# system of differential equations solving in mathcad

I would like to learn to solve system of differential equation in MathCad like this one:

dI(t)/dt = coeff * I(t) * S(t)

dS(t)/dt = -coeff * I(t) * S(t)

N = S(t) + I(t)


where 'N' and 'coeff' are user specified coefficients

In MathCad I did:

            ( coeff * y0 * y1   )
D(t, y) :=  (                   )
( -coeff * y0 * y1  )

( 1 )
ic := (   )
( 9 )


But :

S := rkfixed (ic, 0, 10, 100, D)


returns: D - this function can't be used here.

And I don't know how and where insert N=S(t) + I(t) equation :(

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Looks like Lotka–Volterra equation, and for the constraint, with $S(t) = N- I(t)$ plugged in the second equation you get $dI/dt = cNI(t) - cI^2$, try solving this using a stable marching scheme. – Shuhao Cao Apr 27 '12 at 22:59
As an example I used quite easy system with two variables ... but what should I do when I'll have 4-6 variables with no way to make such substitution? – davs Apr 27 '12 at 23:04
Then this works like a constraint minimization problem numerically rather than a simple "marching in time" problem, does $N$ change with time too? I am not familiar with MathCad, but using rkfixed this kind of Runge Kutta with fixed marching step doesn't seem right. – Shuhao Cao Apr 27 '12 at 23:14
N is constant. variable coeff also sets ones. Following info contains what I'm working with: I - number of infected items and S is number of suspected items. Mentioned above system of DiffEquation is S-I model of virus spreading. N - number of all items (S+I), coeff - rate of infection between items – davs Apr 27 '12 at 23:30
Stackoverflow might be a better place to discuss the problem on such softwares. Best wishes. – Frank Science May 5 '12 at 8:34

## 1 Answer

I can give you an answer for the above system (2 ODEs). For your system equation 3 can be derived from equations 1,2 since:$$\frac{dI(t)}{dt}=coeff\cdot S(t)\cdot I(t) \hspace{0.5cm} (1)$$ $$\frac{dS(t)}{dt}=-coeff\cdot S(t)\cdot I(t) \hspace{0.5cm} (2)$$

Adding $(1)$ and $(2)$, you get:

$$\frac{dI(t)}{dt}+\frac{dS(t)}{dt}=0 \Leftrightarrow I(t)+S(t)=c \ \forall \ t\geq0$$where $c$ is a constant. So, by defining the IC you're defining $N=c=I(0)+S(0)$ too and therefore equation $(3)$ is useless. As far as Mathcad is concerned, since you're trying to march forward in time you correctly chose to use rkfixed. Here is the mathcad code i used for an example problem:

As you can see the sum $S(t)+I(t)$ is constant for every t and equal to 1.1 as defined by the initial condition.

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