"Modular division by 42" is not a well-defined group operation. A group operation has two operands from the group. You may be thinking of either "addition modulo 42" or "multiplication modulo 42", but those are very different groups.

@jgray07m I think that all that you need to do is keep the sliders for birth and death rates at 0. I don't think that dN/dt = 0 will be a problem.

@Joe Riel Thanks, Joe, for the Tally-once idea and for the profiling. I haven't looked at your code yet, but I'm sure that it will be enlightening. I wouldn't have guessed that Tally was the culprit, but it makes sense now. The reason that I count nonzeros rather than zeros is because that's all that the too-vaguely-named ArrayTools:-SearchArray does.

@Joe Riel Using expand will do more than using simplify in this and several related cases.

@Kitonum Yes, I like your interpretation.

Having taught 3-D Calculus with Analytic Geometry, my mental imagery of the problem involves lines and planes. Indeed, I was visualizing it as I fell asleep last night, and I put it into Maple when I awoke.

@radaar To be even more specific topologically, do you mean embeddings^[*1] of [0,1]x[0,1] into R³? So, for example, a mountainous region of the Earth's surface, but not the whole Earth itself?

[*1] An embedding f:A->B is a bicontinuous injection, i.e., a homeomorphism between A and its image in B, (f(A)).

@Mariusz Iwaniuk 

The final condition is D(f)(infinity) = 1, not f(infinity) = 1.

When you put comments in worksheets, they should be marked as comments so that the syntax checker does not interpret them as code, which'll surely produce error messages. There are a few ways to do this. The easiest is to begin the comment with #. Then everything to the end of that line will be ignored by the syntax checker.

It's also possible to set aside whole execution groups (which are somewhat analogous to paragraphs) as fully formatted text, similar to that produced by a word processor. 

@Zeineb 

X_k is the k^th column of A^-1.

@jgray07m So, populations are relative, as in percentages, rather than actual population numbers? Yes, it seems reasonable. I guess then that N is total (alive) population? 

And I need the 5 parameters. Obviously, the v slider should go from 0 to 1.

@jgray07m You can read the help page ?Explore.

Can you come up with some numeric values?

@jgray07m Yes, it totally makes sense now.

There is really only one model because the first one is just a specialization of the second one at v=0. I guess that that's the vaccination rate.

The equations involving N are independent of the first three equations. Thus they can't change anything. So, why are they there?

To get any plots, we need numeric values, or reasonable ranges of values, for all 5 parameters, and we need 3 initial conditions (or reasonable ranges). I can make an Explore app where you can control the parameter values with sliders or knobs, and the plots will change with those.

@ The problem is on page 303 of the aforementioned book (by Steven Strogatz). I understand the differential equations now, and I'll enter them into a Maple worksheet.

@Rouben Rostamian  Nice work; Vote Up. I was on the verge of creating a similar graphic myself before I saw yours.

What you have presented inside the solve command is unintelligible to me as having any relation to differential equations. Where are the derivatives? Do you have two independent variables, phi and psi? There should probably only be one. Or is it an autonomous system with phi and psi as the dependent variables?

Maybe you should just try typing your equations here rather than in Maple. The copy and paste from Maple really mangled your expression.