Birth and death process and Little's law

[LogarithmLuke]

Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible individual the time until the next infection follows an exponential distribution with expected value 1/λ = 100 days and that the durations of the infections follow independent exponential distributions with expected values 1/μ = 7 days.

a) Let Y (t) denote the number of infected individuals in the population at time t measured in days. Specify the birth and death rates of this birth and death process.

The total population contains 5.26 million individuals.

b) Assume the stochastic process {Y (t) : t ≥ 0} has reached its stationary distribution. For each infection, there is a probability of 1% that the infection will result in serious compli- cations that requires hospitalization. On average, the hospitals only have capacity to handle 2000 individuals with complications from a cold. Use Little’s law to calculate the average treatment time required to not exceed the capacity.So for a) the birth rates should decrease with an increase in the amount of infected individuals, however the overall population size remains the same in this model, so I am thinking the birth rate is just 𝜆 and the death rate simply 𝜇? Or should the death rate be 𝜇𝑖 and birth rate 𝜆/𝑖 where 𝑖 is the number of infected individuals?

For b) I am not sure what I need to calculate. From what I see we can just plug straight into Little's law 𝐿=𝜆𝑊 and put 𝐿=2000
and use the 𝜆 given to solve for 𝑊, however the answer obtained that way makes no sense.

 

[StoneTemplePython]

Assume that an individual only has two possible states: susceptible (S) and infected (I). Further, assume that the individuals in the population are independent, and that for each susceptible individual the time until the next infection follows an exponential distribution with expected value 1/λ = 100 days and that the durations of the infections follow independent exponential distributions with expected values 1/μ = 7 days.

a) Let Y (t) denote the number of infected individuals in the population at time t measured in days.
...

So for a) the birth rates should decrease with an increase in the amount of infected individuals, however the overall population size remains the same in this model, so I am thinking the birth rate is just 𝜆 and the death rate simply 𝜇? Or should the death rate be 𝜇𝑖 and birth rate 𝜆/𝑖 where 𝑖 is the number of infected individuals?

For b) I am not sure what I need to calculate. From what I see we can just plug straight into Little's law 𝐿=𝜆𝑊 and put 𝐿=2000
and use the 𝜆 given to solve for 𝑊, however the answer obtained that way makes no sense.

I'm not following your line of thinking here . I don't follow how you got ii in the denominator with $\lambda$. If it isn't guessing -- how do you justify these things?

The right approach is to start with a very simple case and figure out what happens when you convolve two independent exponentials...
- - - -
I do see $n=5.26$ million and

$\lambda \cdot (n−i)$ as the birth/infection rate and
$\mu \cdot i$ as the death/ get healthy rate

How can these things be justified? Also have you drawn a picture of the underlying jump chain?

As for (b) what is your understanding of Little's Law?

 

[LogarithmLuke]

The reason i proposed $$\frac{\lambda}{i}$$is because the birth rate becomes lower as more people get infected. But with that reasoning $$\lambda * (n-i)$$for the birth rate, and $$\mu i$$ for the death rate as you propose also makes sense. Yes I have sketched the jump chain, the states can either go from i to i +1 or i-1 (birth and death process).

My understanding of Little's law is that the long term average number L of customers in a stationary system is equal to the long term average arrival rate times the average serving time W.

So I am guessing I have to find the the long term average arrival rate? I was given the hint that I could use the long-run mean fraction of time per year that a single individual has a cold (which I calculated to be 7/107) instead of calculating the stationary distribution for Y(t).

 

[StoneTemplePython]

The reason i proposed $$\frac{\lambda}{i}$$is because the birth rate becomes lower as more people get infected. But with that reasoning $$\lambda * (n-i)$$for the birth rate, and $$\mu i$$ for the death rate as you propose also makes sense. Yes I have sketched the jump chain, the states can either go from i to i +1 or i-1 (birth and death process).

I may be overly attuned to linearity and non-linearities but here's what doesn't sit well with me.

You are right that the birth rate is bigger when small number of infected people and smaller when large number of infected.

But how is it calculated? If we ignore the getting healthy rate, which by memorylessness, we can (for now),

then the time until next infection at state i is given by the first arrival of $n-i$ poisson processes. That is there are $(n-i)$ people who aren't sick, but 'the clock is ticking' as to time until when they get sick... If you know how to combine (superpose) and split Poisson processes you should recognize that this merged process has parameter $(n-i)\lambda$ -- i.e. the rate parameter scales linearly with the number of processes. I'm not sure what text you are using, but the book by Blitzstein and Hwang does a good job treating this in chapter 13. It is freely available here:

https://projects.iq.harvard.edu/stat110/home

Little's Law is also in chapter 13...

(note: this seems to indicate that your problems are undergrad level homework, and hence may be better located in the homework forums)

My understanding of Little's law is that the long term average number L of customers in a stationary system is equal to the long term average arrival rate times the average serving time W.

So I am guessing I have to find the the long term average arrival rate? I was given the hint that I could use the long-run mean fraction of time per year that a single individual has a cold (which I calculated to be 7/107) instead of calculating the stationary distribution for Y(t).

give it a shot and show me your work, and what you end up with here. I think if you explore it a bit you'll end up with a correct answer

 

[LogarithmLuke]

So if the birth rate is $$\lambda(n-i)$$ and we know that approx 0.065 of the population are infected. So our arrival rate becomes $$ \lambda(5.26*10^6 - 0.065*5.26*10^6)*0.01$$ since only 1% of arrivals require hospitalization. This gives $$W = \approx 4 days$$ which seems like a reasonable answer?

 

[StoneTemplePython]

So if the birth rate is $$\lambda(n-i)$$ and we know that approx 0.065 of the population are infected

I'm not sure where this bold section is coming from -- they weren't stated in the original problem and you haven't shown any work.

b) Assume the stochastic process {Y (t) : t ≥ 0} has reached its stationary distribution. For each infection, there is a probability of 1% that the infection will result in serious compli- cations that requires hospitalization. On average, the hospitals only have capacity to handle 2000 individuals with complications from a cold. Use Little’s law to calculate the average treatment time required to not exceed the capacity.

by the way, I'm not totally comfortable with the wording here. Little's Law deals with averages -- i.e. that the expected number of customers in a queue is equal to the product of the arrival rate and the expected time each customer is in the queue... but the actual thing you are supposed to calculate (which I italicized) really has little to do with averages and technically is worded as a ruin problem. My guess is that its just poorly worded and isn't meant to be a ruin problem but I don't like having to guess these things.

 

[LogarithmLuke]

My apologies. So i calculated that the long-run mean fraction of time a single individual has a cold is $7/107$ so i figured $7/107$ or about $0.065$ of the population will be infected in the long term. Does the calculation seem correct?