基于多特征参数的OMI遥感产品气溶胶分类研究——以广东省为例

郑仰成; 黎丽莉; 王云鹏

doi:10.6054/j.jscnun.2021060

基于多特征参数的OMI遥感产品气溶胶分类研究——以广东省为例

郑仰成^{1, 2, 3},
黎丽莉^4, ,,
王云鹏^{1, 2}

1.
中国科学院广州地球化学研究所有机地球化学国家重点实验室，广州 510640
2.
中国科学院深地科学卓越创新中心，广州 510640
3.
中国科学院大学，北京 100049
4.
仲恺农业工程学院资源与环境学院，广州 510225

基金项目:

国家自然科学基金项目 42007205

广东省自然科学基金项目 2021A1515011375

详细信息

通讯作者:
黎丽莉，Email: lilili@gig.ac.cn

中图分类号: X87
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出版历程
- 收稿日期: 2020-12-07
- 网络出版日期: 2021-09-02
- 刊出日期: 2021-08-24

The Multi-feature Parameter Classification of Aerosol Based on OMI Remote Sensing Data: A Case Study in Guangdong Province

ZHENG Yangcheng^{1, 2, 3},
LI Lili^4, ,,
WANG Yunpeng^{1, 2}

1.
Guangzhou Institute of Geochemistry, Chinese Academy of Sciences, Guangzhou 510640, China
2.
Center for Excellence in Deep Earth Science, Chinese Academy of Sciences, Guangzhou 510640, China
3.
College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 510640, China
4.
College of Environmental Science and Engineering, Zhongkai University of Agricultural and Engineering, Guangzhou 510225, China

摘要

摘要: 为了解决气溶胶分类精度低和特征参数冗杂的问题，基于OMI(Ozone Monitoring Instrument)遥感产品的气溶胶特征参数，利用随机森林算法，将广东省2014年的气溶胶类型划分为沙尘型气溶胶(Desert Dust, DST)、生物质燃烧型含碳气溶胶(Carbonaceous Aerosols Associated with Biomass Burning, CRB)和硫酸盐型城镇-工业气溶胶(Sulfate-based Urban-industrial Aerosols，SLF)3种类型. 并统计分析随机森林以及特征参数的重要性，将分类结果的空间分布与OMI气溶胶类型产品的空间分布进行对比. 结果表明：(1)随机森林算法仅需少量训练样本点即可达到97%以上的总体分类精度. (2)通过计算不同气溶胶特征参数在随机森林分类过程中的重要性高低，得到重要性排名前六的特征参数依次为α指数、UVAI、RI₃₈₈、RI₃₅₄、SSA₅₀₀、AAOD₅₀₀，表明在分类过程中，气溶胶粒径分布和吸收能力起到了最关键的作用. (3)3种气溶胶类型的空间分布显示，SLF型气溶胶为广东省最主要的气溶胶类型；DST型和CRB型气溶胶在珠三角地区占比最高，在粤东、粤北地区的占比最低.
- 气溶胶分类 /
- 随机森林 /
- 臭氧层监测仪 /
- 特征参数 /
- α指数 /
- UVAI
Abstract: In order to solve the problems of low precision and redundant feature parameters in the process of aerosol classification, several significant aerosol feature parameters were extracted from the OMI (Ozone Monitoring Instrument) remote sensing products and Random Forest (RF) algorithm was used for aerosol type classification and verification. Aerosols of Guangdong Province in 2014 were divided into three types: desert dust (DST), carbonaceous aerosols associated with biomass burning (CRB) and sulfate-based urban-industrial aerosols (SLF). The Random Forest classification results and the importance of feature parameters were analyzed. The spatial distribution of cla-ssification results were compared to that of OMI aerosol type products. The following results are obtained. First, with the RF algorithm, a total precision of over 97% can be reached with a few training samples. Second, calculating the importance of different aerosol feature parameters in RF shows that the most important feature parameters are angstrom exponent, UVAI, RI₃₈₈, RI₃₅₄, SSA₅₀₀ and AAOD₅₀₀ in turn, indicating that size distribution and absorption ability of aerosols play the key roles. Third, the spatial distribution of three aerosol types shows that sulfate-based urban-industrial aerosols are dominant in the Pearl River Delta. Proportion of biomass burning and desert dust aerosols are highest in the Pearl River Delta and lowest in the eastern and northern Guangdong.
- aerosol classification /
- random forest /
- OMI /
- feature parameter /
- angstrom exponent /
- UVAI

HTML全文

研究流行病的传播规律极为重要^[1-3].目前, 许多学者利用数学工具描述动力学模型, 进一步预测流行病传播^[4-6].疫苗接种是控制流行病传播的重要手段之一^[7-9], 为了研究疫苗接种对流行病动力学行为的影响, LAHROUZ等^[8]提出了如下带有疫苗接种的SIRS传染病模型:

$\left\{ \begin{array}{l} {\rm{d}}S = \left( {(1 - p)b - \mu S - \frac{{\beta SI}}{{\varphi (I)}} + \gamma R} \right){\rm{d}}t,\\ {\rm{d}}I = \left( { - (\mu + c + \alpha )I + \frac{{\beta SI}}{{\varphi (I)}}} \right){\rm{d}}t,\\ {\rm{d}}R = (pb - (\mu + \gamma )R + \alpha I){\rm{d}}t, \end{array} \right.$

(1)

其中, S、I、R分别代表易感者、感染者、恢复者的人数；所有参数均非负, 其生物学意义如下:b为出生率, p (0≤p≤1)为疫苗接种率, μ为自然死亡率, c为因病死亡率, β为感染系数, α为恢复率, γ为恢复者免疫力丧失率, φ(I)为正函数, 且φ(0)=1, φ′(I)≥0.该模型给出了无病平衡点和地方病平衡点, 讨论了2个平衡点的全局渐近稳定性.

在疾病爆发期间, 信息干预(宣传、报道、教育等)会对人们的行为产生影响, 从而影响流行病的传播速度.近年来, 有许多学者研究了信息干预对流行病传播的影响^[10-12].文献[6]指出媒体报道虽然不能使感染者恢复, 但可以降低感染者人数的峰值.因此, 将媒体报道因素考虑到模型中是很有必要的.

