The mathematics behind modelling

This is my first time using MATLAB and despite reading up on tutorials I am still confused in regards to how to utilise MATLAB. I am trying to simulate a SEIR model, which consists of a system of differential equations, for the spread of dengue fever in MATLAB with the following equations and parameters:

Thank you!!!

[John D'Errico]

Please stop posting the identical question every hour just because you are in a hurry. I've now deleted most of your replicate questions.

[Walter Roberson]

There was a Mathworks server problem this morning that prevented people from telling that their question had been posted.

Answer 1

[Josh Meyer]

You have a system of 7 coupled ODEs. You will need to code the equations into a function, define the initial conditions and interval for integration, then use an ODE solver such as ODE45 to solve the equations numerically. I got you started on your function, but you'll need to fill in the gaps and double check it:

function dSdt = denguefeverODE(t,S)
% Define parameters
Nh = 
Nm = 
uh = 
um = 
Pmh = 
Phm = 
beta = 
nu_h = 
epsilon_m = 
tau_h = 
f = 

% Define the equations. Each element in the output contains the answer for
% one equation, so there are 7 components. For ex. S(1) is Sh while dSdt(1)
% is dSh/dt, and S(7) is Im while dSdt(7) is dIm/dt.
dSdt = zeros(7,1);
dSdt(1) = uh*Nh - (beta*Pmh*(S(7)/Nh)+uh)*S(1);
dSdt(2) = beta*Pmh*(S(7)/Nh)*S(1) - (tau_h+uh)*S(2);
.
.
.
dSdt(7) = epsilon_m*S(6) - um*S(7);

Once you are ready to solve, the solver syntax is

tspan = [t0 tf];        % Change to initial and final times
y0 = [a b c d e f g];   % Need 7 initial conditions, 1 for each variable
[t,y] = ode45(@denguefeverODE, tspan, y0)

Then you can see all of the solution components with

plot(t,y)

More information here: http://www.mathworks.com/help/matlab/ordinary-differential-equations.html

[Muse Riveria]

I am tremendously thankful for your insightful answer, I hadn't approached the question in this manner before. I followed your steps and filled in the gaps, concluding the function with:

tspan = [to tf];        % Change to initial and final times
to = 0;
tf =100;
y0 = [5535002 50 50 0 0 0 0 ];   % Need 7 initial conditions, 1 for each variable
[t,S] = ode45(@denguefeverODE, tspan, y0)
plot(t,S)

Unfortunately, an error came up saying "undefined function or variable 't'". I thought I had defined 't' when I inputted to=0 and tf=100?

Additionally, is there a method to separate the 2 systems dSdt(1) to (4) (human population) and dSdt(5) to (7) (mosquito population) on two separate graphs?

Thank you so much for your help.

[Walter Roberson]

We need to see your denguefeverODE code.

[Torsten]

You will have to define t0 and tf before you define tspan:

to = 0;
tf =100;
tspan = [to tf];        % Change to initial and final times
y0 = [5535002 50 50 0 0 0 0 ];   % Need 7 initial conditions, 1 for each variable
[t,S] = ode45(@denguefeverODE, tspan, y0)
plot(t,S)

Best wishes

Torsten.

[Muse Riveria]

Hi! Thank you for your reply, but even after I defined t0 and tf before defining tspan the error stating undefined function or variable t still appears.

[Walter Roberson]

We need your updated code including the code for denguefeverODE, and you should also post the complete error message including the traceback showing where the error is occurring.

[Muse Riveria]

    >> denguefeverODE(t, S)
    Undefined function or variable 't'.

Answer 2

[Torsten]

function main

to = 0;
tf = 100;
tspan = [to tf];        
y0 = [5535002 50 50 0 0 0 0 ];  
[t,S] = ode45(@denguefeverODE, tspan, y0);
plot(t,S)
title('Human Population Without Control')
xlabel('Time')
ylabel('Susceptible, Exposed, Infected, Recovered')
legend('Susceptible', 'Exposed', 'Infected', 'Recovered')

function dSdt = denguefeverODE(t,S)
Nh = 5535002;
Nm = 33210012;
uh = 0.0045;
um = 0.02941;
Pmh = 0.375;
Phm = 0.750;
beta = 1;
nu_h = 0.1666;
epsilon_m = 0.1;
tau_h = 0.1176;
f = 6;
dSdt = zeros(7,1);
dSdt(1) = uh*Nh - (beta*Pmh*(S(7)/Nh)+uh)*S(1);
dSdt(2) = beta*Pmh*(S(7)/Nh)*S(1) - (tau_h+uh)*S(2);
dSdt(3) = tau_h*S(2)-(nu_h+uh)*S(3);
dSdt(4) = nu_h*S(3)-uh*S(4);
dSdt(5) = um*Nm - (beta*Phm*(S(3)/Nh)+um)*S(5);
dSdt(6) = beta*Phm*(S(3)/Nh)*S(5);
dSdt(7) = epsilon_m*S(6) - um*S(7);

Best wishes

Torsten.

[Muse Riveria]

Thank you so much!!! In order to have two separate graphs, is it possible to compute the human population and vector population separately? I know I'm unable to change y0 = [5535002 50 50 0 0 0 0 ]; to y0 = [5535002 50 50 0 ];to just include the first 4 equations. But beyond this I'm unsure how to approach it.

[Torsten]

You mean two plot-commands :

figure(1);
plot(t,S(:,1),t,S(:,2),t,S(:,3),t,S(:,4));
figure(2);
plot(t,S(:,5),t,S(:,6),t,S(:,7));

?

Best wishes

Torsten.

[Muse Riveria]

Thank you so much!!!!

[Muse Riveria]

Would it be possible to incorporate a constant control strategy (c = 0.084) to solve for the equations mentioned above? Whereby the control strategy (c = 0.084) is constant, with application of 8.4% capacity of insecticide 24 hours per day all the time. Following this administration, the number of infected mosquitos and humans should reduce.

Please refer here on the last few pages if need be: http://repositorium.sdum.uminho.pt/bitstream/1822/15161/3/199Dengue_disease_basic_reproduction_number_and_control.pdf

[Torsten]

So you ask how to implement the equations of the article which already contain a control strategy ? Or do you ask how to modify your equations to include a control strategy ?

I ask this because the equations for the mosquitos in the article seem to differ from the equations you use.

Best wishes

Torsten.

[Muse Riveria]

How to modify the equation, the article was used for reference as the data isn't completely applicable to the geographical location that used for this differential system. Would it be a simple task, or would it require the formation of more complicated subsequent equations?

[Star Strider]

I doubt the DEs would change, since the epidemiology would be essentially the same, but the parameters likely would. (Islands in the Caribbean might be similar enough to not require any significant changes.) If you’re using them for a more northerly latitude in response to global warming, there are several changes you would have to consider. The human epidemiology would be the same, but you might have to consult with an entomologist with a particular interest in Aedes aegypti to determine what would have to change about the vectors.

Answer 3

[MOUSSA DOUMBIA]

Can anybody provide me a sample of an optimal control problem with 3 different control functions?