## Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission

Jing Xiaojie, Zhao Aimin,, Liu Guirong

 基金资助: 国家自然科学基金山西省自然科学基金the NSFCthe Natural Science Foundation Program of ShanxiProvince

 Fund supported: 国家自然科学基金山西省自然科学基金the NSFCthe Natural Science Foundation Program of ShanxiProvince

Abstract

In this paper, a measles epidemic model with partial immunity and environmental transmission is considered, and the basic reproduction number R0 is obtained. By constructing Lyapunov functions, we prove the global asymptotic stability of the infection-free equilibrium and the endemic equilibrium. When R0 < 1, the infection-free equilibrium is globally asymptotically stable, which implies that measles dies out eventually; when R0 > 1, the model has a unique endemic equilibrium, which is globally asymptotically stable, that is the transmission of measles keeps a steady state. Finally, the simulations are carried to verify the rationality of the results. This work has practical significance for guiding us to prevent and control the measles spread.

Keywords： Partial immunity ; Environmental transmission ; The basic reproduction number ; Lyapunov function ; Global stability

Jing Xiaojie, Zhao Aimin, Liu Guirong. Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission. Acta Mathematica Scientia[J], 2019, 39(4): 909-917 doi:

## 2模型的建立

$$$\left\{\begin{array}{ll} S' = \lambda(1-\eta)-\beta SI-\alpha SB-\mu S, \\ V' = \lambda\eta-\varepsilon\beta VI-\varepsilon\alpha VB-\mu V-\delta V, \\ E' = \beta SI+\alpha SB+\varepsilon\beta VI+\varepsilon\alpha VB-\mu E-pE, \\ I' = pE-\mu I-\gamma I, \\ R' = \gamma I-\mu R+\delta V, \\ B' = kI-\tau B. \end{array}\right.$$$

$$$\left\{\begin{array}{ll} S' = \lambda(1-\eta)-\beta SI-\alpha SB-\mu S, \\ V' = \lambda\eta-\varepsilon\beta VI-\varepsilon\alpha VB-\mu V-\delta V, \\ E' = \beta SI+\alpha SB+\varepsilon\beta VI+\varepsilon\alpha VB-\mu E-pE, \\ I' = pE-\mu I-\gamma I, \\ B' = kI-\tau B. \end{array}\right.$$$

 参数 取值 意义 来源 $\lambda$ $1.728\times 10^{7}$ 新生儿的出生率 国家统计局(2017) $\mu$ 0.0131 自然死亡率 国家统计局(2017) $\eta$ 0.8216 新生儿接种疫苗的比例 文献[9] $\beta$ $9.6696\times 10^{-9}$ 人与人的感染率 估计 $\alpha$ $1.1352\times 10^{-9}$ 环境对人的感染率 估计 $\varepsilon$ 0.15 无效的接种率 中国疾病预防控制中心(2017) $\delta$ 0.95 接种疫苗后获得免疫并成为恢复者的转移率 中国疾病预防控制中心(2017) $p$ 26 潜伏者变为染病者的比例 中国疾病预防控制中心(2017) $\gamma$ 20 染病者的恢复率 中国疾病预防控制中心(2017) $k$ 8 染病者排放到环境中的病毒的速率 假设 $\tau$ 1.6 环境中病毒的失效率 假设