@sheriph05 I don't think that a search for a Lyapunov function for these equations will be fruitful.  I suggest that you focus on the analysis of local stability of equilibria through linearization, which is a more attainable goal.  You may worry about global stability afterward.

As Preben has pointed out, a general method for constructing Lyapunov functions is not known, and I seriously doubt whether such a thing is possible at all.  Nevertheless, this 2014 article appears to claim that it offers a method for doing just that.  I don't understand the mathematics in it, and so I cannot pass judgment.

Why do you want underline to those objects?  You may achieve better results through font shape (italic?), weight (bold?),  size (large?), and color (red?).

Decades ago, when people used typewriters to write on paper, they used underlining to emphasize text because there were not many other options—there was only one font.  If you look around in printed media today—newspapers, magazines, books—you will not see much underlining.  One uses a variety of fonts to get one's ideas across.

@ Carlos I assume that by x-value you mean y-value since you specify the last x-value yourself as 812.

To see the solution at x=812, type dsol(812).  Or dsol(x) for any numerical value of x.

@Carlos As Preben Alsholm has correctly pointed out, the option method=rkf45 does not do what you have asked.  Change the line dsol=...  to what he has suggested.

@acer As long as you are doing this, you might as well reqest the addition of a linecolor=mycolor option in the plottools package.  Right now there is no access to the default black border color of objects created by plottools, hence the awkward use of [ two plots + display ] in the examples given by me and kitonum.

@alfarunner I have shown my worksheet in its entirety.  For your reference, here it is as a link: mw.mw. There is no SetCoordinates() in it. It works for me in Maple 2016 and 2017.  Upload your worksheet if you are having trouble with it.

@acer Thanks for your thorough explanation, as usual. That unexpected behavior is not a big deal, and if it is done that way for the sake of efficiency, I am all for it.

@rlopez Hello Robert, thanks for pointing out the TNBFrame().  I didn't know about it.  It's good to know.

Happy new year!

g@sheriph05 You ask: 'Can disease-free equilibrium be obtained in a similar way as the endemic equilibrium point?",

I don't quite understand what that is asking, but I have several comments which may be relevant.

• We have seen that there are two equilibria for your system of equations.  One of them has I=0, which is a good thing—if the solution go there, then the disease dies out.  In your previous messages you seemed to want to dismiss that solution.  I don't quite see why.
• Referring to the two solutions noted above, the one with nonzero I does not necessarily indicate an acceptable solution because if I is negative, then it is not meaningful in the epidemics context.

  I suppose that your interest in Descartes's Rule of Signs stems from a desire to determine whether that I is negative or positive.

• Having established that there is an equilibrium with a positive I value does not necessarily mean that the disease will take hold and stay forever. If the equilibrium is unstable, then its existence is pretty much irrelevant because solutions will not go toward it.

• Similarly, if the equilibrium with I=0 is unstable, then its existence is pretty much irrelevant.  On the other hand, it it is stable, that's great news because it says that the disease eventually dies out.

In conclusion, you have two tasks ahead of you. (a) determine which equilibria are of interest, that is, identify those whose S, M, V, I values are either zero or positive, and (b) analyze the stability of those equilibria.  Only then you may be able to make interesting statements about them.

A primary method for analyzing the stability of  equilibria the Routh-Hurwitz criterion which you may be familiar with already.  If not, then Wikipedia's article on Routh-Hurwitz may be a good starting point.

@Kitonum That last line should be
plot3d(F, x=0..5, y=0..5);  # Visualization

@wswain 

Bill,

Maple's user interface is quite customizable.  It offers a Document Mode and a Worksheet Mode.  It offers a 1-D Input Mode and 2-D Input Mode.  Their combinations already yield four possible configurations.  There are many more options.

Maple's default configuration is the Document Mode and 2-D Input Mode.  I, along with many long-time Maple users, favor the configuration consisting of the 1-D Input Mode and the Worksheet Mode.  You may want to give that a try.  See https://userpages.umbc.edu/~rostamia/math481/config/maple.html for instructions.

Please note that the new configuration will not convert any of your previous documents. It will apply only to new worksheets that you make after this configuration.

Having done that, and with the help of a good guidebook to show the way, the basic operations in Maple are not too difficult to learn.  You shouldn't be groping in the dark.  I suggest Ian Thompson's Understanding Maple.  I must admit that I haven't read the book myself.  My recommendation is based (A) on having viewed a few sample pages on amazon's website; and (B) customer reviews, especially the negative ones.

That last comment may seem puzzling, so let me explain. The negative comments say that Thompson focuses on the 1-D Input Mode and the Worksheet Mode.  I say that's exactly the way it should be.  I wouldn't recommend a book that does otherwise.

I suggest that you begin reading the book from the beginning, and wend your way forward.  Should you run into difficulties, ask here and I am sure there will be many who will offer help.

@sheriph05 The attached worksheet shows that the equilibrium I is expressible in terms of the roots of a cubic polynomial.  You may apply Decartea's Rule of Signs to determine whether there are positive roots.  That requires determining the signs of the cubic's coefficients.  I have isolated those coefficients within the worksheet but I have no idea about the meanings of the numerous parameters of your equations, and therefore cannot make any statement about the signs of the coefficients.

calculations.mw

@wswain Bill, I can see the point of doing things your way if your purpose is to practice with Maple's commands. If, however, your goal is merely to arrive at expressions for the moment and shear, there are better ways.

The attached worksheet does the calculations twice: first, the short way through Maple's dsolve(), and second, the long way, manually.  Of course the first method is the normal way.

manual-calculation.mw

@wswain 

Bill, you wrote: "just trying to understand the Maple capability, approach and language/function effort the computer can/will do to resolve this dependencies".

That, it seems to me, is giving Maple too much undue credit.  Euler figured out the differential equation of the deflection of a beam.  That was the hard part. The equation turned out to be quite elementary.  It doesn't take any special powers to solve it.  A freshman can do it with paper and pencil.  There is nothing magical in what Maple does in this case.

The magic lies in the beam's differential equation.  Understanding and formulating the boundary conditions requires some thought.  Solving it is quite trivial.

Regarding a torsional spring joint, it is a matter of replacing the boundary condition (D@@2)(u)(1)=0 in the worksheet with (D@@2)(u)(1)=-K*D(u)(1), where K the the desired stiffness of the spring.  Experiment with various values of K.  Try K=0, 1, 5, 10.