The SIR model

[CoreyyV]

I'm not sure how many of you would be familiar with the SIR Model of Epidemics, but I'm having trouble trying to adjust some equations given a new parameter.

The original parameters are: t=time, N=total population, b= a fixed # of contacts/day that spread the disease is the other person is susceptible, k =1/avg length of disease (average # of days a person is infected, S(t)=# of susceptible people, I(t)=# of infected people, and I(t)=# of infected people. Where s(t)=S(t)/N, i(t)=I(t)/N, and r(t)=R(t)/N and are fractions of the total population.
It also assumes that when a person has recovered, they do not become susceptible again and that N is constant (no new people arrive and no one dies)

The equation for the rate of change for s(t) is: ds/dt = −i(t)bs(t)
The equation for the rate of change for i(t) is : di/dt = i(t)bs(t)−ki(t)
And the equation for the rate of change for r(t) is: dr/dt = ki(t)

The question says to consider a disease where immunity is lost and to adjust these ROC equations to the new parameter n, which is the amount of days after recovery when the person because susceptible again.

I'm sorry for any confusion, but any help on adjusting these equations would be awesome!

 

[Deleted member 4993]

I'm not sure how many of you would be familiar with the SIR Model of Epidemics, but I'm having trouble trying to adjust some equations given a new parameter.

The original parameters are: t=time, N=total population, b= a fixed # of contacts/day that spread the disease is the other person is susceptible, k =1/avg length of disease (average # of days a person is infected, S(t)=# of susceptible people, I(t)=# of infected people, and I(t)=# of infected people. Where s(t)=S(t)/N, i(t)=I(t)/N, and r(t)=R(t)/N and are fractions of the total population.
It also assumes that when a person has recovered, they do not become susceptible again and that N is constant (no new people arrive and no one dies)

The equation for the rate of change for s(t) is: ds/dt = −i(t)bs(t)
The equation for the rate of change for i(t) is : di/dt = i(t)bs(t)−ki(t)
And the equation for the rate of change for r(t) is: dr/dt = ki(t)

The question says to consider a disease where immunity is lost and to adjust these ROC equations to the new parameter n, which is the amount of days after recovery when the person because susceptible again.

I'm sorry for any confusion, but any help on adjusting these equations would be awesome!

This is your 25 th post on 7 different threads at this forum. By this time you know that, in the absence of display of substantial effort on your side, you would be waiting for a while.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[CoreyyV]

This is your 25 th post on 7 different threads at this forum. By this time you know that, in the absence of display of substantial effort on your side, you would be waiting for a while.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

I'm sorry. I've tried to write out different equations that I think could possibly make sense, but I don't know if any of them make any actual sense and am unsure how to know if I did do anything right. I've attached to this post my attempts at the new equations.

I've tried rewriting these equations with the new parameter n in different ways, but after I come up with a new equation, I don't know how to tell if I did it correctly

 

[Dr.Peterson]

I'm not sure how many of you would be familiar with the SIR Model of Epidemics, but I'm having trouble trying to adjust some equations given a new parameter.

The original parameters are: t=time, N=total population, b= a fixed # of contacts/day that spread the disease is the other person is susceptible, k =1/avg length of disease (average # of days a person is infected, S(t)=# of susceptible people, I(t)=# of infected people, and I(t)=# of infected people. Where s(t)=S(t)/N, i(t)=I(t)/N, and r(t)=R(t)/N and are fractions of the total population.
It also assumes that when a person has recovered, they do not become susceptible again and that N is constant (no new people arrive and no one dies)

The equation for the rate of change for s(t) is: ds/dt = −i(t)bs(t)
The equation for the rate of change for i(t) is : di/dt = i(t)bs(t)−ki(t)
And the equation for the rate of change for r(t) is: dr/dt = ki(t)

The question says to consider a disease where immunity is lost and to adjust these ROC equations to the new parameter n, which is the amount of days after recovery when the person because susceptible again.

I'm sorry for any confusion, but any help on adjusting these equations would be awesome!

