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

• ∗ Corresponding author: Yong Li
• 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.

Mathematics Subject Classification: Primary: 37N25, 34H05; Secondary: 34K13.

 Citation: • • 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

Table 1.  Parameter values

 Parameter Value Source $\mu$ $0.0143~year^{{-1}}$  $\varepsilon$ $6~year^{{-1}}$  $\alpha$ $0.0015~year^{{-1}}$  $c$ $0.05~year^{{-1}}$ Assumed $\gamma$ $0.4055~year^{{-1}}$ Assumed $\beta$ $0.4945$ Assumed
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