How am I getting solutions to the ODE when I don't have a parameter defined?

I was trying to produce a graph that plotted bacterial populations at a fixed time vs. the parameter called treatment. In an m-file, I had this so that I could use the ode45 solver :

function dX = ode(~,X,T)       
  bs = 8e-3; bl = 4e-3; bh = 6.4e-3; % per-capita growth rates of bacteria pop
  N = sum(X); K = 1e8;               % N-total population, K-carrying capacity
  m1 = 2e-5; m2 = 9e-9; p = 5e-13;   % mutation rates
  I = 1e-3;                          % immune system clearing rate of bacteria
  % T = 1e-3;                        % Treatment: I was told to comment this out
  a = 0;                             % level of resistance to antibiotics
  dX = [X(1) * (bs * (1 - N/K) - I - T - m1) - p * X(1) * (X(2) + X(3));
        X(2) * (bl * (1 - N/K) - I - a*T - m2) + m1 * X(1) + p * X(2) * (X(1) - X(3));
        X(3) * (bh * (1 - N/K) - I - a*T) + m2 * X(2) + p * X(3) * (X(1) + X(2))];
end

Then, in the command window, I type the following for-loop:

>> for treatment = 1e-4:1e-5:1e-3
     [t,X] = ode45(@ode, [0 4.5e4], [1e4 0 0], [] , treatment); 
     plot(treatment,X(end,1), treatment,X(end,2), treatment,X(end,3))
     xlabel('Treatment')
     ylabel('Bacteria Population at 1 month')
     legend('Sensitive', 'R. Low-Fitness', 'R. High-Fitness')
     hold on
   end

The point was to plot the bacterial population after one month of initial infection with different levels of treatment. The code kind of gives me a graph of what I wanted (I get dots when I wanted solid lines), but I don't see how the ode45 solver is working because I don't have T defined in my m-file. Can any of you tell me how the ode45 solver is still working even though it's solving the system that has an undefined parameter?

[Anthony]

I have spaces in between the operators of dX because it's easier to read in MATLAB.

Answer 1

[Star Strider]

Interesting. I suspect t is provided primarily to allow the ODE function to use it in ODE functions that require it in the DEs themselves, and to define time-varying parameters, events, and such. In most ODE functions I’ve seen, the ODE functions don’t actually use t. I tested omitting it as an argument on another ODE that did not use t in its calculations, and it too integrated without error.