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Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
A SEIR model for control of infectious diseases with constraints
1. | Faculdade de Engenharia da Universidade do Porto, DEEC and ISR-Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal, Portugal, Portugal |
References:
[1] |
F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Springer-Verlag. New York, (2001).
|
[2] |
F. Clarke, Optimization and Nonsmooth Analysis,, John Wiley, (1983).
|
[3] |
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control,, Springer-Verlag, (2013).
doi: 10.1007/978-1-4471-4820-3. |
[4] |
F. Clarke and MdR de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., 48 (2010), 4500.
doi: 10.1137/090757642. |
[5] |
M. d. R. de Pinho, M. M. Ferreira, U. Ledzewicz and H. Schaettler, A model for cancer chemotherapy with state-space constraints,, Nonlinear Analysis, 63 (2005). |
[6] |
M. d. R. de Pinho, P. Loewen and G. N. Silva, A weak maximum principle for optimal control problems with nonsmooth mixed constraints,, Set-Valued and Variational Analysis, 17 (2009), 203.
doi: 10.1007/s11228-009-0108-1. |
[7] |
E. Demirci, A. Unal and N. Ozalp, A fractional order seir model with density dependent death rate,, MdR de Pinho, 40 (2011), 287.
|
[8] |
P. Falugi, E. Kerrigan and E. van Wyk, Imperial College London Optimal Control Software User Guide (ICLOCS),, Department of Electrical and Electronic Engineering, (2010). |
[9] |
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.
doi: 10.1137/1037043. |
[10] |
M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, $2^{nd}$ Edition (405 pages), (1980). |
[11] |
H. W. Hethcote, The basic epidemiology models: models, expressions for $R_0$, parameter estimation, and applications,, In Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, (2008), 1.
doi: 10.1142/9789812834836_0001. |
[12] |
W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics,, Bulletin of Mathematical Biology, 53 (1991), 35. |
[13] |
H. Maurer and S. Pickenhain, Second order sufficient conditions for optimal control problems with mixed control-state constraints,, J. Optim. Theory Appl., 86 (1995), 649.
doi: 10.1007/BF02192163. |
[14] |
Helmut Maurer and H.J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach,, SIAM J. Control Optm., 41 (2002), 380.
doi: 10.1137/S0363012900377419. |
[15] |
N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary And Sufficient Optimality Conditions In Calculus Of Variations And Optimal Control,, SIAM Advances in Design and Control, 24 (2012).
doi: 10.1137/1.9781611972368. |
[16] |
D. S. Naidu, T. Fernando and K. R. Fister, Optimal control in diabetes,, Optim. Control Appl. Meth., 32 (2011), 181.
doi: 10.1002/oca.990. |
[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.
|
[18] |
L.T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear Systems,, Project Report, (2013). |
[19] |
O. Prosper, O. Saucedo, D. Thompson, G. T. Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza,, Mathematical Biosciences and Engineering, 8 (2011), 141.
doi: 10.3934/mbe.2011.8.141. |
[20] |
P. Shi and L. Dong, Dynamical models for infectious diseases with varying population size and vaccinations,, Journal of Applied Mathematics, 2012 (2012), 1.
doi: 10.1155/2012/824192. |
[21] |
H. Schäettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples,, Springer, (2012).
doi: 10.1007/978-1-4614-3834-2. |
[22] |
C. Sun and Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination,, Applied Mathematical Modelling, 34 (2010), 2685.
doi: 10.1016/j.apm.2009.12.005. |
[23] | |
[24] |
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.
doi: 10.1007/s10107-004-0559-y. |
show all references
References:
[1] |
F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Springer-Verlag. New York, (2001).
|
[2] |
F. Clarke, Optimization and Nonsmooth Analysis,, John Wiley, (1983).
|
[3] |
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control,, Springer-Verlag, (2013).
doi: 10.1007/978-1-4471-4820-3. |
[4] |
F. Clarke and MdR de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., 48 (2010), 4500.
doi: 10.1137/090757642. |
[5] |
M. d. R. de Pinho, M. M. Ferreira, U. Ledzewicz and H. Schaettler, A model for cancer chemotherapy with state-space constraints,, Nonlinear Analysis, 63 (2005). |
[6] |
M. d. R. de Pinho, P. Loewen and G. N. Silva, A weak maximum principle for optimal control problems with nonsmooth mixed constraints,, Set-Valued and Variational Analysis, 17 (2009), 203.
doi: 10.1007/s11228-009-0108-1. |
[7] |
E. Demirci, A. Unal and N. Ozalp, A fractional order seir model with density dependent death rate,, MdR de Pinho, 40 (2011), 287.
|
[8] |
P. Falugi, E. Kerrigan and E. van Wyk, Imperial College London Optimal Control Software User Guide (ICLOCS),, Department of Electrical and Electronic Engineering, (2010). |
[9] |
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.
doi: 10.1137/1037043. |
[10] |
M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, $2^{nd}$ Edition (405 pages), (1980). |
[11] |
H. W. Hethcote, The basic epidemiology models: models, expressions for $R_0$, parameter estimation, and applications,, In Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, (2008), 1.
doi: 10.1142/9789812834836_0001. |
[12] |
W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics,, Bulletin of Mathematical Biology, 53 (1991), 35. |
[13] |
H. Maurer and S. Pickenhain, Second order sufficient conditions for optimal control problems with mixed control-state constraints,, J. Optim. Theory Appl., 86 (1995), 649.
doi: 10.1007/BF02192163. |
[14] |
Helmut Maurer and H.J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach,, SIAM J. Control Optm., 41 (2002), 380.
doi: 10.1137/S0363012900377419. |
[15] |
N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary And Sufficient Optimality Conditions In Calculus Of Variations And Optimal Control,, SIAM Advances in Design and Control, 24 (2012).
doi: 10.1137/1.9781611972368. |
[16] |
D. S. Naidu, T. Fernando and K. R. Fister, Optimal control in diabetes,, Optim. Control Appl. Meth., 32 (2011), 181.
doi: 10.1002/oca.990. |
[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.
|
[18] |
L.T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear Systems,, Project Report, (2013). |
[19] |
O. Prosper, O. Saucedo, D. Thompson, G. T. Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza,, Mathematical Biosciences and Engineering, 8 (2011), 141.
doi: 10.3934/mbe.2011.8.141. |
[20] |
P. Shi and L. Dong, Dynamical models for infectious diseases with varying population size and vaccinations,, Journal of Applied Mathematics, 2012 (2012), 1.
doi: 10.1155/2012/824192. |
[21] |
H. Schäettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples,, Springer, (2012).
doi: 10.1007/978-1-4614-3834-2. |
[22] |
C. Sun and Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination,, Applied Mathematical Modelling, 34 (2010), 2685.
doi: 10.1016/j.apm.2009.12.005. |
[23] | |
[24] |
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.
doi: 10.1007/s10107-004-0559-y. |
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