Problems with code / function, microscopic traffic modeling

I want to setup an IDM model. I have the following code, but it won`t work. It should handle a variable amount of cars and with different car lengths L. Has anyone a solution for this? here are the mathematical expressions from (https://www.researchgate.net/publication/323400960_Comparison_of_Vehicle_Dynamics_of_Microscopic_Car_Following_Models_Optimal_Velocity_and_Intelligent_Driver_Model):

function idm_new
% Zeit, Geschwindigkeit und Position des ersten Fahrzeugs
%zeit_Vektor = 1:1:60;
simulationtime = 60; % in Sekunden
vel_first = (35/3.6);
numberofcars = 3; 

        % X entspricht Wegstrecke
        
% Linearer Verlauf der Geschwindigkeit - Ort, gleiche Beschleunigung
X = zeros(simulationtime,1);
for i = 1:1:simulationtime+1
  
  X(i) =  vel_first * (i-1);  
    
  time_vector(i) = (i-1); 
  velocity_vector (i) = [vel_first];
  
  distance_vector (i) = X(i); 
end
time_vector = time_vector.';
velocity_vector = velocity_vector.';
distance_vector = distance_vector.';

initial_values = zeros(numberofcars,1);

for i = 1:numberofcars
    
    initial_values(i) = initial_values(i) + 2*i;
    
end

% System Loesung
Zeitspanne=0:.001:simulationtime;
[Th,Y] = ode45( @(Zeit,y)idm(Zeit,y,numberofcars,distance_vector,time_vector,velocity_vector),Zeitspanne,initial_values);


% Diagramm
figure(1);
hold on;
name_cell = {};
name_cell{1} = ['Car ' num2str(0) ];
for k = 1:numberofcars
plot(Th,Y(:,k),'Color',rand(1,3),'LineWidth',.5*k)
name_cell{k+1} = ['Car ' num2str(k) ];
end



function dy = idm(Zeit,y,numberofcars,distance_vector,time_vector,velocity_vector)
dy = zeros(numberofcars,1);
% Parameter
a =     1.0;                    
b =     1.5;                    
delta = 4;                      
T =     1.0;                    
L =     [4.5,3.0, 12.4];        
s_0 =   2.0;                    
v_0 =   15;                     

Term = 1/(2*sqrt(a*b));

% first car
%x_0 = 
distance_interp = interp1(time_vector, distance_vector, Zeit);
velocity_interp = interp1(time_vector, velocity_vector, Zeit);
 
    % calculate Delta_v for the first car
Delta_v(1) = y(numberofcars+1)-velocity_interp;

    % calculate s
s(1) = distance_interp-y(1) - L(1);
    
    % Desired distance to the first vehicle
s_stern(1) = s_0 + max(0,y(numberofcars)*T + y(numberofcars)*Delta_v(1)*Term);


% Folgefahrzeuge
for i = 2:numberofcars
    Delta_v(i)  = y(numberofcars+i)-y(numberofcars+i-1);
    s(i)        = y(i-1)-y(i) - L(i);
    s_star(i)  = s0 + max(0,y(numberofcars+i)*T+y(numberofcars+i)*Delta_v(i)*Term);
end

for i=1:numberofcars
dy(i) = y(i+numberofcars);
end
for i=1:numberofcars
dy(i+numberofcars) = a*(1 - (y(i+numberofcars)/v_0).^delta - (s_star(i)/s(i)).^2);
end

end
end