalsteuerungsprobleme bei Epidemiemodellen Master Thesis, Department of Mathematics, University of Bayreuth in Bayreuth, 2014. |
| [23] |
P.-F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance Mathématique et Physique, 10 (1838), 113-121. |
| [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. |
| [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. |
| [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. |
Figure 2.
Scenario 0: The progress of the population without control
Figure 3.
cenario 1: The progress of the population for WM = ∞
Figure 4.
Scenario 1: Discrete and verified optimal control for WM = ∞
Figure 5.
Scenario 2: "Almost bang-bang" control for WM = 2 500 when vaccination is stopped
Figure 6.
Scenario 2: The infected and the recovered population. The latter exhibits an "almost kink" when vaccination is stopped
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.
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
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]
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
Figure 10.
Scenario 5: The approximate candidate optimal control: the verification test yields $ 8.5 \cdot 10^{-8} $ , hence indicating again a perfect coincidence
Figure 12.
Scenario 6: Discounted functional. The progress of the population for $ W_M = \infty $
Figure 13.
Scenario 6: Discounted functional. Discrete and verified optimal control for $ W_M = \infty $
| Parameters | Definitions | Units | Values |
| natural birth rate | unit of time | 0.525 | |
| | natural death rate | unit of time | 0.5 |
| | incidence coefficient | | 0.001 |
| | exposed to infectious rate | unit of time | 0.5 |
| | natural recovery rate | unit of time | 0.1 |
| | disease induced death rate | unit of time | 0.1 |
| | maximum vaccination rate | unit of time | 1 |
| | maximum available vaccines | unit of capita | various |
| | upper bound in Eq. 34 | | various |
| | upper bound in Eq. 38 | unit of capita | various |
| | weight parameter | | 0.1 |
| | weight parameter | unit of money | 1 |
| | initial time | unit of time (years) | 0 |
| | final time | unit of time (years) | 20 |
| | initial susceptible population | unit of capita | 1000 |
| | initial exposed population | unit of capita | 100 |
| | initial infected population | unit of capita | 50 |
| | initial recovered population | unit of capita | 15 |
| | initial total population | unit of capita | 1165 |
| | initial vaccinated population | unit of capita | 0 |
| Parameters | Definitions | Units | Values |
| natural birth rate | unit of time | 0.525 | |
| | natural death rate | unit of time | 0.5 |
| | incidence coefficient | | 0.001 |
| | exposed to infectious rate | unit of time | 0.5 |
| | natural recovery rate | unit of time | 0.1 |
| | disease induced death rate | unit of time | 0.1 |
| | maximum vaccination rate | unit of time | 1 |
| | maximum available vaccines | unit of capita | various |
| | upper bound in Eq. 34 | | various |
| | upper bound in Eq. 38 | unit of capita | various |
| | weight parameter | | 0.1 |
| | weight parameter | unit of money | 1 |
| | initial time | unit of time (years) | 0 |
| | final time | unit of time (years) | 20 |
| | initial susceptible population | unit of capita | 1000 |
| | initial exposed population | unit of capita | 100 |
| | initial infected population | unit of capita | 50 |
| | initial recovered population | unit of capita | 15 |
| | initial total population | unit of capita | 1165 |
| | initial vaccinated population | unit of capita | 0 |
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