本文将信息干预引入到模型(1)中, 得到了新的基于信息干预和疫苗接种的SIRS传染病模型, 研究了新模型平衡点的存在性及其渐近稳定性, 给出了基本再生数, 并通过数值模拟验证了理论结果.

1. 主要结果及证明

将信息干预引入到模型(1)中, 得到了新的基于信息干预和疫苗接种的SIRS传染病模型:

$\left\{ \begin{array}{l} {\rm{d}}S = \left( {\left( {1 - p} \right)b - \mu S - \frac{{\beta SI}}{{\varphi \left( I \right)}} + \gamma R - {\mu _1}mZS} \right){\rm{d}}t,\\ {\rm{d}}I = \left( { - \left( {\mu + c + \alpha } \right)I + \frac{{\beta SI}}{{\varphi \left( I \right)}}} \right){\rm{d}}t,\\ {\rm{d}}R = \left( {pb - (\mu + \gamma )R + \alpha I + {\mu _1}mZS} \right){\rm{d}}t,\\ {\rm{d}}Z = \left( {\frac{{aI}}{{1 + {a_1}I}} - {a_0}Z} \right){\rm{d}}t, \end{array} \right.$

(2)

其中, Z为信息密度, m、μ₁、a、a₁和a₀分别表示信息干预率、信息强度、信息增长率、饱和常数和信息自然消亡率.

本文的主要结果如下:

定理1 (ⅰ)如果ℜ₀ < 1, 则无病平衡点E₀是局部渐近稳定的；如果ℜ₀>1，则E₀是不稳定的.

(ⅱ)模型(2)有1个地方病平衡点E^*, 且当ℜ₀>1并满足A₁B₁>C₁及A₁(B₁C₁-A₁C₁)>C₁²时, E^*是局部渐近稳定的.

定理2 当ℜ₀=1时, 模型(2)有1个前向分支.

定理3 当ℜ₀ < 1时, 模型(2)的无病平衡点U₀=(X₀, 0)是全局渐近稳定的.

定理4 若ℜ₀>1且

${a^2} < \frac{{{a_0}}}{3}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),$

$\begin{array}{l} \mu _1^2{m^2} < \min \left\{ {\frac{{{a_0}{\gamma ^2}}}{{12{\mu ^2}{S^{*2}}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),\frac{{{a_0}\alpha {\mu ^2}(\mu + \gamma )}}{{c{b^2}}},} \right.\\ \;\;\;\;\;\;\;\;\left. {\frac{{8{\mu ^2}(\mu + \gamma )\alpha }}{{3c\gamma {Z^{*2}}}},\frac{{{a_0}\gamma }}{{6{S^{*2}}}},\frac{{2\gamma a_0^2{{\left( {\mu + {a_1}b} \right)}^2}}}{{9{a^2}{b^2}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right)} \right\}, \end{array}$

则模型(2)的地方病平衡点E^*是全局渐近稳定的.

1.1 平衡点和正解的存在性

由文献[13]的方法, 可以得到模型(2)的基本再生数:

${\Re _0} = \frac{{\beta (1 - p)(\mu + \gamma ) + \beta \gamma p}}{{(\mu + \gamma )(\mu + c + \alpha )}}.$

(3)

下面考虑模型(2)的平衡点的存在性.令模型(2)右端等于零, 即

$\left\{ \begin{array}{l} \left( {1 - p} \right)b - \mu S - \frac{{\beta SI}}{{\varphi \left( I \right)}} + \gamma R - {\mu _1}mZS = 0,\\ - \left( {\mu + c + \alpha } \right)I + \frac{{\beta SI}}{{\varphi \left( I \right)}} = 0,\\ pb - \left( {\mu + \gamma } \right)R + \alpha I + {\mu _1}mZS = 0,\\ \frac{{aI}}{{1 + {a_1}I}} - {a_0}Z = 0. \end{array} \right.$

(4)

求解方程组(4), 可得模型(2)存在2个平衡点:(1)无病平衡点 $E_{0}=\left(\frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}, 0, \frac{p b}{\mu+\gamma}, 0\right)$ . (2)当ℜ₀>1时，存在地方病平衡点E^*=(S^*, I^*, R^*, Z^*), 其中

${S^*} = \frac{{\left( {\mu + c + \alpha } \right)\varphi \left( {{I^*}} \right)}}{\beta },$

${R^*} = \frac{1}{{\mu + \gamma }}\left[ {pb + \alpha {I^*} + \frac{{{\mu _1}ma\left( {\mu + c + \alpha } \right)\varphi \left( {{I^*}} \right){I^*}}}{{{a_0}\beta \left( {1 + {a_1}{I^*}} \right)}}} \right],$

${Z^*} = \frac{{a{I^*}}}{{{a_0}\left( {1 + {a_1}{I^*}} \right)}},$

且I^*是以下方程的唯一正根:

$\begin{array}{l} H\left( I \right) = \left( {1 - p} \right)b - \frac{{\left( {\mu + c + \alpha } \right)\varphi \left( I \right)}}{\beta } - \left( {\mu + c + \alpha } \right)I + \frac{{\gamma pb}}{{\mu + \gamma }} + \\ \;\;\;\;\;\;\;\;\frac{{\alpha \gamma I}}{{\mu + \gamma }} + \frac{{{\mu _1}ma\gamma \left( {\mu + c + \alpha } \right)\varphi (I)I}}{{{a_0}\beta \left( {\mu + \gamma } \right)\left( {1 + {a_1}I} \right)}} = 0, \end{array}$

事实上, 如果ℜ₀>1, 则H(0)>0, H′(I) < 0, 那么lim_I→∞ H(I)=-∞, 即ℜ₀>1当且仅当H(I)=0有唯一的正解.

