Solve this system of differential equations:
dS/dt = -\beta S I
dI/dt = \beta S I

The given system of differential equations is:

\[
\frac{dS}{dt} = -\beta S I
\]
\[
\frac{dI}{dt} = \beta S I
\]

### Solution:

1. **Sum of the equations:**
   \[
   \frac{dS}{dt} + \frac{dI}{dt} = -\beta S I + \beta S I = 0
   \]
   This implies:
   \[
   \frac{d(S + I)}{dt} = 0
   \]
   Therefore, \( S + I \) is a constant, say \( C \).

   \[
   S(t) + I(t) = C
   \]

2. **Express \( S \) in terms of \( I \):**
   \[
   S(t) = C - I(t)
   \]

3. **Substitute into one of the original equations:**
   Substitute \( S(t) = C - I(t) \) into \( \frac{dI}{dt} = \beta S I \):
   \[
   \frac{dI}{dt} = \beta (C - I) I
   \]
   This simplifies to:
   \[
   \frac{dI}{dt} = \beta C I - \beta I^2
   \]

4. **Separate the variables:**
   \[
   \frac{dI}{I(C - I)} = \beta dt
   \]

5. **Partial fraction decomposition:**
   \[
   \frac{1}{I(C - I)} = \frac{1}{C} \left(\frac{1}{I} + \frac{1}{C-I}\right)
   \]
   So the equation becomes:
   \[
   \frac{1}{C} \left(\frac{1}{I} + \frac{1}{C - I}\right) dI = \beta dt
   \]
   Integrating both sides:
   \[
   \frac{1}{C} \left(\ln|I| - \ln|C - I|\right) = \beta t + \text{constant}
   \]
   \[
   \ln\left(\frac{I}{C - I}\right) = \beta C t + \text{constant}
   \]
   \[
   \frac{I}{C - I} = A e^{\beta C t}
   \]
   where \( A \) is a constant that can be determined by initial conditions.

   Solving for \( I(t) \):
   \[
   I(t) = \frac{A C e^{\beta C t}}{1 + A e^{\beta C t}}
   \]
   where \( A \) depends on the initial condition \( I(0) \).

   Knowing \( S(t) = C - I(t) \), we have:
   \[
   S(t) = \frac{C}{1 + A e^{\beta C t}}
   \]

### Conclusion:
The solutions are:
\[
S(t) = \frac{C}{1 + A e^{\beta C t}}
\]
\[
I(t) = \frac{A C e^{\beta C t}}{1 + A e^{\beta C t}}
\]

where \( A \) and \( C \) are constants determined by the initial conditions.

Would you like more details on any step, or have any questions? 

Here are some related questions:
1. What are the biological interpretations of \( S(t) \) and \( I(t) \)?
2. How do initial conditions affect the behavior of \( S(t) \) and \( I(t) \)?
3. Can this system be used to model any real-world scenarios?
4. What happens to \( S(t) \) and \( I(t) \) as \( t \) approaches infinity?
5. How would the solution change if the differential equations had additional terms, such as a recovery rate?

**Tip:** Understanding how constants are determined by initial conditions is crucial in solving differential equations applied to real-world scenarios.

Learn how to solve the system of differential equations dS/dt = -βSI, dI/dt = βSI with a step-by-step solution. This problem involves concepts from epidemiology and mathematical modeling, discussing the behavior of S(t) and I(t) over time. Suitable for advanced undergraduate students and researchers.

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