Given the additional information:

• $S(t) + I(t) = N_0$ (which tells us that the total population $N_0$ is constant over time)
• $I(0) = 1$

We can find the specific solutions for $S(t)$ and $I(t)$.

Step 1: Use $S(t) + I(t) = N_0$

Since $S(t) + I(t) = N_0$, we have: $S(t) = N_0 - I(t)$ We substitute this into the differential equation for $\frac{dI}{dt}$:

$\frac{dI}{dt} = \beta S I = \beta (N_0 - I) I$

Step 2: Solve the differential equation

The equation becomes: $\frac{dI}{dt} = \beta N_0 I - \beta I^2$ This can be rewritten as: $\frac{dI}{I(N_0 - I)} = \beta dt$

Step 3: Perform partial fraction decomposition

We decompose $\frac{1}{I(N_0 - I)}$ as: $\frac{1}{I(N_0 - I)} = \frac{1}{N_0} \left(\frac{1}{I} + \frac{1}{N_0 - I}\right)$ Thus, the equation becomes: $\frac{1}{N_0} \left(\frac{1}{I} + \frac{1}{N_0 - I}\right) dI = \beta dt$ Integrating both sides: $\frac{1}{N_0} \left(\ln|I| - \ln|N_0 - I|\right) = \beta t + C_1$ $\ln\left(\frac{I}{N_0 - I}\right) = \beta N_0 t + C_2$ where $C_2$ is a constant of integration.

Step 4: Express $I(t)$

Exponentiating both sides: $\frac{I}{N_0 - I} = A e^{\beta N_0 t}$ where $A = e^{C_2}$ is a constant.

Rearranging to solve for $I(t)$: $I(t) = \frac{A N_0 e^{\beta N_0 t}}{1 + A e^{\beta N_0 t}}$

Step 5: Determine the constant $A$ using the initial condition $I(0) = 1$

At $t = 0$: $I(0) = \frac{A N_0}{1 + A} = 1$ So, $A N_0 = 1 + A$ $A(N_0 - 1) = 1$ $A = \frac{1}{N_0 - 1}$

Step 6: Write the final expressions for $S(t)$ and $I(t)$

Now, substituting $A$ back into the equation for $I(t)$: $I(t) = \frac{\frac{N_0}{N_0 - 1} e^{\beta N_0 t}}{1 + \frac{1}{N_0 - 1} e^{\beta N_0 t}}$ Simplifying: $I(t) = \frac{N_0 e^{\beta N_0 t}}{N_0 - 1 + e^{\beta N_0 t}}$

And since $S(t) = N_0 - I(t)$: $S(t) = \frac{N_0 (N_0 - 1)}{N_0 - 1 + e^{\beta N_0 t}}$

Final Answer:

$I(t) = \frac{N_0 e^{\beta N_0 t}}{N_0 - 1 + e^{\beta N_0 t}}$ $S(t) = \frac{N_0 (N_0 - 1)}{N_0 - 1 + e^{\beta N_0 t}}$

Would you like more details on this derivation or have any questions?

Here are some related questions:

1. How do the initial conditions affect the spread of $I(t)$ over time?
2. What happens to $I(t)$ as $t$ approaches infinity?
3. How would a change in $\beta$ affect the dynamics of $S(t)$ and $I(t)$?
4. Can this model be applied to real-world epidemic modeling?
5. How would you interpret the relationship between $S(t)$ and $I(t)$ at any time $t$?

Tip: When dealing with differential equations in models, always check how initial conditions influence the long-term behavior of the solutions.