Linear Algebra: Non Linear Systems Of Differential Equations

[Wint]

Homework Statement

We are working on homework for a population model affected by an endemic disease (smallpox). There are 3 types of individuals in the population, S (susceptible), I (infected), and R (recovered).

Homework Equations

The total population N = S+ I + R. The three differential equations we are given are:

N'=rN(1-N)-amI

S'=rN(1-N)-abSI+N-S

I'=abSI-aI

rN(1-N) represents the standard growth rate of the population, with r being the maximum relative growth rate (all newborns are assumed to be susceptible).

amI represents the additional death rate from the disease, with m the fraction of disease victims who die and a the ratio of the average lifespan of the population to the duration of the illness.

abSI represents the rate at which susceptibles become infected, with b measuring the average number of susceptibles who get infected by each infective.

The Attempt at a Solution

The first question we were asked to answer was to compute the Jacobian of the system, which I have done (hopefully correctly ):

The second question is to find the critical point that corresponds to the entire population dying, and this is where I get hung up on these kinds of questions. Once I figure out what the values should be, it's rather simple to compare the resulting characteristic polynomial to the Rauth-Hurwitz criteria to check stability, but figuring out what the values should be is what gets me.

What I've decided so far is that rN(1-N), the standard growth rate, and amI, the additional death rate to the disease should be zero, since the population is not growing and neither is the death rate since everyone is dead. Not sure if that's the correct assumption here.

Also, abSI, the rate at which susceptibles are being infected would be 1 I'm guessing, or 100% since everyone is getting infected and dying.

I'm wondering if that is the correct tack, or if I should be looking at the values in the the Jacobian like r, a, m, and b and trying to decide what those should be instead, since it's those values that I have to use to determine what the Jacobian comes out as and then use that to get the characteristic polynomial.

Edit:

After a little more noodling on the problem, I was thinking that I should be looking at r, a, m, and b instead. What I've come up with so far is that:

r = max relative growth rate = 0 (no growth rate since everyone died)
a = ratio of avg lifespan / duration of illness = ? (avg lifespan is 0 since everyone eventually dies from the illness dies?)
m = fraction of disease victims who die = 1 (since everyone dies)
b = ? (not sure if it matters since b is always dependent on a, which I can't decide on a value for yet).

Any insight that can be offered would be great, thanks.

 

[mighty2000]

Hello,

Thank you for your post. It seems like you are on the right track with your approach to finding the critical point for the entire population dying. Let me offer some additional insights to help guide you in the right direction.

Firstly, your assumptions about rN(1-N) and amI being zero are correct. Since the entire population is dying, there is no growth rate and everyone is dying from the disease, so the additional death rate is also zero.

Next, let's consider abSI. This represents the rate at which susceptibles are being infected. In this scenario, since everyone is getting infected and dying, we can assume that this rate is 100%, as you mentioned. This means that b must be equal to 1, since it is the average number of susceptibles who get infected by each infective.

Now, let's look at the remaining parameters - r, a, and m. As you mentioned, r is the maximum relative growth rate and in this scenario, it is equal to 0 since there is no growth. Next, we have a, which is the ratio of average lifespan to the duration of illness. Since everyone is dying from the disease, we can assume that the average lifespan is equal to the duration of the illness, so a must be equal to 1. Finally, we have m, which is the fraction of disease victims who die. In this scenario, since everyone is dying from the disease, m must also be equal to 1.

With these values, you should be able to find the critical point for the entire population dying. I hope this helps. Good luck with your homework!