Calculating sensitivity indices of R0

Questions
1

I am currently working on a class project where I am supposed to analyze some results in few select papers. I want to calculate the sensitivity indices of the parameters with respect to the basic reproduction number (R0) using the formula S(p)=(p/R0)*(∂R0/∂p) where p is a parameter and S(p) is the sensitivity index of p. I have already calculated the indices manually, but I am interested in learning how to automate the process using Wolfram. The formula for R0 and the parameter values are given below.

betas = 0.274,
alphaa = 0.4775,
alphau = 0.695,
mu = 0.062,
qi = 0.078,
gammaa = 0.29,
1/etai = 0.009,
1/etau = 0.05

R0 = (betas*alphaa)/(gammaa+mu) + (betas*alphau*gammaa*(1-qi))/((gammaa+mu)*(etau+mu))

The answers given in the paper are (βs, μ, γa, αa, ηu, αu, qi) = (1.000, − 0.6060, −0.18464, 0.22348, −0.34666, 0.77652, −0.06474). In fact, I am beginner to Wolfram. So, a comprehensive answer would be highly appreciated. Thanks a lot in advance!

Source: pubmed.ncbi.nlm.nih.gov

2

Hi Mad,

The underscore character _ represents Blank, so it cannot be used in a symbol name. Try using valid symbol names.

3

Hi Rohit, I just removed the underscores. Thanks for that!

4

Quite a few more syntax errors in your code: separate operations are separated by semi-colons, not commas. And you can’t do something like 1/etau = 0.05: you need etau = 1/0.05.

Here is a better way to do what you posted:

Clear["Global`*"]

parameters = {betas -> 0.274, mu -> 0.062, gammaa -> 0.29, 
   alphaa -> 0.4775, etau -> 1/0.05, alphau -> 0.695, qi -> 0.078, 
   etai -> 1/0.009};

baseR0 = (betas*alphaa)/(gammaa + mu) + (betas*alphau*
      gammaa*(1 - qi))/((gammaa + mu)*(etau + mu)) /. parameters

Now, assuming everything is correct, this does what you describe:

In[119]:= Table[
 oldParameter = parameters[[parameter, 2]];
 newParameter = oldParameter*1.001;
 deltaParameter = newParameter - oldParameter;
 newR0 = (betas*alphaa)/(gammaa + mu) + (betas*alphau*
       gammaa*(1 - qi))/((gammaa + mu)*(etau + mu)) /. 
   ReplacePart[parameters, {parameter, 2} -> newParameter];
 deltaR0 = newR0 - baseR0;
 {parameters[[parameter, 
    1]], (oldParameter/baseR0)*(deltaR0/deltaParameter)}, {parameter, 
  1, Length[parameters] - 1}]

Out[119]= {{betas, 1.}, {mu, -0.176164}, {gammaa, -0.804172}, {alphaa,
   0.980971}, {etau, -0.0189516}, {alphau, 
  0.0190293}, {qi, -0.00160985}}

As you see though, it doesn’t give the same values you posted. So it’s possible there is something wrong with either your formula for R0 value, or for the sensitivity.

5

Hi Gareth, thanks a lot for taking the time to answer my question. It was so helpful and I really appreciate your support. Could you please help me understand the line “newParameter = oldParameter*1.001” in your code?

6

It’s just a way to create a small (0.001) change in each parameter value.

7

Hey, on a hunch I changed the parameters to etai = 0.009 and etau = 0.05 (instead of 1/those values), and then I get the same sensitivity values that you posted. So if that is right, the definitions of the parameters in the original post are incorrect. (Because if 1/etau = 0.05, then etau = 1/0.05).

8

Hey, thanks a lot for the clarification. I see what you mean. I do sincerely appreciate your kind gesture.

9

I have also been struggling with replicating this graph from the same paper. Any support would be highly appreciated. The authors plot R0 against gamma_a and beta_s.

betas = 0.274,
alphaa = 0.4775,
alphau = 0.695,
mu = 0.062,
qi = 0.078,
gammaa = 0.29,
etai = 0.009,
etau = 0.05

R0 = (betas*alphaa)/(gammaa+mu) + (betas*alphau*gammaa*(1-qi))/((gammaa+mu)*(etau+mu))

1.jpg
1.jpg1159×538 93.1 KB

Source: pubmed.ncbi.nlm.nih.gov