數學建模學習筆記(十七)傳染病模型(SIER)
技術標籤:數學建模學習筆記
傳染病模型講解比較清楚的是知乎這位博主,文章連結戳這在家宅著也能抵抗肺炎!玩一玩SEIR傳染病模型
本文基於這篇文章進行記錄和整理
對於一般傳染病來說,都具備潛伏者(E),因此直接記錄傳統的SIER模型:
模型公式:
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\left\{ \begin{array}{l} \frac{{dS}}{{dt}} = - \frac{{r\beta IS}}{N}\\ \\ \frac{{dE}}{{dt}} = \frac{{r\beta IS}}{N} - \sigma E\\ \\ \frac{{dI}}{{dt}} = \sigma E - \gamma I\\ \\ \frac{{dR}}{{dt}} = \gamma I \end{array} \right.
迭代公式:
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\left\{ \begin{array}{l} {S_n} = {S_{n - 1}} - \frac{{r\beta {I_{n - 1}}{S_{n - 1}}}}{N}\\ \\ {E_n} = {E_{n - 1}} + \frac{{r\beta {I_{n - 1}}{S_{n - 1}}}}{N} - \sigma {E_{n - 1}}\\ \\ {I_n} = {I_{n - 1}} + \sigma {E_{n - 1}} - \gamma {I_{n - 1}}\\ \\ {R_n} = {R_{n - 1}} + \gamma {I_{n - 1}} \end{array} \right.
引入潛伏者傳染概率,改進SEIR模型,
公式為
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\left\{ \begin{array}{l} {\frac{{dS}}{{dt}} = - \frac{{r\beta IS}}{N} - \frac{{{r_2}{\beta _2}ES}}{N}}\\ {}\\ {\frac{{dE}}{{dt}} = \frac{{r\beta IS}}{N} - \sigma E + \frac{{{r_2}{\beta _2}ES}}{N}}\\ {}\\ {\frac{{dI}}{{dt}} = \sigma E - \gamma I}\\ {}\\ {\frac{{dR}}{{dt}} = \gamma I} \end{array} \right.
迭代公式為:
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\left\{ \begin{array}{l} {S_n} = {S_{n - 1}} - \frac{{r\beta {I_{n - 1}}{S_{n - 1}}}}{N} - \frac{{{r_2}{\beta _2}{E_{n - 1}}{S_{n - 1}}}}{N}\\ \\ {E_n} = {E_{n - 1}} + \frac{{r\beta {I_{n - 1}}{S_{n - 1}}}}{N} - \sigma {E_{n - 1}} + \frac{{{r_2}{\beta _2}{E_{n - 1}}{S_{n - 1}}}}{N}\\ \\ {I_n} = {I_{n - 1}} + \sigma {E_{n - 1}} - \gamma {I_{n - 1}}\\ \\ {R_n} = {R_{n - 1}} + \gamma {I_{n - 1}} \end{array} \right.
⎩⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎧Sn=Sn−1−NrβIn−1Sn−1−Nr2β2En−1Sn−1En=En−1+NrβIn−1Sn−1−σEn−1+Nr2β2En−1Sn−1In=In−1+σEn−1−γIn−1Rn=Rn−1+γIn−1
matlab程式碼:
原始碼:
clear;clc;
%--------------------------------------------------------------------------
% 引數設定
%--------------------------------------------------------------------------
N = 12700000; %人口總數
E = 0; %潛伏者
I = 1; %傳染者
S = N - I; %易感者
R = 0; %康復者
r = 20; %感染者接觸易感者的人數
B = 0.03; %傳染概率
a = 0.1; %潛伏者轉化為感染者概率
y = 0.1; %康復概率
T = 1:140;
for idx = 1:length(T)-1
S(idx+1) = S(idx) - r*B*S(idx)*I(idx)/N;
E(idx+1) = E(idx) + r*B*S(idx)*I(idx)/N-a*E(idx);
I(idx+1) = I(idx) + a*E(idx) - y*I(idx);
R(idx+1) = R(idx) + y*I(idx);
end
plot(T,S,T,E,T,I,T,R);grid on;
xlabel('天');ylabel('人數')
legend('易感者','潛伏者','傳染者','康復者')
稍作改進,反應每日新增病例情況:
%--------------------------------------------------------------------------
% 初始化
%--------------------------------------------------------------------------
clear;clc;
%--------------------------------------------------------------------------
% 引數設定
%--------------------------------------------------------------------------
N = 29000; %人口總數
E = 0; %潛伏者
I = 1; %傳染者
S = N - I; %易感者
R = 0; %康復者
m=1;
r = 25; %感染者接觸易感者的人數
B = 0.03; %傳染概率
a = 0.1; %潛伏者轉化為感染者概率
r2 = 3; %潛伏者接觸易感者的人數
B2 = 0.03; %潛伏者傳染正常人的概率
y = 0.1; %康復概率
T = 1:182;
for idx = 1:length(T)-1
S(idx+1) = S(idx) - r*B*S(idx)*I(idx)/N(1) - r2*B2*S(idx)*E(idx)/N;
E(idx+1) = E(idx) + r*B*S(idx)*I(idx)/N(1)-a*E(idx) + r2*B2*S(idx)*E(idx)/N;
I(idx+1) = I(idx) + a*E(idx) - y*I(idx);
R(idx+1) = R(idx) + y*I(idx);
m(idx+1) = E(idx+1) + I(idx+1);
end
x=1:182;
plot(x,m);grid on;
xlabel('day');ylabel('Demand for drugs')