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.