接下来讨论模型(2)的正解.由模型(2)可得

${\left. {\frac{{{\rm{d}}S}}{{{\rm{d}}t}}} \right|_{S = 0}} = \left( {1 - p} \right)b + \gamma R \ge 0,{\left. {\frac{{{\rm{d}}I}}{{{\rm{d}}t}}} \right|_{I = 0}} \ge 0,$

${\left. {\frac{{{\rm{d}}R}}{{{\rm{d}}t}}} \right|_{R = 0}} = pb + \alpha I + {\mu _1}mZS \ge 0,{\left. {\frac{{{\rm{d}}Z}}{{{\rm{d}}t}}} \right|_{Z = 0}} = \frac{{aI}}{{1 + {a_I}}} \ge 0,$

如果考虑该区域内部, 由模型(2)可知人口总数N=S+I+R满足如下微分方程:

$\frac{{{\rm{d}}N}}{{{\rm{d}}t}} = b - \mu N - cI.$

所以, $\frac{\mathrm{d} N}{\mathrm{d} t} \leqslant b-\mu N$ , 即sup_t→∞ $N \leqslant \frac{b}{\mu}$ .因此, $\frac{b}{\mu}$ 是S、I和R的上界.根据模型(2)的第4个方程和I的界, 有lim sup_t→∞ $Z \leqslant \frac{a b}{a_{0}\left(\mu+a_{1} b\right)}$ .而且, 有以下不变集集合:

$\begin{gathered} \mathit{\Gamma } = \left\{ {(S,I,R,Z) \in \mathbb{R}_ + ^4:S + I + R \leqslant \frac{b}{\mu },0 \leqslant Z \leqslant \frac{{ab}}{{{a_0}\left( {\mu + {a_1}b} \right)}},} \right. \hfill \\ \left. {S \geqslant 0,I \geqslant 0,R \geqslant 0,Z \geqslant 0} \right\}. \hfill \\ \end{gathered}$

综上, 在非负集 $\mathbb{R}_{+}^{4}$ 上, 解将保持在Γ中.

1.2 E₀和E^*的局部稳定性

为了方便, 给出了模型(2)的可变矩阵:

$\mathit{\boldsymbol{D}} = \left[ {\begin{array}{*{20}{c}} { - \mu - \frac{{\beta I}}{{\varphi \left( I \right)}} - c}&{ - \frac{{\beta S}}{{\varphi \left( I \right)}}}&\gamma &{ - \frac{{\beta I}}{{\varphi \left( I \right)}}}\\ {\frac{{\beta I}}{{\varphi \left( I \right)}}}&{ - \left( {pq} \right)}&0&0\\ {{\mu _1}mZ}&\alpha &{ - \left( {\mu + \gamma } \right)}&{{\mu _1}mS}\\ 0&{\frac{a}{{{{\left( {1 + {a_1}I} \right)}^2}}}}&0&{ - {a_0}} \end{array}} \right].$

(5)

对地方病平衡点E^*, 其特征方程为:

${\lambda ^4} + {A_1}{\lambda ^3} + {B_1}{\lambda ^2} + {C_1}\lambda + {D_1} = 0,$

其中,

${A_1} = {a_0} + 2\mu + \gamma + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}} + {\mu _1}m{Z^*},$

$\begin{array}{l} {B_1} = {a_0}(\mu + \gamma ) + \left( {\mu + {a_0}} \right)\left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}} + {\mu _1}m{Z^*}} \right) + \\ \;\;\;\;\;\;\;\;\gamma \left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}} \right) + \frac{{{\beta ^2}{S^*}{I^*}}}{{\varphi {{\left( {{I^*}} \right)}^2}}}, \end{array}$

$\begin{array}{*{20}{c}} {{C_1} = \frac{{{\beta ^2}{S^*}{I^*}}}{{\varphi {{\left( {{I^*}} \right)}^2}}}\left( {{a_0} + \mu + \gamma } \right) + \frac{{a\beta {I^*}{\mu _1}m{S^*}}}{{{{\left( {1 + {a_1}{I^*}} \right)}^2}}} - \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}\alpha {\gamma ^ + }}\\ {{a_0}\mu \left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}} + {\mu _1}m{Z^*}} \right) + {a_0}\gamma \left( {\mu + \frac{{\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}} \right),} \end{array}$

$\begin{array}{*{20}{c}} {{D_1} = \frac{{{a_0}\beta {I^*}}}{{\varphi \left( {{I^*}} \right)}}(\mu (\mu + c + \alpha ) + \gamma (\mu + c)) + }\\ {\frac{{a\beta {I^*}}}{{\varphi \left( {{I^*}} \right){{\left( {1 + {a_1}{I^*}} \right)}^2}}}\mu {\mu _1}m{S^*}.} \end{array}$

定理1的证明 (ⅰ)与文献[13]中定理2的证明类似, 此处略.

(ⅱ)显然A₁>0, D₁>0.由Routh-Hurwitz判据, 若A₁B₁>C₁和A₁(B₁C₁-A₁D₁)>C₁², 则J_E^*的特征方程的所有根要么非负, 要么有负实部.因此, 如果ℜ₀>1, A₁B₁>C₁, A₁(B₁C₁-A₁D₁)>C₁²，则由Hartman-Grobman定理^[14]可知E^*是局部渐近稳定的.

定理2的证明 当ℜ₀=1时, 令x₁=S, x₂=I, x₃=R, x₄=Z, 将φ=β作为分叉参数.于是, 对φ=φ^*=β^*, 由ℜ₀=1可得 $\beta^{*}=\frac{b((1-p) \mu+\gamma)}{\mu(\mu+\gamma)(\mu+c+\alpha)}$ .应用新的变换, 模型(2)可以改写为：

$\left\{ \begin{array}{l} \frac{{{\rm{d}}{x_1}}}{{{\rm{d}}t}} = \left( {1 - p} \right)b - \mu {x_1} - \frac{{\beta {x_1}{x_2}}}{{\varphi \left( {{x_2}} \right)}} + \gamma {x_3} - {\mu _1}m{x_1}{x_4}: = {f_1},\\ \frac{{{\rm{d}}{x_2}}}{{{\rm{d}}t}} = - (\mu + c + \alpha ){x_2} + \frac{{\beta {x_1}{x_2}}}{{\varphi \left( {{x_2}} \right)}}: = {f_2},\\ \frac{{{\rm{d}}{x_3}}}{{{\rm{d}}t}} = pb - (\mu + \gamma ){x_3} + \alpha {x_2} + {\mu _1}m{x_1}{x_4}: = {f_3},\\ \frac{{{\rm{d}}{x_4}}}{{{\rm{d}}t}} = \frac{{a{x_2}}}{{1 + {a_1}{x_2}}} - {a_0}{x_4}: = {f_4}. \end{array} \right.$

