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April  2019, 24(4): 1469-1483. doi: 10.3934/dcdsb.2018216

## Dynamic behavior and optimal scheduling for mixed vaccination strategy with temporary immunity

 a. School of Mathematics, Jilin University, Changchun 130012, China b. School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

∗ Corresponding author: Yong Li

Received  November 2017 Revised  February 2018 Published  April 2019 Early access  June 2018

This paper presents an SEIRVS epidemic model with different vaccination strategies to investigate the elimination of the chronic disease. The mixed vaccination strategy, a combination of constant vaccination and pulse vaccination, is a future development tendency of disease control. Theoretical analysis and threshold conditions for eradicating the disease are given. Then we propose an optimal control problem and solve the optimal scheduling of the mixed vaccination strategy through the combined multiple shooting and collocation (CMSC) method. Theoretical results and numerical simulations can help to design the final mixed vaccination strategy for the optimal control of the chronic disease once the new vaccine comes into use.

Citation: Siyu Liu, Xue Yang, Yingjie Bi, Yong Li. Dynamic behavior and optimal scheduling for mixed vaccination strategy with temporary immunity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1469-1483. doi: 10.3934/dcdsb.2018216
##### References:
 [1] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991. [2] B. E. Asri, Deterministic minimax impulse control in finite horizon: The viscosity solution approach, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 63-77.  doi: 10.1051/cocv/2011200. [3] G. Barles, Deterministic impulse control problems, SIAM Journal on Control and Optimization, 23 (1985), 419-432.  doi: 10.1137/0323027. [4] A. Bensoussan and J. L. Lions, Impulse control and quasi-variational inequalities, Fruit Growing Research, 1984. [5] L. T. Biegler, Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Computers & Chemical Engineering, 8 (1984), 243-247.  doi: 10.1016/0098-1354(84)87012-X. [6] P. Clayden, S. Collins, C. Daniels, M. Frick, M. Harrington, T. Horn, R. Jefferys, K. Kaplan, E. Lessem, L. McKenna and T. Swan, 2014 Pipeline Report: HIV, Hepatitis C Virus (HCV) and Tuberculosis Drugs, Diagnostics, Vaccines, Preventive Technologies, Research Toward a Cure, and Immune-Based and Gene Therapies in Development, New York, 2014. [7] W. A. Coppel, Stability, Asymptotic Behavior of Differential Equations, American Mathematical Monthly, 1965. [8] A. R. D. Cruz, R. T. N. Cardoso and R. H. C. Takahashi, Multi-objective design with a stochastic validation of vaccination campaigns, IFAC Proceedings Volumes, 42 (2009), 289-294.  doi: 10.3182/20090506-3-SF-4003.00053. [9] A. d'Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Mathematical Biosciences, 179 (2002), 57-72.  doi: 10.1016/S0025-5564(02)00095-0. [10] A. d'Onofrio, Mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times, Applied Mathematics and Computation, 151 (2004), 181-187.  doi: 10.1016/S0096-3003(03)00331-X. [11] P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [12] S. Jana, P. Haldar and T. K. Kar, Mathematical analysis of an epidemic model with isolation and optimal controls, International Journal of Computer Mathematics, 94 (2017), 1318-1336.  doi: 10.1080/00207160.2016.1190009. [13] T. Khan, G. Zaman and M. I. Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis B, Journal of Biological Dynamics, 11 (2017), 172-189.  doi: 10.1080/17513758.2016.1256441. [14] J. Li, The spread and prevention of tuberculosis, Chinese Remedies and Clinics, 13 (2013), 482-483. [15] S. Liu, Y. Li, Y. Bi and Q. Huang, Mixed vaccination strategy for the control of tuberculosis: A case study in China, Mathematical Biosciences and Engineering, 14 (2017), 695-708.  doi: 10.3934/mbe.2017039. [16] Z. Lu, X. Chi and L. Chen, The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Mathematical and Computer Modelling, 36 (2002), 1039-1057.  doi: 10.1016/S0895-7177(02)00257-1. [17] A. Mubayi, C. Zaleta, M. Martcheva and C. Castillo-Chávez, A cost-based comparison of quarantine strategies for new emerging diseases, Mathematical Biosciences and Engineering, 7 (2010), 687-717.  doi: 10.3934/mbe.2010.7.687. [18] National Bureau of Statistics of China, Statistical Data of Category A and B Infectious Diseases 2011-2015. Available from: http://data.stats.gov.cn/easyquery.htm?cn=C01. [19] National Bureau of Statistics of China, China Statistical Yearbook 2016, Birth Rate, Death Rate and Natural Growth Rate of Population, 2016. Available from: http://www.stats.gov.cn/tjsj/ndsj/2016/indexch.htm. [20] K. E. Nelson and C. M. Williams, Early histroy of infectious disease: epidemiology and control of infectious diseases, in Infectious Disease Epidemiology: Theory and Practice, Jones and Bartlett Learning, (2014), 3-18. [21] D. J. Nokes and J. Swinton, The control of childhood viral infections by pulse vaccination, IMA Journal of Mathematics Applied in Medicine & Biology, 12 (1995), 29-53.  doi: 10.1093/imammb/12.1.29. [22] B. Song, C. Castillo-Chávez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts, Mathematical Biosciences, 180 (2002), 187-205.  doi: 10.1016/S0025-5564(02)00112-8. [23] O. V. Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Annals of Operations Research, 37 (1992), 357-373.  doi: 10.1007/BF02071065. [24] J. Tamimi and P. Li, A combined approach to nonlinear model predictive control of fast systems, Journal of Process Control, 20 (2010), 1092-1102.  doi: 10.1016/j.jprocont.2010.06.002. [25] E. Verriest, F. Delmotte and M. Egerstedt, Control of epidemics by vaccination, Proceedings of the American Control Conference, 2 (2005), 985-990.  doi: 10.1109/ACC.2005.1470088. [26] Y. Yang, S. Tang, X. Ren, H. Zhao and C. Guo, Global stability and optimal control for a tuberculosis model with vaccination and treatment, Discrete and Continuous Dynamical Systems - Series B, 21 (2016), 1009-1022.  doi: 10.3934/dcdsb.2016.21.1009. [27] Y. Yang, Y. Xiao and J. Wu, Pulse HIV vaccination: feasibility for virus eradication and optimal vaccination schedule, Bulletin of Mathematical Biology, 75 (2013), 725-751.  doi: 10.1007/s11538-013-9831-8. [28] Y. Zhou, J. Wu and M. Wu, Optimal isolation strategies of emerging infectious diseases with limited resources, Mathematical Biosciences and Engineering, 10 (2013), 1691-1701.  doi: 10.3934/mbe.2013.10.1691.

