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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!

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!