对于无病平衡点x^*, 可以得到(x^*, β^*)的雅可比矩阵:

$\begin{array}{l} {\mathit{\boldsymbol{D}}_{{x^ * }}}\left( {{\beta ^ * }} \right) = \\ \left[ {\begin{array}{*{20}{c}} { - \mu }&{ - \left( {\mu + c + \alpha } \right)}&\gamma &{\frac{{ - {\mu _1}m\left[ {\left( {1 - p} \right)\left( {\mu + \gamma } \right) + \gamma p} \right]}}{{\varphi \left( {x_2^*} \right)\left( {\mu + \gamma } \right)}}}\\ 0&0&0&0\\ 0&\alpha &{ - \left( {\mu + \gamma } \right)}&{\frac{{{\mu _1}m\left[ {\left( {1 - p} \right)\left( {\mu + \gamma } \right) + \gamma p} \right]}}{{\varphi \left( {x_2^*} \right)\left( {\mu + \gamma } \right)}}}\\ 0&a&0&{ - {a_0}} \end{array}} \right]. \end{array}$

当ℜ₀=1时, D_x^*(β^*)有1个零特征值, 且其他特征值都是非负的. D_x^*(β^*)相应于零特征值的右特征向量为y=(y₁, y₂, y₃, y₄)′，其中

$\left\{ \begin{array}{l} {y_1} = \frac{1}{\mu }\left\{ {\frac{{\gamma \alpha }}{{\left( {\mu + \gamma } \right)a}} - \frac{{\mu + c + \alpha }}{a} - \frac{{\mu {\mu _1}m[(1 - p)(\mu + \gamma ) + \gamma p]}}{{\varphi \left( {x_2^*} \right){a_0}{{(\mu + \gamma )}^2}}}} \right\},\\ {y_2} = \frac{1}{a},\\ {y_3} = \frac{1}{{(\mu + \gamma )}}\left\{ {\frac{\alpha }{a} + \frac{{{\mu _1}m[(1 - p)(\mu + \gamma ) + \gamma p]}}{{{a_0}\varphi \left( {x_2^*} \right)(\mu + \gamma )}}} \right\},\\ {y_4} = \frac{1}{{{a_0}}}. \end{array} \right.$

类似地, 可以得到D_x^*(β^*)相应于零特征值的左特征向量为z=(z₁, z₂, z₃, z₄), 其中z₁=0, z₂=1, z₃=0, z₄=0.由文献[12]可知，当ℜ₀=1时, 常数a₂和b₂可决定无病平衡点的稳定性.下面计算a₂和b₂. f=(f₁, f₂, f₃, f₄)在(x^*, β^*)处的非零二阶偏导为：

$\frac{{{\partial ^2}{f_2}}}{{\partial {x_2}\partial {x_1}}} = {\beta ^*},\frac{{{\partial ^2}{f_2}}}{{\partial {x_1}\partial {x_2}}} = {\beta ^*},\frac{{{\partial ^2}{f_2}}}{{\partial {x_2}\partial \beta }} = x_1^*,$

则

$\begin{array}{l} {a_2} = \frac{{ - 2((1 - p)(\mu + \gamma ) + \gamma p)}}{{{a_0}{a^2}\mu {{(\mu + \gamma )}^3}(\mu + c + \alpha )\varphi \left( {x_2^*} \right)}}\left\{ {{a_0}\varphi \left( {x_2^*} \right) \times } \right.\\ \;\;\;\;\;\;\;(\mu + c + \alpha )(\mu + \gamma ) + {a_0}\varphi \left( {x_2^*} \right)\gamma (\mu + c)(\mu + \gamma ) + \\ \;\;\;\;\;\;\;\left. {a\mu {\mu _1}m[(1 - p)(\mu + \gamma ) + \gamma p]} \right\}, \end{array}$

${b_2} = \frac{{(\mu + \gamma )(1 - p) + \gamma p}}{{a\varphi \left( {x_2^*} \right)(\mu + \gamma )}}.$

显然, a₂ < 0, b₂>0.得证.

1.3 E₀和E^*的全局稳定性

假设模型(2)可改写为^[15]:

$\frac{{{\rm{d}}\mathit{\boldsymbol{X}}}}{{{\rm{d}}t}} = F(\mathit{\boldsymbol{X}},Y),\frac{{{\rm{d}}Y}}{{{\rm{d}}t}} = G(\mathit{\boldsymbol{X}},Y),G(\mathit{\boldsymbol{X}},0) = 0,$

其中, $\boldsymbol{X} \in \mathbb{R}^{3}$ 、 $Y \in \mathbb{R}$ 分别代表未感染者和感染者的人数.令U₀=(X₀, 0)为无病平衡点.

引理1^[15] 若ℜ₀ < 1且以下条件满足：

(ⅰ)对 $\frac{\mathrm{d} \boldsymbol{X}}{\mathrm{d} t}=F(\boldsymbol{X}, 0), \boldsymbol{X}_{0}$ 是全局渐近稳定的;

(ⅱ)∀(X, Y)∈Γ, 有G(X, Y)=D_YG(X₀, 0)Y- $\hat{G}(\boldsymbol{X}, Y), \hat{G}(\boldsymbol{X}, Y) \geqslant 0$ , 其中, D_YG(X₀, 0)为M-矩阵, 则模型(2)的无病平衡点U₀=(X₀, 0)是全局渐近稳定的.