This is very hard to read; that's one reason we ask you, if possible, to show us the problem exactly as given to you. Let me at least spread things out so I can see them:

The original parameters are:​

t=time,​

N=total population,​

b= a fixed # of contacts/day that spread the disease is the other person is susceptible,​

k =1/avg length of disease (average # of days a person is infected,​

S(t)=# of susceptible people,​

I(t)=# of infected people, and​

I(t)=# of infected people.​

Where​

s(t)=S(t)/N,​

i(t)=I(t)/N, and​

r(t)=R(t)/N​

are fractions of the total population.​

It also assumes that when a person has recovered, they do not become susceptible again and that N is constant (no new people arrive and no one dies)​

​

The equation for the rate of change for s(t) is: ds/dt = −i(t)bs(t)​

The equation for the rate of change for i(t) is : di/dt = i(t)bs(t)−ki(t)​

And the equation for the rate of change for r(t) is: dr/dt = ki(t)​

I'm sorry. I've tried to write out different equations that I think could possibly make sense, but I don't know if any of them make any actual sense and am unsure how to know if I did do anything right. I've attached to this post my attempts at the new equations.

I've tried rewriting these equations with the new parameter n in different ways, but after I come up with a new equation, I don't know how to tell if I did it correctly

Now, I also have trouble following your work. Are the vertical bars absolute values, or something else, perhaps separating different attempts?

It will help if you use words, not just symbols. Pick one of these equations, maybe the first one, and tell us what you are thinking when you write what you do. If I were doing this, I would start by explaining why the equation as given is what it is, and then talking about how the addition of n changes what it has to say. Only then would I try writing a modified equation. This is the way to know that they make sense: whether you can explain the reason for what you wrote.

 

[CoreyyV]

This is very hard to read; that's one reason we ask you, if possible, to show us the problem exactly as given to you. Let me at least spread things out so I can see them:

The original parameters are:​

t=time,​

N=total population,​

b= a fixed # of contacts/day that spread the disease is the other person is susceptible,​

k =1/avg length of disease (average # of days a person is infected,​

S(t)=# of susceptible people,​

I(t)=# of infected people, and​

I(t)=# of infected people.​

Where​

s(t)=S(t)/N,​

i(t)=I(t)/N, and​

r(t)=R(t)/N​

are fractions of the total population.​

It also assumes that when a person has recovered, they do not become susceptible again and that N is constant (no new people arrive and no one dies)​

​

The equation for the rate of change for s(t) is: ds/dt = −i(t)bs(t)​

The equation for the rate of change for i(t) is : di/dt = i(t)bs(t)−ki(t)​

And the equation for the rate of change for r(t) is: dr/dt = ki(t)​

Now, I also have trouble following your work. Are the vertical bars absolute values, or something else, perhaps separating different attempts?

It will help if you use words, not just symbols. Pick one of these equations, maybe the first one, and tell us what you are thinking when you write what you do. If I were doing this, I would start by explaining why the equation as given is what it is, and then talking about how the addition of n changes what it has to say. Only then would I try writing a modified equation. This is the way to know that they make sense: whether you can explain the reason for what you wrote.

Sorry, the vertical bars were my attempt at separating the equations (I can see that me doing it that way is confusing, so sorry for that)

And I see I made a mistake in that I said "I(t)" twice, when the second one was supposed to be "R(t)" and = # of recovered people.

In the first equation, in my attempt furthest to the right, my thought process behind it was since I now have to take into consideration that after a certain amount of days, those who were recovered become susceptible once again, I thought that adding (n-k) would make sense because taking the difference of the average amount of days until susceptible again and average amount of days until recovered and multiplying it by the original equation seemed logical.

But as I'm typing this out, another equation comes to my mind that makes a little more sense. And it's: -i(t)*b*s(t)*nr(t). My reasoning behind this is that ds/dt would stay the same and you would just multiply that equation by nr(t) because after n days, those who are recovered become susceptible again, so the average amount of days until susceptible times recovered people would seem to make sense.

(sorry for the late reply. I was called into work unexpectedly)

 

[Dr.Peterson]

I'm not sure what's expected; but your reasoning sounds somewhat like guesses. Why would adding this, or multiplying that, do what you want to do?

As I suggested, a way to start is to learn from what you are given. Looking at the original equation for ds/dt, the RHS shows what fraction of people move from susceptible to infected, which corresponds to the first term for di/dt; the second term of the latter is the fraction that move from infected to recovered, which seems somewhat similar to what you may need to add to ds/dt to represent those who move from recovered to susceptible, since both are based on a number of days. Maybe you can base your answer on that.

 

[CoreyyV]

I'm not sure what's expected; but your reasoning sounds somewhat like guesses. Why would adding this, or multiplying that, do what you want to do?

As I suggested, a way to start is to learn from what you are given. Looking at the original equation for ds/dt, the RHS shows what fraction of people move from susceptible to infected, which corresponds to the first term for di/dt; the second term of the latter is the fraction that move from infected to recovered, which seems somewhat similar to what you may need to add to ds/dt to represent those who move from recovered to susceptible, since both are based on a number of days. Maybe you can base your answer on that.

