# How to verify if the stationary states of a biological model are correct?

I'm mathematically modeling yellow fever, and yet, after assembling the model, I need to find the stationary states of the model. I had to do it by hand because wolfram software could not get the necessary values.

My question is, how do I check if my stationary states are in line with my biological model?

I was told that there is a way to verify this by performing a substitution of the s values of the points within my own template code. But I have no idea how to do it.

The code used is taking forever to get back to me as this data behaves in my model

function ypsir = ypsirtry(t,y)
betah = .01;
muh = 0.01;
sigmah = 0.05;
alphah = 0.06;
deltah = 0.05;
mui = 0.07;
mum = 0.07;
betai = 0.03;
lambdai = 0.06;
lambdah = 0.5;
k = 500;

ypsir = zeros(5,1);
ypsir(1) = sigmah*y(3) - muh*y(1) - betah*y(1)*y(5) + lambdah*(y(2)+ y(3) +
y(1));
ypsir(2) = -deltah*y(2) - y(2)*(alphah + muh) + betah*y(5)*y(1);
ypsir(3) = deltah*y(2) - sigmah*y(3);
ypsir(4) = -mui*y(4) - mum*y(4) - betai*y(2)*y(4) + lambdai*(y(4) + y(5))*(1
- (y(4) + y(5))/k);
ypsir(5) = betai*y(2)*y(4) - mui*y(5) - mum*y(5);
end


Along with this calling program

clear;
to = 0;
tf =50;
yo = [99 1 0 1 3];
[t y] = ode45(@(t,y)ypsirtry(t,y),[to tf],yo);
plot(t,y(:,1),t,y(:,2),t,y(:,3),t,y(:,4),t,y(:,5))
title('First Model of Yellow Fever')
xlabel('time')
ylabel('susceptible, infected, recovered,Mosquito Non-carrier, Carrier
mosquito')


My stationary states are:

S = (1/betai)*[(muh+alphah)+(sigmah/deltah)*deltah]*(mui+mum)/betai*N*I);

I = sigmah*R/deltah;

R = ((muh/betah + lambdah)*(muh+alphah))/(sigmah+lamdah)-(muh+alphah)*
(deltah/sigmah*deltah)-(betah/muh - lambdah)*(betai*N*I)*
(delta/sigmah*deltah));

N = -k/lambdai((-lamdai+(lambdai/k))*P/-betai*I + lambdai - (2k/lambdai*P)-
mui-mum);

P = betai*N*I/(mui+mum);


At where S = susceptible I = infected R = retweeted N = non-carrier mosquito P = carrier mosquito

• I think your question is a good question, but we would need more details in order to answer them. How did you code your model? Which stationary states? (Why do you use the biomart tag?) Which data do you have to verify your models (if it is not purely theoretical)? What do you mean by "performing a schedule" ? You can edit it to include this or other information.
– llrs
Mar 16 '18 at 15:46
• I'm already going to make the issues, sorry for the delay! I collared biomat because I thought it would be mathematical modeling in biology. I'm new to the site and I was a little lost on which site to put. I will post the code with the respective data. But when putting it to compile it is dying a lot of time. But I will add other information. Do you know how I can put on latex around here? to make it easier to see formulas in my edits. Sorry English, it is not my mother tongue how can you perceive rs ...
– wgrt
Mar 16 '18 at 21:09
• Wouldn't it be sufficient to determine this by running it until the system elicits stationary behavior and then comparing the values? If you did that for a variety of values for the constants (betah and so on) then that would also be a good test. Mar 16 '18 at 22:31
• @wgrt See the tour page to learn about the site and the help center page to know how it works. How much is a lot of time: 3 days? Do you know if it can be optimized? (I don't know enough matlab to propose it myself). Also you left some questions from my previous comment unanswered
– llrs
Mar 17 '18 at 10:25
• @Llopis You're not going to optimize an RK45 method that's implemented in C much. If this really is slow to compute then presumably there are some very drastic changes in values (RK45 shrinks its steps in such cases). Mar 17 '18 at 18:25