定理3的证明 类似文献[12]中定理2的证明, 有

$\begin{array}{l} F(\mathit{\boldsymbol{X}},Y) = \left( {(1 - p)b - \mu S - \frac{{\beta SI}}{{\varphi (I)}} + \gamma R - {\mu _1}mZS,} \right.\\ \left. {pb - (\mu + \gamma )R + \alpha I + {\mu _1}mZS,\frac{{aI}}{{1 + {a_1}I}} - {a_0}Z} \right), \end{array}$

$G(\mathit{\boldsymbol{X}},Y) = - (\mu + c + \alpha )I + \frac{{\beta SI}}{{\varphi (I)}},G(\mathit{\boldsymbol{X}},0) = 0,$

其中, X=(S, R, Z)′, Y=I.无病平衡点U₀=E₁=(X₀, 0)且 $\boldsymbol{X}_{0}=\left(\frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}, \frac{p b}{\mu+\gamma}, 0\right)$ .取t→∞, 则有 $\boldsymbol{X} \rightarrow\left(\frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}, \frac{p b}{\mu+\gamma}, 0\right)$ .显然X₀是全局渐近稳定的.更进一步地,

$\begin{array}{l} G(\mathit{\boldsymbol{X}},Y) = - (\mu + c + \alpha )\left( {1 - {\Re _0}} \right)I - \\ \;\;\;\;\;\;\;\frac{{\beta I}}{{\varphi (I)}}\left[ {\frac{{\varphi (I)(\mu (1 - p)b + \gamma b)}}{{\mu (\mu + \gamma )}} - S} \right]. \end{array}$

若 $S \leqslant \frac{\mu(1-p) b+\gamma b}{\mu(\mu+\gamma)}$ , 则

$\hat G(\mathit{\boldsymbol{X}},Y) = \frac{{\beta I}}{{\varphi (I)}}\left[ {\frac{{\varphi (I)(\mu (1 - p)b + \gamma b)}}{{\mu (\mu + \gamma )}} - S} \right] \ge 0.$

所以条件(ⅰ)、(ⅱ)满足.因此, 当ℜ₀ < 1时, 模型(2)的无病平衡点是全局渐近稳定的.

定理4的证明 在区间Γ上, 考虑函数V:

$\begin{array}{l} V(S,I,R,Z) = \frac{1}{2}{\left[ {\left( {S - {S^*}} \right) + \left( {I - {I^*}} \right) + \left( {R - {R^*}} \right)} \right]^2} + \\ \;\;\;\;{m_1}\left( {I - {I^*} - {I^*}\log \frac{I}{{{I^*}}}} \right) + \frac{{{m_2}}}{2}{\left( {S - {S^*} + I - {I^*}} \right)^2} + \frac{{{m_3}}}{2}{\left( {R - {R^*}} \right)^2} + \\ \;\;\;\;\frac{1}{2}{\left( {Z - {Z^*}} \right)^2} = {V_1} + {m_1}{V_2} + {m_2}{V_3} + {m_3}{V_4} + {V_5}, \end{array}$

(6)

其中，m₁、m₂、m₃都是正常数.可以得到V_i(i=1, 2, …, 5)的导数如下:

$\begin{array}{l} {{\dot V}_1} = - \mu {\left( {S - {S^*}} \right)^2} - (\mu + c){\left( {I - {I^*}} \right)^2} - \mu {\left( {R - {R^*}} \right)^2} - \\ \;\;\;\;\;\;(2\mu + c)\left( {S - {S^*}} \right)\left( {I - {I^*}} \right) - 2\mu \left( {S - {S^*}} \right)\left( {R - {R^*}} \right) - \\ \;\;\;\;\;\;(2\mu + c)\left( {I - {I^*}} \right)\left( {R - {R^*}} \right), \end{array}$

$\begin{array}{l} {{\dot V}_2} = - \frac{{\beta S}}{{\varphi (I)\varphi \left( {{I^*}} \right)}}\left( {\varphi (I) - \varphi \left( {{I^*}} \right)} \right)\left( {I - {I^*}} \right) + \\ \;\;\;\;\;\;\;\frac{\beta }{{\varphi \left( {{I^*}} \right)}}\left( {S - {S^*}} \right)\left( {I - {I^*}} \right), \end{array}$

$\begin{array}{l} {{\dot V}_3} = - \mu {\left( {S - {S^*}} \right)^2} - \frac{{\beta {S^*}}}{{\varphi \left( {{I^*}} \right)}}{\left( {I - {I^*}} \right)^2} + \gamma \left( {S - {S^*}} \right)\left( {R - {R^*}} \right) - \\ \;\;\;\;\;{\mu _1}m{S^*}\left( {S - {S^*}} \right)\left( {Z - {Z^*}} \right) - {\mu _1}mZ{\left( {S - {S^*}} \right)^2} - \\ \;\;\;\;\;\left( {\mu + \frac{{\beta {S^*}}}{{\varphi \left( {{I^*}} \right)}}} \right)\left( {S - {S^*}} \right)\left( {I - {I^*}} \right) + \gamma \left( {I - {I^*}} \right)\left( {R - {R^*}} \right) - \\ \;\;\;\;\;{\mu _1}mZ\left( {I - {I^*}} \right)\left( {S - {S^*}} \right) - {\mu _1}m{S^*}\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right), \end{array}$

$\begin{array}{l} {{\dot V}_4} = - (\mu + \gamma ){\left( {R - {R^*}} \right)^2} + \alpha \left( {R - {R^*}} \right)\left( {I - {I^*}} \right) + \\ \;\;\;\;\;{\mu _1}m{Z^*}\left( {R - {R^*}} \right)\left( {S - {S^*}} \right) + {\mu _1}mS\left( {R - {R^*}} \right)\left( {Z - {Z^*}} \right), \end{array}$

${{\dot V}_5} = \frac{{a\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right)}}{{\left( {1 + {a_1}I} \right)\left( {1 + {a_1}{I^*}} \right)}} - {a_0}{\left( {Z - {Z^*}} \right)^2}.$

取 $m_{1}=\frac{\varphi\left(I^{*}\right)}{\beta}\left((2 \mu+c)+\frac{2 \mu}{\gamma}\left(\mu+\frac{\beta S^{*}}{\varphi\left(I^{*}\right)}\right)\right)$ , $m_{2}=\frac{2 \mu}{\gamma}$ , $m_{3}=\frac{c}{\alpha}$ , 可得到：

