How to simulate an ODE when one of the parameters is a function of time

I am simulation a system of ODEs in which one parameter is a function of time (similar to a switch function) and others are constants. Below is the MWE and it gives me various errors.

function dxdt = Eq(~,x,par,h)
     
    t2 = 10;
    counter =1;
    for t=0:1:20
        if  t>0 && t<t2
            h(:,counter) = 1;
        else
            h(:,counter) = 0;
        end
        counter = counter+1;
    end
    disp(size(h))
        dxdt = NaN(3,1);
        
        S = x(1);
        I = x(2);
        V = x(3);
    
        % creating a vector of parameters
        K1 = par(1);
        k2 = par(2);
        k3 = par(3);
        k4 = par(4);
        U = par(5);
        
        dxdt(1) = -(h*U*K1+k2)*S; %S
        dxdt(2) = h*U*K1*S - k3*I; %I
        dxdt(3) = k3*I - k4*V;% V
           
    end

Here is the solver file:

clear all
close all
 
Tfinal = 20;
dt = 1;
t = 0:dt:Tfinal;
disp(size(t))
 
K1 = 0.5;
k2 = 0.1;
k3 = 0.1;
k4 = 0.05;
U = 1;
t1 = 1;
t2 = 10;
 
counter =1;
    for t=0:1:20
        if t>0 && t<t2
            h(:,counter) = 1;
        else
            h(:,counter) = 0;
        end
        counter = counter+1;
    end
    disp(size(h))
    
par = [k1 k2 k3 k4 U];
 
S0 = 1;
I0 = 0;
V0 = 0;
 
x0 = [S0 I0 V0]; % you also have to add it to the IC vector
sol = ode23(@Eq,[0 Tfinal],x0,par,h); % solving the model
% size(sol)
x1 = deval(sol,t);
x = x1';
% size(x)
S = x(:,1);
I = x(:,2);
V = x(:,3);
 

Answer 1

[darova]

Read how to Pass Extra Parameters in help ode23:

sol = ode23(@(t,x)Eq(t,x,par),[0 Tfinal],x0); % solving the model

Passing trash parameter?

function dxdt = Eq(~,x,par,h)           % why do you pass 'h' parameter if you declare it in the function

Why this part of code is repeated in main code? I suggest you to remove it from main code and change it in the function

% for t=0:1:20
%     if  t>0 && t<t2
%         h(:,counter) = 1;
%     else
%         h(:,counter) = 0;
%     end
%     counter = counter+1;
% end

        % replace with this
h = t>0 && t<t2;

Also read about function handle. If you have simple function you can write it in main code

Eq = @(t,x) [-((0<t&&t<t2)*U*K1+k2)*x(1); ...%S
               (0<t&&t<t2)*U*K1*x(1)-k3*x(2); ...%I
               k3*x(2) - k4*x(3)];% V
sol = ode23(@(t,x)Eq(t,x,par),[0 Tfinal],x0); % solving the model

[Rose]

I have a similar model and same situation. I almost understoof all your points but how does

h = t>0 && t<t2;

give a step function h(t) which is 1 when 0<t<10 and otherwise 0?

[darova]

't' variable is a scalar in a function. Imagine that Eq function repeats and 't' variable changes every time so 'h' variable would be different