Tricky 10th grade Math Problem of 3 Equation

[aeronautical]

Homework Statement

Epidemic. Consider the following model of how an epidemic spreads through a population. First we will introduce some definitions:

N = number of individuals in population
Mk = the number of susceptible after k weeks
Sk = number of infectious after k weeks
Ik = the number of immune after k weeks
d = disease duration in weeks
k = constant, which describes how easily the disease is infecting

Then we can formulate our mathematical model

Mk +1 = Mk - k * Sk * Mk (1)
Sk + 1 = Sk + k * Sk * Mk - Sk/d (2)
Ik + 1 = Ik + Sk/d (3)

Recognize the first model equations in words! Then examine how Mk, Sk and Ik developed week by week until the epidemic is over. You can use the values N = 1000, S0 = 1, k = 0.002 and d = 1.

The Attempt at a Solution

I have tried to combine all three equations into one, and I know that the problem is solved once the number of immunes Ik, equal the total number of individual in the population. Yet, all I get is a constant equation that repeats itself. I assume that N=Mk+Ik+Sk. Can somebody help me out with this tricky problem?

 

[Mark44]

Aren't your equations recursive definitions?
M_k+1 = M_k - k * S_k * M_k
S_k+1 = S_k + k * S_k * M_k - S_k/d
I_k+1 = I_k + S_k/d

You might be running into trouble if you are thinking that you have M_k + 1 rather than M_k+1.

 

[aeronautical]

Aren't your equations recursive definitions?
M_k+1 = M_k - k * S_k * M_k
S_k+1 = S_k + k * S_k * M_k - S_k/d
I_k+1 = I_k + S_k/d

You might be running into trouble if you are thinking that you have M_k + 1 rather than M_k+1.

Well I don't know how to begin solving this problem...can you pls help?

 

[berkeman]

Well I don't know how to begin solving this problem...can you pls help?

Using Mark44's corrected form of the equations, tell us in your own words what the first equation represents. You should be able to do that.

Next, I'd recommend making a quick Excel spreadsheet with these equations, and seeing how the numbers play out after some number of weeks. Use the numbers you are given, to plug into the equations. Once you see how the equations are working, then you can decide if you can find a closed-form solution for if/when an outbreak will end...