# Modeling epidemics - solving differential equation

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.

haruspex
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in which I increases.
Express that algebraically.
N = S + R + I
Which tells you what in terms of rate of change of I?

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
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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.

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
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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.

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
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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.

Right.

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.

haruspex
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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.