%% Section 8.1 SIR model
% The model is 
% 
% $$S'=-aSI,\quad I'=aSI-bI, \quad R'=bI.$$
% 
% If we take $a=b=1$, we can use *ode45* to solve and graph the components.
% Let $x(1)=S,\ x(2)=I,\ x(3)=R$ so the system becomes $x'=f(t,x)$ where
% $x=(x(1),x(2),x(3))^T$ and 
%%
 f = @(t,x) [-x(1)*x(2);x(1)*x(2)-x(2);x(2)]
%%
% We solve with $S(0)=4,\ I(0)=0.1,\ R(0)=0.$ and then plot the components
% of the solution.

[t,xa]=ode45(f,[0 6], [4 0.1 0]);
 plot(t,xa(:,1))
 hold on
 plot(t,xa(:,2),'k')
 plot(t,xa(:,3),'r')
 hold off
 
 %% 
 % The blue curve is the population which has not yet had the disease, the
 % black curve is the infected population and the red curve is the
 % population which has recovered. We've now reproduced Figure 1 in
 % Polking, Boggess and Arnold.