show all references

##### References:
 [1] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991. [2] B. E. Asri, Deterministic minimax impulse control in finite horizon: The viscosity solution approach, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 63-77.  doi: 10.1051/cocv/2011200. [3] G. Barles, Deterministic impulse control problems, SIAM Journal on Control and Optimization, 23 (1985), 419-432.  doi: 10.1137/0323027. [4] A. Bensoussan and J. L. Lions, Impulse control and quasi-variational inequalities, Fruit Growing Research, 1984. [5] L. T. Biegler, Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Computers & Chemical Engineering, 8 (1984), 243-247.  doi: 10.1016/0098-1354(84)87012-X. [6] P. Clayden, S. Collins, C. Daniels, M. Frick, M. Harrington, T. Horn, R. Jefferys, K. Kaplan, E. Lessem, L. McKenna and T. Swan, 2014 Pipeline Report: HIV, Hepatitis C Virus (HCV) and Tuberculosis Drugs, Diagnostics, Vaccines, Preventive Technologies, Research Toward a Cure, and Immune-Based and Gene Therapies in Development, New York, 2014. [7] W. A. Coppel, Stability, Asymptotic Behavior of Differential Equations, American Mathematical Monthly, 1965. [8] A. R. D. Cruz, R. T. N. Cardoso and R. H. C. Takahashi, Multi-objective design with a stochastic validation of vaccination campaigns, IFAC Proceedings Volumes, 42 (2009), 289-294.  doi: 10.3182/20090506-3-SF-4003.00053. [9] A. d'Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model, Mathematical Biosciences, 179 (2002), 57-72.  doi: 10.1016/S0025-5564(02)00095-0. [10] A. d'Onofrio, Mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times, Applied Mathematics and Computation, 151 (2004), 181-187.  doi: 10.1016/S0096-3003(03)00331-X. [11] P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [12] S. Jana, P. Haldar and T. K. Kar, Mathematical analysis of an epidemic model with isolation and optimal controls, International Journal of Computer Mathematics, 94 (2017), 1318-1336.  doi: 10.1080/00207160.2016.1190009. [13] T. Khan, G. Zaman and M. I. Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis B, Journal of Biological Dynamics, 11 (2017), 172-189.  doi: 10.1080/17513758.2016.1256441. [14] J. Li, The spread and prevention of tuberculosis, Chinese Remedies and Clinics, 13 (2013), 482-483. [15] S. Liu, Y. Li, Y. Bi and Q. Huang, Mixed vaccination strategy for the control of tuberculosis: A case study in China, Mathematical Biosciences and Engineering, 14 (2017), 695-708.  doi: 10.3934/mbe.2017039. [16] Z. Lu, X. Chi and L. Chen, The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Mathematical and Computer Modelling, 36 (2002), 1039-1057.  doi: 10.1016/S0895-7177(02)00257-1. [17] A. Mubayi, C. Zaleta, M. Martcheva and C. Castillo-Chávez, A cost-based comparison of quarantine strategies for new emerging diseases, Mathematical Biosciences and Engineering, 7 (2010), 687-717.  