$\begin{array}{l} \dot V \le - \mu {\left( {S - {S^*}} \right)^2} - \left( {\mu + c} \right){\left( {I - {I^*}} \right)^2} - \mu {\left( {R - {R^*}} \right)^2} - \\ \;\;\;\;\;\frac{{2\mu }}{\gamma }\left( {\mu {{\left( {S - {S^*}} \right)}^2} + \frac{{\beta {S^*}}}{{\varphi \left( {{I^*}} \right)}}{{\left( {I - {I^*}} \right)}^2} + {\mu _1}mZ{{\left( {S - {S^*}} \right)}^2}} \right) - \\ \;\;\;\;\;\frac{{2\mu }}{\gamma }\left( {{\mu _1}m{S^*}\left( {S - {S^*}} \right)\left( {Z - {Z^*}} \right) + {\mu _1}mZ\left( {I - {I^*}} \right)\left( {S - {S^*}} \right) + } \right.\\ \;\;\;\;\;\left. {{\mu _1}m{S^*}\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right)} \right) - \frac{c}{\alpha }\left( {\mu + \gamma } \right){\left( {R - {R^*}} \right)^2} + \\ \;\;\;\;\;\frac{c}{\alpha }\left( {{\mu _1}m{Z^*}\left( {R - {R^*}} \right)\left( {S - {S^*}} \right) + {\mu _1}mS\left( {R - {R^*}} \right)\left( {Z - {Z^*}} \right)} \right) + \\ \;\;\;\;\;\frac{{a\left( {I - {I^*}} \right)\left( {Z - {Z^*}} \right)}}{{\left( {1 + {a_1}I} \right)\left( {1 + {a_1}{I^*}} \right)}} - {a_0}{\left( {Z - {Z^*}} \right)^2}. \end{array}$

在区间Γ上, 取S和Z的上界, 若满足

${a^2} < \frac{{{a_0}}}{3}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),$

$\begin{array}{l} \mu _1^2{m^2} < \min \left\{ {\frac{{{a_0}{\gamma ^2}}}{{12{\mu ^2}{S^{*2}}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right),\frac{{{a_0}\alpha {\mu ^2}(\mu + \gamma )}}{{c{b^2}}},} \right.\\ \;\;\;\left. {\frac{{8{\mu ^2}(\mu + \gamma )\alpha }}{{3c\gamma {Z^{*2}}}},\frac{{{a_0}\gamma }}{{6{S^{*2}}}},\frac{{2\gamma a_0^2{{\left( {\mu + {a_1}b} \right)}^2}}}{{9{a^2}{b^2}}}\left( {c + \frac{{2\mu \beta {S^*}}}{{\gamma \varphi \left( {{I^*}} \right)}}} \right)} \right\}, \end{array}$

易得LV≤0.因此, 可知在Γ区间, 如果S=S^*, I=I^*, R=R^*, Z=Z^*, 则有LV < 0或LV=0.最后, 由LaSalle不变原理可知E^*在Γ区间是全局渐近稳定的.

2. 数值模拟

本节将通过一系列的数值例子来验证模型(2)的无病平衡点和地方病平衡点的稳定性.

例1 模型(2)的参数取值如下:

p=0.5, b=1, β=0.01, μ=0.01, μ₁=0.01, m=0.017, c=0.005, γ=0.001, α=0.8, a=0.01, a₀=0.045, a₁=1.令φ(I)=1+I², 可得ℜ₀=0.669 3 < 1, E₀=(54.545 5, 0, 45.454 5, 0)且E^*不存在.显然, 图 1A验证了定理3.

图 1 模型(2)中疾病的灭绝性分析

Figure 1. The extinction analysis of disease for model (2)

下载: 全尺寸图片幻灯片

此外, 给出了信息强度μ₁对传染病的影响.对于模型(2), 给出了关于I(t)的不同参数μ₁(μ₁=0, μ₁=0.10, μ₁=0.20)的时间序列(图 1B).可知在疾病爆发期间, 若增加信息强度μ₁, 则可以减少感染者的数量并加速疾病的灭绝.

例2 模型(2)的参数取值如下:

p=0.5, b=4, β=0.04, μ=0.02, μ₁=0.009, m=0.01, c=0.005, γ=0.01, α=0.7, a=0.02, a₀=0.045, a₁=1.令φ(I)=1+I², 可知ℜ₀=3.174 6>1, a²=4.000 0×10^-4 < 0.033 1, μ₁²m²=8.100 00×10^-9 < min{6.634 0×10^-7, 1.771 9×10^-5, 13.889 7, 2.701 6×10^-8, 0.050 5}且E₀=(93.333 3, 0, 40.000 0, 0), E^*=(74.512 9, 1.238 7, 57.371 4, 0.245 9).显然, 图 2A验证了定理4.

图 2 模型(2)中疾病的持久性分析

Figure 2. The persistence analysis of disease for model (2)

下载: 全尺寸图片幻灯片

类似地, 给出了在不同的信息强度μ₁(μ₁=0, μ₁=0.10, μ₁=0.20)的时间序列I(t)(图 2B).可知若增加信息强度μ₁, 则可降低感染者数量.

3. 结论

本文研究了基于信息干预和疫苗接种的SIRS传染病模型(2), 分析了该模型中无病平衡点和地方病平衡点的局部、全局稳定性.研究结果表明:

(1) 通过疫苗接种率ℜ₀, 可以进一步知道疫苗接种如何影响疾病传播.事实上, ℜ₀可以被写成如下形式:

${\Re _0} = - p\frac{{\beta \mu }}{{(\mu + \gamma )(\mu + c + \alpha )}} + \frac{\beta }{{\mu + c + \alpha }}.$

如果增加疫苗接种率p, 则基本再生数ℜ₀将会减少.换言之, 疫苗接种率的增加可以抑制疾病的传播.

(2) 疾病爆发时, 信息强度的增加可以降低感染者的数量并且加速疾病的灭亡.

