# Modeling epidemics - solving differential equation

• green-beans
In summary, the conversation discusses a modified SIR model with equations for the rate of decrease of susceptibles and for the increase of those who have been removed or recovered. The goal is to find the region in the S-I plane where the number of infectives, I, increases over time. This is found through solving the differential equation and determining that the shaded region should be under the x-axis and some area above it but below the parabola.
green-beans
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.

green-beans said:
in which I increases.
Express that algebraically.
green-beans said:
N = S + R + I
Which tells you what in terms of rate of change of I?

haruspex said:
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.

green-beans said:
If I express I algebraically
No, I mean express algebraically the statement that I increases.
green-beans said:
I am considering dI/dS
Why? The question asks in which region of the I-S space I increases with time.

haruspex said:
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.

green-beans said:
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.
green-beans said:
α*S*√(I) - β*I>0
You are almost there! Just simplify.

haruspex said:
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).

green-beans said:
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.

haruspex said:
Right.

haruspex said:
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.

green-beans said:
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.

## 1. How do differential equations play a role in modeling epidemics?

Differential equations are used in modeling epidemics because they can help us understand and predict the spread of infectious diseases. These equations take into account factors such as the number of infected individuals, the rate of transmission, and the recovery rate to create a mathematical model of an epidemic.

## 2. What is the purpose of modeling epidemics?

The purpose of modeling epidemics is to provide a better understanding of how infectious diseases spread and to predict their future course. This can help public health officials make informed decisions about disease control and prevention measures.

## 3. How accurate are models of epidemics?

The accuracy of models of epidemics depends on the quality of data used and the assumptions made in the model. While models cannot predict the exact course of an epidemic, they can provide valuable insights and inform decision-making.

## 4. What are some challenges in modeling epidemics?

Some challenges in modeling epidemics include limited data, making accurate assumptions about the disease and its transmission, and accounting for human behavior and interventions that can affect the spread of the disease. Models also need to be regularly updated as new data becomes available.

## 5. How can modeling epidemics help in controlling outbreaks?

Modeling epidemics can help in controlling outbreaks by providing insights into the effectiveness of different control measures, such as quarantine and social distancing. These models can also help predict the potential impact of different interventions, allowing for more informed decision-making.

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