December 2018, 11(6): 1179-1199. doi: 10.3934/dcdss.2018067

Optimal control of non-autonomous SEIRS models with vaccination and treatment

1. 

Research Unit for Inland Development (UDI), Polytechnic Institute of Guarda, 6300-559 Guarda, Portugal

2. 

Centro de Matemática e Aplicações da Universidade da Beira Interior (CMA-UBI), Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, Portugal

3. 

Departamento de Matemática and Instituto de Telecomunicações (IT), Universidade da Beira Interior, 6201-001 Covilhã, Portugal

4. 

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

* Corresponding author: delfim@ua.pt

Received  April 2017 Revised  June 2017 Published  June 2018

Fund Project: Mateus was partially supported by FCT through CMA-UBI (project UID/MAT/00212/2013), Rebelo by FCT through CMA-UBI (project UID/MAT/00212/2013), Rosa by FCT through IT (project UID/EEA/50008/2013), Silva by FCT through CMA-UBI (project UID/MAT/00212/2013), and Torres by FCT through CIDMA (project UID/MAT/04106/2013) and TOCCATA (project PTDC/EEI-AUT/2933/2014 funded by FEDER and COMPETE 2020)

We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.

Citation: Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067
References:
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R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991.

[2]

I. AreaF. NdaïrouJ. J. NietoC. J. Silva and D. F. M. Torres, Ebola model and optimal control with vaccination constraints, J. Ind. Manag. Optim., 14 (2018), 427-446. doi: 10.3934/jimo.2017054.

[3]

E. R. Avakov, The maximum principle for abnormal optimal control problems, Soviet Math. Dokl., 37 (1988), 231-234.

[4]

Z. Bai and Y. Zhou, Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Anal. Real World Appl., 13 (2012), 1060-1068. doi: 10.1016/j.nonrwa.2011.02.008.

[5]

A. CoriA. ValleronF. CarratG. Scalia TombaG. Thomas and P. Boëlle, Estimating influenza latency and infectious period durations using viral excretion data, Epidemics, 4 (2012), 132-138. doi: 10.1016/j.epidem.2012.06.001.

[6]

C. Ding, N. Tao and Y. Zhu, A mathematical model of Zika virus and its optimal control, Proceedings of the 35th Chinese Control Conference, July 27-29, 2016, Chengdu, China. IEEE Xplore, (2016), 2642-2645. doi: 10.1109/ChiCC.2016.7553763.

[7]

S. EdlundJ. KaufmanJ. LesslerJ. DouglasM. BrombergZ. KaufmanR. BassalG. ChodickR. MaromV. ShalevY. MesikaR. Ram and A. Leventhal, Comparing three basic models for seasonal influenza, Epidemics, 3 (2011), 135-142. doi: 10.1016/j.epidem.2011.04.002.

[8]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron. J. Differential Equations, 1998 (1998), 12 pp.

[9]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975.

[10]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492. doi: 10.3934/mbe.2009.6.469.

[11]

S. GaoL. Chen and Z. Teng, Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear Anal. Real World Appl., 9 (2008), 599-607. doi: 10.1016/j.nonrwa.2006.12.004.

[12]

H. W. HethcoteM. A. Lewis and P. van den Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64. doi: 10.1007/BF00276080.

[13]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539.

[14]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57.

[15]

T. Kuniya and Y. Nakata, Permanence and extinction for a nonautonomous SEIS epidemic model, Appl. Math. Comput., 218 (2012), 9321-9331. doi: 10.1016/j.amc.2012.03.011.

[16]

A. P. Lemos-PaiãoC. J. Silva and D. F. M. Torres, An epidemic model for cholera with optimal control treatment, J. Comput. Appl. Math., 318 (2017), 168-180. doi: 10.1016/j.cam.2016.11.002.

[17]

M. Y. LiH. L. Smith and L. Wang, Global dynamics an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58-69. doi: 10.1137/S0036139999359860.

[18]

W. M. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[19]

M. Martcheva, A non-autonomous multi-strain SIS epidemic model, J. Biol. Dyn., 3 (2009), 235-251. doi: 10.1080/17513750802638712.

[20]

J. P. Mateus and C. M. Silva, A non-autonomous SEIRS model with general incidence rate, Appl. Math. Comput., 247 (2014), 169-189. doi: 10.1016/j.amc.2014.08.078.

[21]

Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363 (2010), 230-237. doi: 10.1016/j.jmaa.2009.08.027.

[22]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.

[23]

A. Rachah and D. F. M. Torres, Dynamics and optimal control of Ebola transmission, Math. Comput. Sci., 10 (2016), 331-342. doi: 10.1007/s11786-016-0268-y.

[24]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X.

[25]

M. A. Safi and S. M. Garba, Global stability analysis of SEIR model with Holling type Ⅱ incidence function, Comput. Math. Methods Med., 2012 (2012), Art. ID 826052, 8 pp. doi: 10.1155/2012/826052.

