Modeling epidemics - solving differential equation

[green-beans]

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!

 

[haruspex]

in which I increases.

Express that algebraically.

N = S + R + I

Which tells you what in terms of rate of change of I?

 

[green-beans]

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.

 

[haruspex]

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.

 

[green-beans]

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.

 

[haruspex]

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.

 

[green-beans]

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).

 

[haruspex]

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.

 

[green-beans]

Right.

Thank you for your help!

 

[green-beans]

Right.

Actually, I have just realized 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.

 

[haruspex]

Actually, I have just realized 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.