How can I plot a system of nonlinear ODEs with an added term used to model chemotherapy?

So this is the system of ODES (originally just dx/dt and dy/dt), the dz/dt is the chemotherapy term and the added exponential function on x and y relates to this. It has a right hand solution of z(t^+) = z(t) + u2. This subsytem of dz/dt and z(t^+) gives a periodic solution. I want to show how it makes an effect on the y term (tumour cells).

Solution to subsystem:

We have that TR_{CI} is: (where 1.400mg is u2)

I want to recreate these graphs, please help.

[Rakeeza Malik]

https://www.researchgate.net/publication/295685214_Mathematical_Modelling_and_Analysis_of_the_Tumor_Treatment_Regimens_with_Pulsed_Immunotherapy_and_Chemotherapy/fulltext/56f169b908ae1cb29a3d0e26/Mathematical-Modelling-and-Analysis-of-the-Tumor-Treatment-Regimens-with-Pulsed-Immunotherapy-and-Chemotherapy.pdf - you can find the paper and information here. I'm looking to recreate the graphs to do with single chemotherapy

[darova]

What about ode45? What are those values? Do you have them?

[Rakeeza Malik]

Yes I have those values,

and these are the values for the TR

[Rakeeza Malik]

I have used ODE45 however my problem is that I don't know how to incorporate the condition of z(t^+) = z(t) + u2 to help simulate the system of ODES. Also how to incorporate it on different time intervals, as the value of z changes.

Answer 1

[darova]

Here is an idea:

function main
t1 = 0.01;                  % start period
t2 = tau-0.01;              % end period
hold on
for i = 1:numOfPeriods
    [t,u] = ode45(@f,[t1 t2],u0);
    t1 = i*tau+0.01;
    t2 = (i+1)*tau-0.01;
    u0 = u(end,:);          % initial condition for next period
    plot(t,u(:,1))          % plot x(t)
end
hold off
end

function du = f(t,u)
x = u(1); y = u(2); z = u(3);
du(1,1) = ...   % eq for dx
end

Do you have exact number of tau for your simulation?

Let me know if it's clear

[darova]

Maybe μ value should be added each cycle/period

u0 = u(end,:);          % initial condition for next period
u0(3) = u0(3) + mu;

[Rakeeza Malik]

Thanks for this suggestion! I have tried it but can't seem to make it work, u0 is coming up undefined. Also the inital value of tau for the first graph is 21, however for the other graphs tau changes, as you can see where I provided the values for u, the values of tau are next to them.

Also under du(1,1) would i then have the equations for all dx, dy, dz?

[darova]

• Also under du(1,1) would i then have the equations for all dx, dy, dz?

Yes, but you should write it

du(1,1) = ...   % eq for dx
du(2,1) = ...   % eq for dy
du(3,1) = ...   % eq for dz

• I have tried it but can't seem to make it work

Can i see it?