[26]

R. Shope, Global climate change and infectious diseases, Environ. Health Perspect, 96 (1991), 171-174. doi: 10.1289/ehp.9196171.

[27]

C. SilvaH. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337. doi: 10.3934/mbe.2017021.

[28]

D. F. M. Torres, Lipschitzian regularity of the minimizing trajectories for nonlinear optimal control problems, Math. Control Signals Systems, 16 (2003), 158-174. doi: 10.1007/s00498-003-0132-x.

[29]

P. van den Driessche, Deterministic Compartmental Models: Extensions of Basic Models, In: Mathematical epidemiology, vol. 1945 of Lecture Notes in Math., Springer, Berlin, (2008), 147-157. doi: 10.1007/978-3-540-78911-6_5.

[30]

F. J. S. Wang and W. R. Derrick, On deterministic epidemic models, Bull. Inst. Math. Acad. Sinica, 6 (1978), 73-84.

[31]

A. WeberM. Weber and P. Milligan, Modeling epidemics caused by respiratory syncytial virus (RSV), Math. Biosci., 172 (2001), 95-113. doi: 10.1016/S0025-5564(01)00066-9.

[32]

T. ZhangJ. Liu and Z. Teng, Existence of positive periodic solutions of an SEIR model with periodic coefficients, Appl. Math., 57 (2012), 601-616. doi: 10.1007/s10492-012-0036-5.

[33]

T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69 (2007), 2537-2559. doi: 10.1007/s11538-007-9231-z.

[34]

T. Zhang and Z. Teng, Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model, Chaos Solitons Fractals, 39 (2009), 2411-2425. doi: 10.1016/j.chaos.2007.07.012.

[35]

Y. ZhouD. Xiao and Y. Li, Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, Chaos Solitons Fractals, 32 (2007), 1903-1915. doi: 10.1016/j.chaos.2006.01.002.

show all references

References:
[1]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991.

[2]

I. AreaF. NdaïrouJ. J. NietoC. J. Silva and D. F. M. Torres, Ebola model and optimal control with vaccination constraints, J. Ind. Manag. Optim., 14 (2018), 427-446. doi: 10.3934/jimo.2017054.

[3]

E. R. Avakov, The maximum principle for abnormal optimal control problems, Soviet Math. Dokl., 37 (1988), 231-234.

[4]

Z. Bai and Y. Zhou, Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Anal. Real World Appl., 13 (2012), 1060-1068. doi: 10.1016/j.nonrwa.2011.02.008.

[5]

A. CoriA. ValleronF. CarratG. Scalia TombaG. Thomas and P. Boëlle, Estimating influenza latency and infectious period durations using viral excretion data, Epidemics, 4 (2012), 132-138. doi: 10.1016/j.epidem.2012.06.001.

[6]

C. Ding, N. Tao and Y. Zhu, A mathematical model of Zika virus and its optimal control, Proceedings of the 35th Chinese Control Conference, July 27-29, 2016, Chengdu, China. IEEE Xplore, (2016), 2642-2645. doi: 10.1109/ChiCC.2016.7553763.

[7]

S. EdlundJ. KaufmanJ. LesslerJ. DouglasM. BrombergZ. KaufmanR. BassalG. ChodickR. MaromV. ShalevY. MesikaR. Ram and A. Leventhal, Comparing three basic models for seasonal influenza, Epidemics, 3 (2011), 135-142. doi: 10.1016/j.epidem.2011.04.002.

[8]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron. J. Differential Equations, 1998 (1998), 12 pp.

[9]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975.

[10]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492. doi: 10.3934/mbe.2009.6.469.

[11]

S. GaoL. Chen and Z. Teng, Pulse vaccination of an SEIR epidemic model with time delay, Nonlinear Anal. Real World Appl., 9 (2008), 599-607. doi: 10.1016/j.nonrwa.2006.12.004.

[12]

H. W. HethcoteM. A. Lewis and P. van den Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64. doi: 10.1007/BF00276080.

[13]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539.

[14]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57.

[15]

T. Kuniya and Y. Nakata, Permanence and extinction for a nonautonomous SEIS epidemic model, Appl. Math. Comput., 218 (2012), 9321-9331. doi: 10.1016/j.amc.2012.03.011.

[16]

A. P. Lemos-PaiãoC. J. Silva and D. F. M. Torres, An epidemic model for cholera with optimal control treatment, J. Comput. Appl. Math., 318 (2017), 168-180. doi: 10.1016/j.cam.2016.11.002.

[17]

M. Y. LiH. L. Smith and L. Wang, Global dynamics an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58-69. doi: 10.1137/S0036139999359860.

[18]

W. M. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[19]

M. Martcheva, A non-autonomous multi-strain SIS epidemic model, J. Biol. Dyn., 3 (2009), 235-251. doi: 10.1080/17513750802638712.

[20]

J. P. Mateus and C. M. Silva, A non-autonomous SEIRS model with general incidence rate, Appl. Math. Comput., 247 (2014), 169-189. doi: 10.1016/j.amc.2014.08.078.