图 1 气溶胶分类流程图

Figure 1. The flow chart of aerosol type classification

下载: 全尺寸图片幻灯片

图 2 气溶胶分类精度与训练样本数量关系

Figure 2. The relationship between the training sample size and the classification precision

下载: 全尺寸图片幻灯片

图 3 广东省的气溶胶类型占比的空间分布(随机森林分类结果)

Figure 3. The spatial distribution of aerosol types proportion in Guangdong (results of RF classification)

下载: 全尺寸图片幻灯片

图 4 广东省的气溶胶类型占比的空间分布(OMI产品标签数据)

Figure 4. The spatial distribution of aerosol type proportion in Guangdong (label data from OMI product)

下载: 全尺寸图片幻灯片

表 1 各项气溶胶特征参数的重要性

Table 1 The importance of each aerosol feature parameter

特征参数	重要性
AAOD λ=354 nm	0.034
λ=388 nm	0.019
λ=500 nm	0.045
AOD λ=354 nm	0.008
λ=388 nm	0.009
λ=500 nm	0.008
SSA λ=354 nm	0.038
λ=388 nm	0.020
λ=500 nm	0.079
RI λ=354 nm	0.141
λ=388 nm	0.128
经度	0.004
纬度	0.004
UVAI	0.218
α指数	0.245

下载: 导出CSV

表 2 3类气溶胶特征参数的均值、标准差、最大值及最小值

Table 2 The mean value, standard deviation, and maximum and minimum values of three aerosol feature parameter

气溶胶类型	特征参数	均值	标准差	最大值	最小值
DST	α指数	0.605	0.000	0.604	0.602
	UVAI	1.233	0.416	3.862	0.800
	RI₃₈₈	0.007	0.004	0.028	0.003
	RI₃₅₄	0.019	0.010	0.071	0.007
	SSA₅₀₀	0.950	0.026	0.982	0.829
	AAOD₅₀₀	0.039	0.018	0.155	0.011
CRB	α指数	1.614	0.059	1.740	1.544
	UVAI	1.168	0.460	5.604	0.801
	RI₃₈₈	0.001	0.001	0.016	0.000
	RI₃₅₄	0.002	0.002	0.022	0.000
	SSA₅₀₀	0.985	0.010	0.997	0.869
	AAOD₅₀₀	0.016	0.007	0.082	0.005
SLF	α指数	1.853	0.013	1.872	1.779
	UVAI	0.106	0.385	0.800	-2.186
	RI₃₈₈	0.004	0.003	0.019	0.000
	RI₃₅₄	0.004	0.003	0.019	0.000
	SSA₅₀₀	0.964	0.021	1.000	0.853
	AAOD₅₀₀	0.026	0.014	0.082	0.000

下载: 导出CSV

表 3 广东省各城市3种气溶胶类型占比

Table 3 The proportion of three aerosol types of each city in Guangdong %

城市	OMI产品标签数据			随机森林分类结果
城市	DST	CRB	SLF	DST	CRB	SLF
潮州市	6.6	2.5	90.9	6.6	4.1	89.3
东莞市	23.9	1.2	74.9	19.6	4.3	76.1
佛山市	16.4	1.3	82.3	19.6	2.2	78.2
广州市	17.1	1.8	81.1	17.1	2.1	80.8
河源市	7.8	1.2	9.1	7.8	1.9	90.3
惠州市	9.6	1.0	89.4	9.6	2.5	87.9
江门市	13.8	1.6	84.6	13.7	1.9	84.4
揭阳市	7.3	1.7	91.0	7.3	1.9	90.8
茂名市	12.8	0.7	86.5	12.8	0.7	86.5
梅州市	5.9	1.7	92.4	5.9	3.0	91.1
清远市	10.8	0.5	88.7	10.6	0.5	88.9
汕头市	8.6	8.6	82.8	10.6	11.6	77.8
汕尾市	6.9	4.0	89.1	6.9	3.9	89.2
韶关市	9.6	1.8	88.6	8.1	1.9	90.0
深圳市	10.1	2.9	87.0	10.1	5.0	84.9
阳江市	7.7	2.1	90.2	8.2	2.1	89.7
云浮市	5.3	1.1	93.6	8.2	1.9	89.9
湛江市	12.7	3.2	84.1	16.7	6.9	76.4
肇庆市	10.5	0.9	88.6	10.5	0.9	88.6
中山市	8.4	3.1	88.5	10.8	4.1	85.1
珠海市	8.7	3.3	88.0	10.8	4.1	85.1

下载: 导出CSV

参考文献(24)

