Model for spread of infection, with vaccination
Let's repeat the results so far (Eenoku & callculus):
$$
N = 1000000 \quad ; \quad k = 0.08 \quad ; \quad I_0 = 3000 \quad ; \quad H = 600 \\
\frac{dI(t)}{dt}=k \left[N-I(t)\right]-H
$$
Ansatz (suppose the solution has the following form):
$$
I(t) = C e^{-k\, t} + D
$$
Initial condition:
$$
I(0) = C + D = I_0 \quad \Longrightarrow \quad C = I_0 - D
\quad \Longrightarrow \\I(t) = (I_0 - D) e^{-k\, t} + D = I_0 + D\left[1 - e^{-k\, t}\right]
$$
Substitute into the differential equation:
$$
\frac{dI(t)}{dt}=k \left[N-I(t)\right]-H \quad \Longrightarrow \quad
-k\,C e^{-k\cdot t} = k\,N - k\,C e^{-k\, t} - k\,D - H
\\ \Longrightarrow \quad k\,N - k\,D - H = 0 \quad \Longrightarrow \quad D = N - H/k
$$
Plugging in the numbers:
$$
I(t) = 3000 + 992500 \times \left[ 1 - \exp(-0.08 \times t) \right]
$$
Seems a bit of an undesirable outcome to me. Something wrong with the medication?