I wasn't necessarily guessing. When I was thinking about it, it was making sense to me. I guess I wasn't thinking correctly and wasn't good at explaining my thought process behind it.

I just figured since I'm trying to add another group of people back into the original equation (the newly susceptible people), I would do it by adding an equation for how you find the newly susceptible people, which I thought would be nr(t) (since after n days, those who had recovered become susceptible again, you'd multiply n by the fraction of number of recovered people), to the original equation -i(t)bs(t).

Thank you for your responses by the way

 

[Dr.Peterson]

I wasn't necessarily guessing. When I was thinking about it, it was making sense to me. I guess I wasn't thinking correctly and wasn't good at explaining my thought process behind it.

I just figured since I'm trying to add another group of people back into the original equation (the newly susceptible people), I would do it by adding an equation for how you find the newly susceptible people, which I thought would be nr(t) (since after n days, those who had recovered become susceptible again, you'd multiply n by the fraction of number of recovered people), to the original equation -i(t)bs(t).

The reason I said it "sounded somewhat like guessing" is that you had several contradictory suggestions, which doesn't sound like you were convinced! But guessing can be a valuable part of solving a problem, if you then find a convincing explanation for the guess.

One of your proposals was to add n-k, and an earlier one was to subtract n/k. Neither of these makes sense dimensionally. Did you observe that k is defined not as a number of days, but as 1 over a number of days? (That seems odd to me, as I'd think they'd just define a parameter d as a number of days and use 1/d in the equations, but they can do what they choose.) Since s is a dimensionless fraction, ds/dt has dimension day^-1, and it might make sense to subtract k from it, but not to add n, a number of days.

So for me the key is to analyze the equations you were given and learn from them. When they write the third equation, dr/dt = ki(t), they are saying that the fraction newly recovered per day is the number currently infected divided by the number of days one remains infected, which makes sense, as one person can be said to recover per that many days.

You need to do something similar for the number susceptible. Your nr(t) is close, but has the wrong dimensions; and what you wrote previously was -i(t)*b*s(t)*nr(t), which means multiplying by that rather than adding it as you say now.

 

[CoreyyV]

The reason I said it "sounded somewhat like guessing" is that you had several contradictory suggestions, which doesn't sound like you were convinced! But guessing can be a valuable part of solving a problem, if you then find a convincing explanation for the guess.

One of your proposals was to add n-k, and an earlier one was to subtract n/k. Neither of these makes sense dimensionally. Did you observe that k is defined not as a number of days, but as 1 over a number of days? (That seems odd to me, as I'd think they'd just define a parameter d as a number of days and use 1/d in the equations, but they can do what they choose.) Since s is a dimensionless fraction, ds/dt has dimension day^-1, and it might make sense to subtract k from it, but not to add n, a number of days.

So for me the key is to analyze the equations you were given and learn from them. When they write the third equation, dr/dt = ki(t), they are saying that the fraction newly recovered per day is the number currently infected divided by the number of days one remains infected, which makes sense, as one person can be said to recover per that many days.

You need to do something similar for the number susceptible. Your nr(t) is close, but has the wrong dimensions; and what you wrote previously was -i(t)*b*s(t)*nr(t), which means multiplying by that rather than adding it as you say now.

Yeah you’re right, those first few equations that I wrote were kind of guesses, but the most recent one that I was explaining wasn’t. And you’re also right that I previously was multiplying nr(t) to the equation, but before my latest response, I was still trying to think it over and realized that multiplying nr(t) didn’t make sense and adding it did. So I’m sorry for being all over the place, haha.

The reason I thought it was nr(t) was because I thought the way to find newly the susceptible is similar to finding the recovered, ki(t), by multiplying i(t) by 1/(avg # of days a person is infected). Because of that, nr(t) made sense to me to add to the first equation. But now as I’m thinking more about it and because of what you said, I’m thinking that I should be adding 1/n*r(t) rather than nr(t) and since newly susceptible should be grouped with people who’ve been susceptible the whole time, should the equation be: -i(t)b(s(t)+1/n*r(t))

 

[Dr.Peterson]

I think that's right. Now adjust the other equations, and see if they look good as a group.

 

[CoreyyV]

I think that's right. Now adjust the other equations, and see if they look good as a group.

Okay! Thank you so much for your guidance, Dr. Peterson. I really Appreciate it!