doi: 10.3934/mbe.2010.7.687. [18] National Bureau of Statistics of China, Statistical Data of Category A and B Infectious Diseases 2011-2015. Available from: http://data.stats.gov.cn/easyquery.htm?cn=C01. [19] National Bureau of Statistics of China, China Statistical Yearbook 2016, Birth Rate, Death Rate and Natural Growth Rate of Population, 2016. Available from: http://www.stats.gov.cn/tjsj/ndsj/2016/indexch.htm. [20] K. E. Nelson and C. M. Williams, Early histroy of infectious disease: epidemiology and control of infectious diseases, in Infectious Disease Epidemiology: Theory and Practice, Jones and Bartlett Learning, (2014), 3-18. [21] D. J. Nokes and J. Swinton, The control of childhood viral infections by pulse vaccination, IMA Journal of Mathematics Applied in Medicine & Biology, 12 (1995), 29-53.  doi: 10.1093/imammb/12.1.29. [22] B. Song, C. Castillo-Chávez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts, Mathematical Biosciences, 180 (2002), 187-205.  doi: 10.1016/S0025-5564(02)00112-8. [23] O. V. Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Annals of Operations Research, 37 (1992), 357-373.  doi: 10.1007/BF02071065. [24] J. Tamimi and P. Li, A combined approach to nonlinear model predictive control of fast systems, Journal of Process Control, 20 (2010), 1092-1102.  doi: 10.1016/j.jprocont.2010.06.002. [25] E. Verriest, F. Delmotte and M. Egerstedt, Control of epidemics by vaccination, Proceedings of the American Control Conference, 2 (2005), 985-990.  doi: 10.1109/ACC.2005.1470088. [26] Y. Yang, S. Tang, X. Ren, H. Zhao and C. Guo, Global stability and optimal control for a tuberculosis model with vaccination and treatment, Discrete and Continuous Dynamical Systems - Series B, 21 (2016), 1009-1022.  doi: 10.3934/dcdsb.2016.21.1009. [27] Y. Yang, Y. Xiao and J. Wu, Pulse HIV vaccination: feasibility for virus eradication and optimal vaccination schedule, Bulletin of Mathematical Biology, 75 (2013), 725-751.  doi: 10.1007/s11538-013-9831-8. [28] Y. Zhou, J. Wu and M. Wu, Optimal isolation strategies of emerging infectious diseases with limited resources, Mathematical Biosciences and Engineering, 10 (2013), 1691-1701.  doi: 10.3934/mbe.2013.10.1691.
Comparison between the constant vaccination strategy and mixed vaccination strategy with the same cost (w = 3). The red dashed line shows the constant vaccination strategy with $p = 1$. The blue solid line shows optimal mixed vaccination strategy with $p = 0.45, p_{c} = 0.2$ and $T = 5$. All the other parameters are shown in Table 1
Comparison between the constant vaccination strategy and optimal mixed vaccination strategy. The red dashed line shows the constant vaccination strategy with $p = 0.85 (0.6\leq p\leq 0.85)$. The blue solid line shows optimal mixed vaccination strategy with $0.6\leq u_{1}(t)\leq 0.85, 0.1\leq u_{2}(t)\leq 0.3$ and $5\leq N\leq 10$. All the other parameters are shown in Table 1
Optimal mixed vaccination strategy under limited vaccinated individuals with $0.6\leq u_{1}(t)\leq 0.85, 0.1\leq u_{2}(t)\leq 0.3$ and $5\leq N\leq 10$. All the other parameters are shown in Table 1
Parameter values
 Parameter Value Source $\mu$ $0.0143~year^{{-1}}$ [19] $\varepsilon$ $6~year^{{-1}}$ [14] $\alpha$ $0.0015~year^{{-1}}$ [14] $c$ $0.05~year^{{-1}}$ Assumed $\gamma$ $0.4055~year^{{-1}}$ Assumed $\beta$ $0.4945$ Assumed
 Parameter Value Source $\mu$ $0.0143~year^{{-1}}$ [19] $\varepsilon$ $6~year^{{-1}}$ [14] $\alpha$ $0.0015~year^{{-1}}$ [14] $c$ $0.05~year^{{-1}}$ Assumed $\gamma$ $0.4055~year^{{-1}}$ Assumed $\beta$ $0.4945$ Assumed
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