Modeling epidemics - solving differential equation

  • #1
Moved from a technical math section, so no template was used
I am given a modified SIR model in which the rate of decrease of susceptibles S is proportional to the number of susceptibles and the square-root of the number if infectives, I. If the number R of those who have been removed or recovered increases in proportion to the infectives, we have the following equations:
dS/dt = -α*S*√(I)
dR/dt = β*I where α and β are positive constants. If the total population, N = S + R + I does not change over time, shade the region of the S-I plane in which I increases.

To find the region, as far as I understand, I need to first find the differential equation for dS/dt which is
α*S*√(I) - β*I. Then, I need to consider the following differential equation:
dI/dS = -1 + {β*√(I)}/α*S}
which I am not sure how to solve since I cannot separate the variables and the method with integrating factor is not applicable either.
I also tried considering dS/dI but it looked even worse.

Thank you in advance!
 

Answers and Replies

  • #3
Express that algebraically.

Which tells you what in terms of rate of change of I?
Hi, thank you for your reply, I am not quite sure what you mean. If I express I algebraically, then I'll obtain I = N - S - R
But then I am not sure how this is will solve the differential equation since I am considering dI/dS and the expression I = N - S - R also introduces R.
 
  • #4
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,952
7,199
If I express I algebraically
No, I mean express algebraically the statement that I increases.
I am considering dI/dS
Why? The question asks in which region of the I-S space I increases with time.
 
  • #5
No, I mean express algebraically the statement that I increases.

Why? The question asks in which region of the I-S space I increases with time.
Hmmmm... I am sorry but I still do not get what you are trying to say. As far as I understand, since the question asks to shade the region of the I-S space in which I increases with time, I need to obtain the function I(S) or S(I) which I can get by solving the differential equation. However, I will increase with time when dI/dt >0, i.e. α*S*√(I) - β*I>0. So, when -dS/dt - dR/dt>0 which implies that -dS/dt>dR/dt but I am not sure how this can help to find the region.
 
  • #6
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,952
7,199
since the question asks to shade the region of the I-S space in which I increases with time, I need to obtain the function I(S) or S(I)
I do not follow the logic of that.
α*S*√(I) - β*I>0
You are almost there! Just simplify.
 
  • #7
I do not follow the logic of that.

You are almost there! Just simplify.
Ohhh, I see! So, I get that I < {(α)^2 * (S)^2}/{β^2} which is a parabola starting at the origin. Therefore, the shaded region should be under S-axis (if I plot I as y-axis and S as x-axis).
 
  • #8
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,952
7,199
Ohhh, I see! So, I get that I < {(α)^2 * (S)^2}/{β^2} which is a parabola starting at the origin. Therefore, the shaded region should be under S-axis (if I plot I as y-axis and S as x-axis).
Right.
 
  • #10
Right.
Actually, I have just realised that the shaded area should be the area not under negative x-axis but under the parabola. So, it will be the entire area under x-axis and some of the area above it but below the parabola.
 
  • #11
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
36,952
7,199
Actually, I have just realised that the shaded area should be the area not under negative x-axis but under the parabola. So, it will be the entire area under x-axis and some of the area above it but below the parabola.
Ah, yes, I hadn't read your previous reply carefully enough, sorry. Got to the mention of parabola and stopped.
 

Related Threads on Modeling epidemics - solving differential equation

Replies
16
Views
921
Replies
2
Views
757
  • Last Post
Replies
12
Views
878
Replies
1
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
0
Views
2K
Replies
3
Views
1K
  • Last Post
Replies
3
Views
1K
Replies
1
Views
1K
Top