How to find the endemic equilibrium of a mathematical model for Glioma

Can MATLAB do this?

function hdot = glioma0(t,x,Pg, Ps, beta1, beta2, i1, i2, i3, a1, a2, a3, beta3, alpha, zeta, Pr);
    % The time-dependent term is A * sin(w0 * t - theta)
    % H = 1*sin(-1*t);
    % G = x(1)
    hdot = zeros(5,1);
    % H = heaviside(x(5));
    
    u   = 0.0;
    Phi = 200.0;
    
    H   = 0;
    if (x(5)>0)
        H = 1;
        % else if (x(5)<=0);
        % H = 0;
    end
    
    hdot(1) = Pg*x(1)*(1 - x(1))        - beta1*x(1)*(x(2) + x(3))   - (i1*x(1)*x(5))/(a1 + x(1));
    hdot(2) = Ps*x(2)*(1 - x(2) - x(3)) - beta2*x(1)*x(2) - u*x(2)*H - (i2*x(2)*x(5))/(a2 + x(2));
    hdot(3) = Pr*x(3)*(1 - x(1) - x(3)) - beta3*x(1)*x(3) + u*x(2)*H;
     
    if     hdot(1) < 0
        hdot(4) = alpha*hdot(1)*x(4) - (i3*x(4)*x(5))/(a3 + x(4));    
    elseif hdot(1) > 0
        hdot(4) = -(i3*x(4)*x(5))/(a3 + x(4));
    end
    
    hdot(5) = Phi - zeta*x(5);
    
    % hdot = hdot';
    % hdot = [hdot(:);H,h1]; 
    % To make xdot a column
    % End of FUN1.M
end

This is the differential equation.

Answer 1

[Divyam]

Hi @Latifah Hanum,

MATLAB can numerically solve the equilibrium of your model using the 'fsolve' function. You need to write a function that returns the steady state equations for your model and then use 'fsolve' to find the roots. Here are the steps to solve for equilibrium points for your model:

• Set up your system and parameters
• Create a function that returns the vector of equations at steady state
• Choose a reasonable initial guess and supply the function handle and initial guesses to the 'fsolve' function

x0 = [1, 1, 1, 1, 1]; % Initial guess
F = @(x) glioma_equilibrium(<Add parameter values>);
[x_equil, fval, exitflag] = fsolve(F, x0);

• Interpret the equilibria stored in the x_equil variable
• Repeat this process using different initial guesses to arrive at a meaningful solution

Since your model is non-linear, it may have multiple equilibria and hence the initial guess you provide to the 'fsolve' function will determine the chances of convergence. To ensure that you receive meaningful equilibria, make your parameters and initial guesses physically meaningful.

For more information regarding the 'fsolve' function, refer to the following documentation: https://www.mathworks.com/help/optim/ug/fsolve.html