[21]

Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363 (2010), 230-237. doi: 10.1016/j.jmaa.2009.08.027.

[22]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.

[23]

A. Rachah and D. F. M. Torres, Dynamics and optimal control of Ebola transmission, Math. Comput. Sci., 10 (2016), 331-342. doi: 10.1007/s11786-016-0268-y.

[24]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X.

[25]

M. A. Safi and S. M. Garba, Global stability analysis of SEIR model with Holling type Ⅱ incidence function, Comput. Math. Methods Med., 2012 (2012), Art. ID 826052, 8 pp. doi: 10.1155/2012/826052.

[26]

R. Shope, Global climate change and infectious diseases, Environ. Health Perspect, 96 (1991), 171-174. doi: 10.1289/ehp.9196171.

[27]

C. SilvaH. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337. doi: 10.3934/mbe.2017021.

[28]

D. F. M. Torres, Lipschitzian regularity of the minimizing trajectories for nonlinear optimal control problems, Math. Control Signals Systems, 16 (2003), 158-174. doi: 10.1007/s00498-003-0132-x.

[29]

P. van den Driessche, Deterministic Compartmental Models: Extensions of Basic Models, In: Mathematical epidemiology, vol. 1945 of Lecture Notes in Math., Springer, Berlin, (2008), 147-157. doi: 10.1007/978-3-540-78911-6_5.

[30]

F. J. S. Wang and W. R. Derrick, On deterministic epidemic models, Bull. Inst. Math. Acad. Sinica, 6 (1978), 73-84.

[31]

A. WeberM. Weber and P. Milligan, Modeling epidemics caused by respiratory syncytial virus (RSV), Math. Biosci., 172 (2001), 95-113. doi: 10.1016/S0025-5564(01)00066-9.

[32]

T. ZhangJ. Liu and Z. Teng, Existence of positive periodic solutions of an SEIR model with periodic coefficients, Appl. Math., 57 (2012), 601-616. doi: 10.1007/s10492-012-0036-5.

[33]

T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69 (2007), 2537-2559. doi: 10.1007/s11538-007-9231-z.

[34]

T. Zhang and Z. Teng, Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model, Chaos Solitons Fractals, 39 (2009), 2411-2425. doi: 10.1016/j.chaos.2007.07.012.

[35]

Y. ZhouD. Xiao and Y. Li, Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, Chaos Solitons Fractals, 32 (2007), 1903-1915. doi: 10.1016/j.chaos.2006.01.002.

Figure 1.  SEIRS autonomous model (${\rm{per}} = 0$ in (28) and (29)): uncontrolled (dashed lines) versus optimally controlled (continuous lines).
Figure 2.  SEIRS periodic model (${\rm{per}} = 0.8$ in (28) and (29)): uncontrolled (dashed lines) versus optimally controlled (continuous lines).
Figure 3.  SEIRS model subject to optimal control: autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.
Figure 4.  SEIRS model without control measures: autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.
Figure 5.  The optimal controls ${\mathbb{T}}^*$ (5) (treatment) and ${\mathbb{V}}^*$ (6) (vaccination): autonomous (${\rm{per}} = 0$) versus periodic (${\rm{per}} = 0.8$) cases.
Figure 6.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the natural death $\mu \in [0, 0.1]$.
Figure 7.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the rate of recovery $\gamma \in [0, 0.1]$.
Figure 8.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the infectivity rate $\varepsilon \in [0, 0.1]$.
Figure 9.  Variation of infected individuals $I^*(t)$ and optimal measures of treatment ${\mathbb{T}}^*(t)$ and vaccination ${\mathbb{V}}^*(t)$, in both autonomous (${\rm{per}} = 0$) and periodic (${\rm{per}} = 0.8$) cases, with the loss of immunity rate $\eta \in [0, 0.1]$.
Table 1.  Values of the parameters for problem (P) used in Section 6.
NameDescriptionValue
$S_0$Initial susceptible population0.98
$E_0$Initial exposed population0
$I_0$Initial infective population0.01
$R_0$Initial recovered population0.01
$\mu$natural deaths0.05
$\varepsilon$infectivity rate0.03
$\gamma$rate of recovery0.05
$\eta$loss of immunity rate0.041
$k_1$weight for the number of infected1
$k_2$weight for treatment0.01
$k_3$weight for vaccination0.01
$\tau_{\max}$maximum rate of treatment0.1
$v_{\max}$maximum rate of vaccination0.4
NameDescriptionValue
$S_0$Initial susceptible population0.98
$E_0$Initial exposed population0
$I_0$Initial infective population0.01
$R_0$Initial recovered population0.01
$\mu$natural deaths0.05
$\varepsilon$infectivity rate0.03
$\gamma$rate of recovery0.05
$\eta$loss of immunity rate0.041
$k_1$weight for the number of infected1
$k_2$weight for treatment0.01
$k_3$weight for vaccination0.01
$\tau_{\max}$maximum rate of treatment0.1
$v_{\max}$maximum rate of vaccination0.4
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