Probability virus question at different infection rates

[Mark53]

Homework Statement

From various studies, it is known that once an individual is infected with a virus, they become infectious at rate λ. The individual will recover at rate λ, independent of the time it took for them to become infectious. Let X be the total amount of time an individual has this virus.

(a) What is the distribution of X?
(b) What is the probability an infected individual has this virus for less than 1 day?
(c) If λ = 1, what is the probability that after getting the virus, they become infectious in less than one day and then they recover in less than one day?

The Attempt at a Solution

a)
X~Poi(λ)

b)

P(x<1)=P(x=0)

=e^(-λ)λ^0/0!
=e^(-λ)

c)

P(x<1)nP(x<1)
=P(x=0)*P(x=0)
=e^(-λ)*e^(-λ)
=e^(-2)

not sure if this part is correct

 

[haruspex]

I wonder if you have misread the question. X covers the time both to become infectious and then to recover. Each of those has an expected duration of 1/λ. You answer to a) would give E(X)=1/λ.

 

[StoneTemplePython]

Homework Statement

... at rate λ... Let X be the total amount of time an individual has this virus.

(a) What is the distribution of X?

The Attempt at a Solution

a)
X~Poi(λ)

I'm not totally happy with the way this problem is worded. Keep in mind that a Poisson is a discrete distribution (or counting process) characterized by exponentially distributed inter-arrival times. ##X##, here is the total amount of time a given person has the virus (clock starts ticking at infection).

Total time of being sick is a real valued quantity aka it comes from an uncountable source (real line). I think you want the exponential distribution here, not the Poisson for ##X##.
- - - -
I also could not tell whether the rate of becoming infectious (characterized by ##\lambda##) and the rate of getting healthy (characterized by ##\lambda##) are independent or sequential processes? Could a person get healthy and never have the 'arrival' of the infectious state? (Or from a modeling standpoint, the infectious arrival happens after regaining health arrival and hence is moot.) I'm thinking the answer is yes, and this can be thought of as a case of Poisson splitting and combining if you wanted-- and again we're interested specifically in the time until next arrival. The interpretation here is needed for (c), and really (b)... Alternatively, these just happen in a fixed sequence.

Setting aside my interpretation questions, your part b is close, but not quite right... (why? start by looking up the formula of the CDF for an exponential distribution ). You should be comfortable using CDFs -- they are quite powerful.

I'll leave (c) open for now as the (in)dependence of infectiousness and regaining health arrivals is needed first.

 

[Ray Vickson]

I wonder if you have misread the question. X covers the time both to become infectious and then to recover. Each of those has an expected duration of 1/λ. You answer to a) would give E(X)=1/λ.

I agree, and of course ##X## does not have a Poisson distribution, or anything like it.

 

[haruspex]

I agree, and of course ##X## does not have a Poisson distribution, or anything like it.

Funnily enough, I'm not so sure now. I may have misread it.
The problem is the past tense, "took":

independent of the time it took for them to become infectious

That made me think this time starts on becoming infectious. But, of course, as @StoneTemplePython points out, they could recover without becoming infectious, so logically the two times should start on being infected. That makes a) and b) trivial but c) somewhat tricky.

 

[Mark53]

I'm not totally happy with the way this problem is worded. Keep in mind that a Poisson is a discrete distribution (or counting process) characterized by exponentially distributed inter-arrival times. ##X##, here is the total amount of time a given person has the virus (clock starts ticking at infection).

Setting aside my interpretation questions, your part b is close, but not quite right... (why? start by looking up the formula of the CDF for an exponential distribution ). You should be comfortable using CDFs -- they are quite powerful.

I'll leave (c) open for now as the (in)dependence of infectiousness and regaining health arrivals is needed first.

for part b

P(x<1)=intergral from 0 to 1 of λe^(-λx)=1-e^(-λ)

would this be correct for part b now?

how would I get started on part c?

 

[haruspex]

for part b

P(x<1)=intergral from 0 to 1 of λe^(-λx)=1-e^(-λ)

would this be correct for part b now?

how would I get started on part c?

The first thing that has to be resolved is the interpretation of the question. Do you read it that the recovery time starts immediately on being infected (so recovery may occur without ever becoming infectious) or on becoming infectious?
To me, the first makes logical sense in the real world, but the wording of the question suggests the second.

 

[Mark53]

The first thing that has to be resolved is the interpretation of the question. Do you read it that the recovery time starts immediately on being infected (so recovery may occur without ever becoming infectious) or on becoming infectious?
To me, the first makes logical sense in the real world, but the wording of the question suggests the second.

From the question it seems like the second one

 

[haruspex]

From the question it seems like the second one

Ok.
The next problem is units. You are given a numerical value for λ but not the units. If you assume the units are day^-1 then your answer to b) is correct.
For c), you have two conditions to be met. What is the probability of each? Are they independent?

 

[Mark53]

Ok.
The next problem is units. You are given a numerical value for λ but not the units. If you assume the units are day^-1 then your answer to b) is correct.
For c), you have two conditions to be met. What is the probability of each? Are they independent?

the questions says that they are independent of each other

and the probability would be the same for each as in part b but just substitute λ=1 into them

 

[haruspex]

the questions says that they are independent of each other

and the probability would be the same for each as in part b but just substitute λ=1 into them

Right, so what is the answer to c)?

 

[Mark53]

Right, so what is the answer to c)?

P(x<1)*P(x<1)
(1-e^(-1))*(1-e^(-1))
=1-2e^(-1)+e^(-2)
=0.3996

 

[haruspex]

P(x<1)*P(x<1)
(1-e^(-1))*(1-e^(-1))
=1-2e^(-1)+e^(-2)
=0.3996

Looks good to me.

 

[Mark53]

Looks good to me.

thanks for the help!

 

[StoneTemplePython]

So with the interpretation we're going for (Infected -> Infectious -> Cured, and the underlying time scale is days), I think the answer for (c) is correct.

There are some technical nits for (a) and (b) though.

Consider ##Y## which is an exponential random variable with parameter ##\lambda##. ##Y_1## is time from infected --> infectious. Then ##Y_2## is time from infectious --> healthy.

As I understand it total time sick is given by ##X## where ##X = Y_1 + Y_2##. That is, the total time sick is actually the convolution of two exponential r.v.'s -- aka a distribution that is Erlang of order 2 (which you can get from cleverly differentiating the Poisson distribution if you prefer that to convolutions). A quick sanity check tells us ##E[X] = E[Y_1] + E[Y_2] = \frac{2}{\lambda}##, which conforms with the fact that on average it takes ##\frac{1}{\lambda}## for someone to go infected --> infectious, then another ##\frac{1}{\lambda}## to go from infectious --> healthy

(b) What is the probability an infected individual has this virus for less than 1 day?

I take it this means a person is infected (but not yet infectious) and we are interested in the probability they will be totally cured a day later. Thus we are interested in the probability of being 2 arrivals (or more -- after 2 occur they are moot in our model) in a day, or alternatively -- we are interested in the complement of there being only 0 or 1 arrivals in a day -- note that this is the CDF of an Erlang of order 2.

If you haven't heard of an Erlang distribution, apologies -- they very natural come up when discussing Poisson processes and convolutions of exponentials.