Euler's Method

2020 6 7
1 个回答
更新时间：2020 6 12
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Hi!
Model of an epidemic: We will now describe a mathematical model for the spread of an epidemic.
Suppose we have a community of L people that initially contains P infected people and Q uninfected people. Let y (t) be the number of people infected at one time t. If the disease is not very serious, like the common cold, everyone continues to be active and the epidemic spreads.
Since there are P Q possible contacts between people from one group to another, the rate of change of y (t) is proportional to
P Q, so the problem can be modeled using the initial value problem
(a) Taking L = 25000, k = 0.00003 and h = 0.2, with the initial condition y0 = 250, use
Euler's method to approximate the solution of the P.V.I. in the interval [0, 60].
(b) Draw the graph of the approximate solution in part (a).
(c) Estimate the average number of people infected by calculating the arithmetic mean
of section (a).

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Edinson Manga
Edinson Manga 2020-6-7
H = [0.2];
Y_euler = cell(1,numel(H));
for k=1:numel(H)
h = H(k);
x = 0:h:60;
y = zeros(1, numel(x));
y(1) = 250;
n = numel(y);
for i=1:n-1
f = ((0.00003*y(i))*(25000-y(i)));
y(i+1) = y(i) + h * f;
end
Y_euler{k} = [x; y].';
end
syms Y(X)
eq = diff(Y) == (0.00003*Y)*(25000-Y);
ic = Y(0) == 250;
sol = dsolve(eq, ic);
y_sol = double(subs(sol, X, x));
hold on
plot(x, y_sol, 'r', 'DisplayName', 'true solution');
for k=1:numel(H)
plot(Y_euler{k}(:,1), Y_euler{k}(:,2), 'b--', 'DisplayName', ['Euler method h=' num2str(H(k), '%.3f')]);
end
legend
This is my attempt, but I still can't find a way to find the arithmetic mean.

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回答（1 个）

Alan Stevens
Alan Stevens 2020-6-7
编辑：Alan Stevens 2020-6-7

0 个投票

Here's a rather simpler way to do what you want:
% EulerEpidemic.m
% Data
L = 25000; k = 0.00003; h = 0.2;
y0 = 250; tend = 60;
n = tend/h;
% Initial conditions
y(1) = y0;
t(1) = 0;
% Euler
for i = 2:n+1
f = k*y(i-1)*(L - y(i-1));
y(i) = y(i-1) + h*f;
t(i) = (i-1)*h;
end
Average = round(mean(y),0);
% True
i = 1:n+1;
y_true(i) = L./( 1 - (1-L/y0)*exp(-k*L.*t(i)) );
% Graph
plot(t,y,'--',t,y_true,'r'),grid
xlabel('time'),ylabel('infected')
legend('Euler','True')
text(20,L/2,['Average = ', num2str(Average)])

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Alan Stevens
Alan Stevens 2020-6-7
编辑：Alan Stevens 2020-6-7
Edited code to correct small error.
Edinson Manga
Edinson Manga 2020-6-7
编辑：Edinson Manga 2020-6-7
When I execute the code, I get an error
Alan Stevens
Alan Stevens 2020-6-8
编辑：Alan Stevens 2020-6-8
What error? It works perfectly for me! Describe the error message that you get.
darova
darova 2020-6-12
i changed this line
Average = round(mean(y));
and it works ok for me too

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