[1]	EALO M, ALASTUEY A, PEREZ N, et al. Impact of aerosol particle sources on optical properties in urban, regional and remote areas in the north-western Mediterranean[J]. Atmospheric Chemistry and Physics, 2018, 18: 1149-1169. doi: 10.5194/acp-18-1149-2018
[2]	CHEN Q X, SHEN W X, YUAN Y, et al. Verification of aerosol classification methods through satellite and ground-based mea-surements over Harbin, Northeast China[J]. Atmospheric Research, 2019, 216: 167-175. doi: 10.1016/j.atmosres.2018.09.022
[3]	RENARD J B, DULAC F, DURAND P, et al. In situ mea-surements of desert dust particles above the western Mediterranean Sea with the balloon-borne Light Optical Aerosol Counter/sizer (LOAC) during the ChArMEx campaign of summer 2013[J]. Atmospheric Chemistry and Physics, 2018, 18: 3677-3699. doi: 10.5194/acp-18-3677-2018
[4]	CRUTZEN P J, ANDREAE M O. Biomass burning in the tropics: impact on atmospheric chemistry and biogeochemi-cal cycles[J]. Science, 1990, 250: 1669-1678. doi: 10.1126/science.250.4988.1669
[5]	DUBOVIK O, HOLBEN B, ECK T F, et al. Variability of absorption and optical properties of key aerosol types observed in worldwide locations[J]. Journal of the Atmospheric Sciences, 2001, 59: 590-608.
[6]	HAMILL P, GIORDANO M, WARD C, et al. An AERONET-based aerosol classification using the Mahalanobis distance[J]. Atmospheric Environment, 2016, 140: 213-233. doi: 10.1016/j.atmosenv.2016.06.002
[7]	KALAPUREDDY M C R, KASKAOUTIS D G, RAJ P E, et al. Identification of aerosol type over the Arabian Sea in the pre-monsoon season during the Integrated Campaign for Aerosols, Gases and Radiation Budget (ICARB)[J]. Journal of Geophysical Research, 2009, 114: D17203/1-12. http://adsabs.harvard.edu/abs/2009JGRD..11417203K
[8]	SREEKANTH V. On the classification and sub-classification of aerosol key types over south central peninsular India: MODIS-OMI algorithm[J]. Science of the Total Environment, 2014, 468/469: 1086-1092. doi: 10.1016/j.scitotenv.2013.09.038
[9]	白冰, 张强, 陈旭辉, 等. 中国西北干旱半干旱区气溶胶分类及特征[J]. 中国沙漠, 2019, 39(5): 105-110. https://www.cnki.com.cn/Article/CJFDTOTAL-ZGSS201905014.htm BAI B, ZHANG Q, CHEN X H, et al. Classification and characteristics of aerosols in arid and semi-arid areas of northwest China[J]. Journal of Desert Research, 2019, 39(5): 105-110. https://www.cnki.com.cn/Article/CJFDTOTAL-ZGSS201905014.htm
[10]	KUMAR K R, KANG N, YIN Y. Classification of key aerosol types and their frequency distributions based on satellite remote sensing data at an industrially polluted city in the Yangtze River Delta, China[J]. International Journal of Climatology, 2018, 38: 320-336. doi: 10.1002/joc.5178
[11]	CHEN Q X, YUAN Y, SHUAI Y, et al. Graphical aerosol classification method using aerosol relative optical depth[J]. Atmospheric Environment, 2016, 135: 84-91. doi: 10.1016/j.atmosenv.2016.03.061
[12]	TORRES O, TANSKANEN A, VEIHELMANN B, et al. Aerosols and surface UV products from Ozone Monitoring Instrument observations: an overview[J]. Journal of Geophysical Research-Atmospheres, 2007, 112: D24S47/1-14. doi: 10.1029/2007JD008809/full
[13]	BUCHARD V, DA SILVA A M, COLARCO P R, et al. Using the OMI aerosol index and absorption aerosol optical depth to evaluate the NASA MERRA Aerosol Reanalysis[J]. Atmospheric Chemistry and Physics, 2015, 15: 5743-5760. doi: 10.5194/acp-15-5743-2015
[14]	FIELD R D, VAN DER WERF G R, FANIN T, et al. Indonesian fire activity and smoke pollution in 2015 show persistent nonlinear sensitivity to El Nino-induced drought[J]. Proceedings of the National Academy of Sciences of the United States of America, 2016, 113: 9204-9209. doi: 10.1073/pnas.1524888113
[15]	CHEN J M, LI C L, RISTOVSKI Z, et al. A review of biomass burning: emissions and impacts on air quality, health and climate in China[J]. Science of the Total Environment, 2017, 579: 1000-1034. http://www.sciencedirect.com/science/article/pii/S0048969716324561
[16]	ZHENG Y C, LI L L, WANG Y P. An aerosol type classification method based on remote sensing data in Guangdong, China[C]//The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Göttingen, Germany: Copernicus Publications, 2019: 239-243.
[17]	BREIMAN L. Random forests[J]. Machine Learning, 2001, 45: 5-32. doi: 10.1023/A:1010933404324
[18]	耿仁方, 付波霖, 蔡江涛, 等. 基于无人机影像和面向对象随机森林算法的岩溶湿地植被识别方法研究[J]. 地球信息科学学报, 2019, 21(8): 1295-1306. https://www.cnki.com.cn/Article/CJFDTOTAL-DQXX201908016.htm GENG R F, FU B L, CAI J T, et al. Object-based karst wetland vegetation classification method using unmanned aerial vehicle images and random forest algorithm[J]. Journal of Geo-information Science, 2019, 21(8): 1295-1306. https://www.cnki.com.cn/Article/CJFDTOTAL-DQXX201908016.htm
[19]	曹爽, 潘锁艳, 管海燕. 机载多光谱LiDAR的随机森林地物分类[J]. 测绘通报, 2019, 11: 79-84. https://www.cnki.com.cn/Article/CJFDTOTAL-CHTB201911016.htm CAO S, PAN S Y, GUAN H Y. Random forest-based land-use classification using multispectral LiDAR data[J]. Bulletin of Surveying and Mapping, 2019, 11: 79-84. https://www.cnki.com.cn/Article/CJFDTOTAL-CHTB201911016.htm
[20]	马玥, 姜琦刚, 孟治国, 等. 基于随机森林算法的农耕区土地利用分类研究[J]. 农业机械学报, 2016, 47(1): 297-303. https://www.cnki.com.cn/Article/CJFDTOTAL-NYJX201601040.htm MA Y, JIANG Q G, MENG Z G, et al. Classification of land use in farming area based on random forest algorithm[J]. Transactions of the Chinese Society of Agriculture Machinery, 2016, 47(1): 297-303. https://www.cnki.com.cn/Article/CJFDTOTAL-NYJX201601040.htm
[21]	SU H, SHEN W, WANG J, et al. Machine learning and geostatistical approaches for estimating aboveground biomass in Chinese subtropical forests[J]. Forest Ecosystems, 2020, 7: 64/1-20. http://qikan.cqvip.com/Qikan/Article/Detail?id=7103641717
[22]	HUNT D A, TABOR K, HEWSON J H, et al. Review of remote sensing methods to map coffee production systems[J]. Remote Sensing, 2020, 12: 2041/1-23. http://www.researchgate.net/publication/342452963_Review_of_Remote_Sensing_Methods_to_Map_Coffee_Production_Systems
[23]	李宁, 汪丽娜. 基于随机森林回归算法的用水总量影响因素解析——以广东省为例[J]. 华南师范大学学报(自然科学版), 2021, 53(1): 78-84. doi: 10.6054/j.jscnun.2021012 LI N, WANG L N. An analysis of the factors in total water consumption based on random forest regression algorithm: a case study of Guangdong Province[J]. Journal of South China Normal University(Natural Science Edition), 2021, 53(1): 78-84. doi: 10.6054/j.jscnun.2021012
[24]	刘望保, 谢智豪. 位置服务大数据下广州市土地利用类型模拟探讨[J]. 华南师范大学学报(自然科学版), 2019, 51(1): 75-83. doi: 10.6054/j.jscnun.2019013 LIU W B, XIE Z H. Inferring land use of Guangzhou from big data of location service[J]. Journal of South China Normal University(Natural Science Edition), 2019, 51(1): 75-83. doi: 10.6054/j.jscnun.2019013

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1. 主要结果及证明
1.1 平衡点和正解的存在性
1.2 E0和E*的局部稳定性
1.3 E0和E*的全局稳定性
2. 数值模拟
3. 结论

基于多特征参数的OMI遥感产品气溶胶分类研究——以广东省为例

通讯作者: 黎丽莉，Email: lilili@gig.ac.cn

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