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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 & Continuous Dynamical Systems - B, 2019, 24 (4) : 1469-1483. doi: 10.3934/dcdsb.2018216
##### References:

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##### References:
">Figure 1.  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
">Figure 2.  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
">Figure 3.  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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